# Ahlfors-regular conformal dimension and energies of graph maps

Kevin M. Pilgrim  Indiana University
831 E. Third St., Bloomington, Indiana 47405
USA<EMAIL_ADDRESS>and  Dylan P. Thurston  Indiana University
831 E. Third St., Bloomington, Indiana 47405
USA<EMAIL_ADDRESS>
(Date: December 15, 2021)

###### Abstract.

For a hyperbolic rational map $f$ with connected Julia set, we give upper and
lower bounds on the Ahlfors-regular conformal dimension of its Julia set
$J_{f}$ from a family of energies of associated graph maps. Concretely, the
dynamics of $f$ is faithfully encoded by a pair of maps
$\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{1}\rightrightarrows G_{0}$ between finite graphs that satisfies a
natural expanding condition. Associated to this combinatorial data, for each
$q\geq 1$, is a numerical invariant $\overline{E}{}^{q}[\pi,\phi]$, its
asymptotic $q$-conformal energy. We show that the Ahlfors-regular conformal
dimension of $J_{f}$ is contained in the interval where
$\overline{E}{}^{q}=1$.

Among other applications, we give two families of quartic rational maps with
Ahlfors-regular conformal dimension approaching $1$ and $2$, respectively.

###### Contents

  1. 1 Introduction
  2. 2 Topological dynamics
  3. 3 The conformal gauge
  4. 4 Energies of graph maps
  5. 5 Combinatorial modulus
  6. 6 Sandwiching $\operatorname{ARCdim}$
  7. 7 Applications

##  1\. Introduction

###  1.1. Motivation

The iterates of a rational function $f$ define a holomorphic dynamical system
on the Riemann sphere $\widehat{\mathbb{C}}$. Its Julia set, typically
fractal, may be defined as the smallest set $J_{f}$ satisfying
$J_{f}=f^{-1}(J_{f})=f(J_{f})$ and $\\#J_{f}\geq 3$. Figure 1 shows the Julia
set of the rational function
$f(z)=\frac{4}{27}\frac{(z^{2}-z+1)^{3}}{(z(z-1))^{2}}$. It turns out that as
a topological space, $J_{f}$ is a _Sierpiński carpet_ —the complement in the
sphere of a countable collection of Jordan domains whose closures are disjoint
and whose diameters tend to zero.

Figure 1. The Julia set of $f$.

With the spherical metric inherited from the round metric on
$\widehat{\mathbb{C}}$, the Julia set $J_{f}$ becomes a metric space. As a
dynamical system, the map $f$ is _hyperbolic_ —each critical point converges
to an attracting cycle—and _critically finite_ —the orbits of the critical
points are finite. Hyperbolicity is equivalent to the condition that the
restriction $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muJ_{f}\to J_{f}$ is an expanding self-covering map. In this setting,
hyperbolicity is a dynamical regularity condition that leads to strong metric
consequences. As is visually evident, $J_{f}$ is _approximately-self-similar_
(see Definition 3.4). A key invariant of $J_{f}$ is its Hausdorff dimension
$\operatorname{hdim}(J_{f})$. For carpet Julia sets, we have
$1<\operatorname{hdim}(J_{f})<2$; see [PUZ89, HS01] for the lower and upper
bounds, respectively. A less obvious fact (for any hyperbolic rational map) is
that, with $q=\operatorname{hdim}(J_{f})$ and $\mathcal{H}^{q}$ the
corresponding Hausdorff measure on $J_{f}$, then
$0<\mathcal{H}^{q}(J_{f})<\infty$ [PU10, Theorem 9.1.6. Corollary 9.1.7]. In
particular, for any ball $B(x,r)$ with $x\in J_{f}$, and any
$r\leq\operatorname{diam}(J_{f})$we have $\mathcal{H}^{q}(B(x,r))\asymp
r^{q}$, with implicit constants independent of $x$ and $r$. This latter
condition is known as _Ahlfors $q$-regularity_.

A homeomorphism between metric spaces is _quasi-symmetric_ if it does not
distort the roundness of balls too much; see §3.3. The _Ahlfors-regular
conformal gauge_ $\mathcal{G}$ of $J_{f}$ is the set of all metric measure
spaces $(X,d,\mu)$ such that there exists a quasi-symmetric homeomorphism
$(J_{f},d_{Spherical})\to(X,d)$ and, for some $q>0$, the measure $\mu$ is
$q$-Ahlfors regular with respect to $d$; see [Hei01, HP09]. The regularity
assumption on $\mu$ implies that $\mu$ is comparable to the $q$-dimensional
Hausdorff measure $\mathcal{H}^{q}$ on $X$. The _Ahlfors-regular conformal
dimension_ of $J_{f}$ is the infimum over such exponents $q$, i.e.,

$\operatorname{ARCdim}(J_{f})\coloneqq\inf\\{\operatorname{hdim}(X)\mid(X,d,\mu)\in\mathcal{G}(J_{f})\\};$

see [MT10] for an introduction. For approximately self-similar carpets $X$,
such as the Julia set $J_{f}$ of Figure 1, we know
$1<\operatorname{ARCdim}(X)<2$; see [Mac10]. Interest in conformal dimension
stems in part from the following. Limit sets of Kleinian groups acting on the
Riemann sphere and, more generally, boundaries of hyperbolic groups are
another source of approximately self-similar spaces. In that setting, the
conformal dimension, analogously defined, carries significant information
about the group; see [Kle06].

Hyperbolicity and the critically finite property implies, by a rigidity result
of W. Thurston [DH93], that the geometry and dynamics on $J_{f}$ is determined
by combinatorial homotopy-theoretic data: the conjugacy-up-to-isotopy class of
the map, relative to its post-critical set. More precisely: if $g$ is another
rational map, and if there are orientation-preserving homeomorphisms
$\phi_{0},\phi_{1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu(\widehat{\mathbb{C}},P_{f})\to(\widehat{\mathbb{C}},P_{g})$ such that
$\phi_{0}\circ f=g\circ\phi_{1}$ on $P_{f}$ and $\phi_{0}$ is isotopic to
$\phi_{1}$ through homeomorphisms agreeing on $P_{f}$, then we may take
$\phi_{0}=\phi_{1}$ to be a Möbius transformation. Hence the invariant
$\operatorname{ARCdim}(J_{f})$ is determined from combinatorial data.

For general hyperbolic critically finite rational maps with connected Julia
set, our main result, Theorem A, implies an estimate for
$\operatorname{ARCdim}(J_{f})$ in terms of combinatorial data. In concrete
cases, by-hand computations with this data can yield nontrivial rigorous upper
and lower bounds. For the carpet Julia set of Figure 1, such computations
yield
$1.6309\approx\frac{1}{1-\log_{6}2}\leq\operatorname{ARCdim}(J_{f})\leq\frac{2}{1-\log_{6}(10/13)}\approx
1.7445$. See §7 for details.

###  1.2. Combinatorial encoding

Our methods rely on a particular method of combinatorial encoding of rational
maps [Thu20]. We choose a finite graph $G_{0}$, called a _spine_ , onto which
$\widehat{\mathbb{C}}-P_{f}$ deformation retracts. The homotopy type of
$G_{0}$ depends only on $\\#P_{f}$. Letting
$G_{1}=f^{-1}(G_{0})\subset\widehat{\mathbb{C}}-P_{f}$, we obtain two graph
maps $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{1}\rightrightarrows G_{0}$, where $\pi$ and $\phi$ are respectively
the restrictions of $f$ and of the deformation retraction. The data
$(\pi,\phi)$ is a _virtual endomorphism of graphs_ and is well-defined up to a
notion of homotopy equivalence; see [Thu20, Definition 2.2]. We denote by
$[\pi,\phi]$ the homotopy class of $(\pi,\phi)$. Figure 2 illustrates the data
for the above map $f$.

$G_{1}=\vbox{\hbox{$\includegraphics[width=108.405pt]{barycenter-
tri-11}$}}\qquad
G_{0}=\vbox{\hbox{$\includegraphics[width=104.79128pt]{barycenter-tri-10}$}}$
Figure 2. Spines for the map $f$, up to conjugacy-up-to-isotopy. Doubling the
large Euclidean equilateral triangles over their boundaries gives two Riemann
surfaces, each isomorphic to $\widehat{\mathbb{C}}$. The map $f$ sends each
small triangle at left to one of the two large triangles at right and
implements barycentric subdivision. The set of vertices of the large triangles
is $P_{f}$. Half of a spine $G_{0}$ of $\widehat{\mathbb{C}}-P_{f}$ is shown
at right, and its preimage $G_{1}$ under the piecewise-affine map homotopic to
$f$ is shown at left. The map $\pi$ is the covering map preserving colors,
while the map $\phi$ (defined up to homotopy) is induced by the deformation
retraction.

For any iterate $n\in\mathbb{N}$, the Julia set of $f$ is the same as that of
$f^{n}$. It follows from the expanding nature of the dynamics of $f$ that upon
replacing $f$ by a suitable iterate, we may assume the virtual endomorphism
$(\pi,\phi)$ constructed in the previous paragraph is _forward expanding_ or,
synonymously, _backward contracting_ ; see Definition 2.18. The critically
finite property implies that $G_{1}$ and $G_{0}$ are connected and $\phi$ is
surjective on fundamental group. This is a property we call _recurrence_ ; see
Definition 2.21. To summarize: to the dynamics of a critically finite
hyperbolic rational map, we associate a forward-expanding recurrent virtual
graph endomorphism $(\pi,\phi)$.

It turns out (see §2) that any forward-expanding recurrent virtual graph
endomorphism $(\pi,\phi)$ determines, via now-standard constructions, a
dynamical system given by an expanding topological self-cover on a compact
metrizable space $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ (Theorem F). The topological conjugacy class
of $f$ depends only on the homotopy class $[\pi,\phi]$, suitably defined; see
[IS10, Theorem 4.2].

Via other now-standard constructions, there is an associated distinguished
class of Ahlfors-regular metrics $\mathcal{G}(\mathcal{J}[\pi,\phi])$
associated to $[\pi,\phi]$; see §3. If $[\pi,\phi]$ arises from a hyperbolic
rational map $f$, then the spherical metric on $J_{f}$ belongs to
$\mathcal{G}(\mathcal{J}[\pi,\phi])$ (Proposition 3.3).

###  1.3. Asymptotic $q$-conformal energies

A virtual endomorphism of graphs $(\pi,\phi)$ has, for each $q\in[1,\infty]$,
an associated _asymptotic $q$-conformal graph energy_
$\overline{E}{}^{q}(\pi,\phi)$, introduced by the second author [Thu19]. We
summarize some key points; see §4 or the references for more. First,
$\overline{E}{}^{q}(\pi,\phi)$ depends only on the homotopy class
$[\pi,\phi]$, so that these analytic quantities depend only on combinatorial
data. If $(\pi,\phi)$ arises from a rational map $f$, different choices of
spine lead to homotopic graph endomorphisms, so that we may write
unambiguously $\overline{E}{}^{q}(f)$. In the general case, we will also write
$\overline{E}{}^{q}[\pi,\phi]$ to indicate the asymptotic energy depends only
on the homotopy class. The inequality $\overline{E}{}^{\infty}[\pi,\phi]<1$
holds if and only if some iterate of $(\pi,\phi)$ is homotopic to a backward-
contracting virtual graph endomorphism. If $(\pi,\phi)$ arises from a
hyperbolic critically-finite rational map $f$, then
$\overline{E}{}^{2}[\pi,\phi]<1$, and this property characterizes such maps
among the wider class of their topological counterparts, that is, critically
finite self-branched-coverings of the sphere for which each cycle in the post-
critical set contains a critical point [Thu20]. As a function of $q$, the
asymptotic energy $\overline{E}{}^{q}[\pi,\phi]$ is continuous and non-
increasing, so that the level set $\\{q\mid\overline{E}{}^{q}[\pi,\phi]=1\\}$
is an interval $[q_{*}[\pi,\phi],q^{*}[\pi,\phi]]$.

###  1.4. Main result

Our main result relates Ahlfors-regular conformal dimension to these critical
exponents, and implies that the invariant $\overline{E}{}^{q}[\pi,\phi]$
contains useful information for other values of $q$.

######  Theorem A.

For any recurrent, forward-expanding virtual graph endomorphism $(\pi,\phi)$,

$q_{*}[\pi,\phi]\leq\operatorname{ARCdim}(\mathcal{J}[\pi,\phi])\leq
q^{*}[\pi,\phi].$

Equivalently, for $q=\operatorname{ARCdim}(\mathcal{J}[\pi,\phi])$, we have
$\overline{E}{}^{q}[\pi,\phi]=1$.

In fact we expect that $q_{*}=q^{*}$:

######  Conjecture 1.1.

For any recurrent virtual graph endomorphism $[\pi,\phi]$, the function
$q\mapsto\overline{E}{}^{q}[\pi,\phi]$ is either constant or strictly
decreasing.

In particular if $[\pi,\phi]$ is forward-expanding then (since
$\overline{E}{}^{1}[\pi,\phi]\geq 1$ and
$\overline{E}{}^{\infty}[\pi,\phi]<1$), the conjecture would imply that

$q_{*}[\pi,\phi]=q^{*}[\pi,\phi]$

and Theorem A characterizes $\operatorname{ARCdim}$.

One might expect more to be true in Conjecture 1.1, for instance that
$q\mapsto\log\overline{E}{}^{q}[\pi,\phi]$ is a convex function of $q$. More
generally, it would be interesting to know the relationship between our
constructions and more classical constructions in thermodynamic formalism. In
this vein, we remark that Das, Przytycki, Tiozzo, and Urbański [DPT+19] have
developed the thermodynamic formalism in a setting which includes the
topologically coarse expanding conformal maps $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ considered here.

###  1.5. Outline

The bulk of the paper is devoted to developing the technology to prove Theorem
A.

#### Topological dynamics

(§2) We begin with an arbitrary forward-expanding recurrent virtual
endomorphism of graphs $(\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{1}\rightrightarrows G_{0})$. Iteration, suitably defined, gives rise
to

  1. (1)

a sequence of virtual endomorphisms
$(\pi^{n}_{n-1},\phi^{n}_{n-1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{n}\rightrightarrows G_{n-1}),n\in\mathbb{N}$;

  2. (2)

a connected, locally connected, compact space $\mathcal{J}$, the inverse limit
of

$\cdots\overset{\phi^{n+1}_{n}}{\longrightarrow}G_{n}\overset{\phi^{n}_{n-1}}{\longrightarrow}G_{n-1}\overset{\phi^{n-1}_{n-2}}{\longrightarrow}\cdots\overset{\phi^{2}_{1}}{\longrightarrow}G_{1}\overset{\phi^{1}_{0}}{\longrightarrow}G_{0};$

  3. (3)

a positively expansive self-covering map $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ of degree $d\coloneqq\deg(\pi)$;

  4. (4)

the topological conjugacy class of $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ depends only on the homotopy class
$[\pi,\phi]$.

Our development in this section is quite general. We consider pairs of maps
$\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ between finite CW complexes equipped with
length metrics and satisfying natural expansion conditions, and establish
properties of the dynamics on the limit space. The main result, Theorem F,
shows that under these conditions, the construction of the conformal gauges
given in the next section applies.

#### The conformal gauge

(§3) We apply a construction in [HP09] to put a nice metric $d_{\varepsilon}$
on $\mathcal{J}$, called a _visual metric_. It depends on a suitably small but
arbitrary parameter $\varepsilon$, and on the data of a finite cover
$\mathcal{U}_{0}$ of $\mathcal{J}$ by open, connected sets. Changing this data
changes the metric by a special type of quasi-symmetric map called a
_snowflake map_. Equipped with a visual metric, $f$ is positively expansive.
Even better, any ball of sufficiently small radius is sent homeomorphically
and homothetically onto its image, with expansion constant $e^{\varepsilon}$.
The metric space $(\mathcal{J},d_{\varepsilon},\mathcal{H}^{q})$ is Ahlfors
regular of exponent
$q\coloneqq\log(d)/\varepsilon=\operatorname{hdim}(\mathcal{J},d_{\varepsilon})$.
Therefore, the invariant $\operatorname{ARCdim}(\mathcal{J}[\pi,\phi])$ is
well-defined. See Theorem 3.2.

#### Energies of graph maps

(§4) When equipped with natural length metrics, the maps
$\phi^{n}\coloneqq\phi^{n}_{0}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{n}\to G_{0}$ have, for each $q\in[1,\infty]$, a $q$-conformal energy
$E^{q}_{q}[\phi^{n}]$ in the sense introduced by the second author in [Thu19].
The growth rate of this energy as $n$ tends to infinity, namely
$\overline{E}{}^{q}[\pi,\phi]\coloneqq\lim E^{q}_{q}[\phi^{n}]^{1/n}$, is
called the _asymptotic $q$-conformal energy_, depends only on the homotopy
class $[\pi,\phi]$, and is non-increasing in $q$ (Proposition 4.11). The
expansion hypothesis implies $\overline{E}{}^{\infty}[\pi,\phi]<1$, giving the
interval in the statement of Theorem A.

#### Combinatorial modulus

(§5) In this section we recall (and extend slightly) results on a
combinatorial version of modulus in a fairly general setting, and how it is
related to Ahlfors-regular conformal dimension. Although the limit space
$\mathcal{J}$ hardly appears in this section, the ultimate motivation is of
course to estimate its conformal dimension. In more detail, fix
$n\in\mathbb{N}$. There is a natural projection
$\phi^{\infty}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to G_{n}$. The collection $\mathcal{V}_{n}$ of fibers
$(\phi^{\infty}_{n})^{-1}(e)$ of closed edges $e\in E(G_{n})$ gives a covering
of $\mathcal{J}$.111For technical reasons, we actually work with a slightly
different cover $\mathcal{V}_{n}$ given by inverse images of a slightly large
set $\widehat{e}$ called the _star_ of $e$; see Definition 5.8. Given a family
of paths $\mathcal{C}$ in $\mathcal{J}$ and an exponent $q\geq 1$, we get a
numerical invariant, $\operatorname{mod}_{q}(\mathcal{C},\mathcal{V}_{n})$,
the combinatorial modulus of this family.

§§5.1–5.4 develop general properties of combinatorial modulus. We need to
consider a mild generalization: we define combinatorial modulus for families
of _weighted_ curves.

§5.5 continues by relating combinatorial modulus to energies of graph maps.
For this, it is technically convenient to work with the reciprocal of modulus,
namely extremal length. The relation arises via the characterization of graph
map energy in terms of maximum distortion of extremal length. See Theorem 4.7,
which requires a formulation of extremal length in terms of weighted curves.

§5.6 recalls a result of Carrasco [Car13, Theorem 1.3], which was
independently proved by Keith-Kleiner (unpublished). This result states that
for a suitably self-similar space $Z$ and a suitable family of coverings
$(\mathcal{S}_{n})_{n}$ indexed by $\mathbb{N}$, there is a _critical
exponent_ $q$. For a reasonably natural curve family $\Gamma$ in the space,
this critical exponent distinguishes between
$\operatorname{mod}_{p}(\Gamma,\mathcal{S}_{n})\to\infty$ if $p<q$ and
$\operatorname{mod}_{p}(\Gamma,\mathcal{S}_{n})\to 0$ if $p>q$; see Theorem
5.11.

#### Sandwiching the dimension

(§6) The proof of Theorem A applies the developments in §5 and §4. We relate
combinatorial modulus of curve families in the limit space to combinatorial
modulus of curve families on the graphs $G_{n}$, and then to energies of graph
maps.

Here is a brief summary of the proof. The collection
$\\{\mathcal{V}_{n}\\}_{n}$ above forms a family of _snapshots_ of the limit
space, equipped with a visual metric; see §6.1. We show a curve
$c\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muC\to G_{n}$ can be approximately lifted under
$\phi^{\infty}_{n}$ to a curve $c^{\prime}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muC\to\mathcal{J}$ such that the composition
$c^{\prime\prime}\coloneqq\phi^{\infty}_{n}\circ c^{\prime}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muC\to G_{n}$ is homotopic to $c$, with traces of size uniformly bounded
independent of $n$. This implies that combinatorial moduli for $c$ and
$c^{\prime\prime}$ on $G_{n}$ are comparable to that of $c^{\prime}$ on
$\mathcal{J}$ (Lemma 6.4).

With this setup, the upper bound on conformal dimension is straightforward to
verify; see §6.3. The lower bound is more involved and uses in an essential
way the existence of a curve $c\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muC\to G_{n}$ which is extremal for the distortion of extremal length in
the homotopy class of $\phi^{n}_{0}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{n}\to G_{0}$. When projected to $G_{0}$, the strands of
$\phi^{n}_{0}\circ c$ are very long and cross edges of $G_{0}$ many times. We
decompose $c$ into a family of subcurves, each of which projects to an edge of
$G_{0}$, and make the needed estimates. See §6.4.

###  1.6. Applications

In §7, we give several applications of our methods to the calculation of
Ahlfors-regular conformal dimension in the setting of complex dynamics, using
Theorem A to estimate the conformal dimension (above and below) from a forward
expanding recurrent virtual graph endomorphism $(\pi,\phi)$ in the manner
discussed above.

####  1.6.1. Techniques for estimates

Theorem A yields practical methods for estimating the Ahlfors-regular
conformal dimension. If specific $\phi$ and $q$ are given with
$E^{q}_{q}(\phi)<1$, the submultiplicativity of energy under composition
yields $\overline{E}{}^{q}[\pi,\phi]<1$ and thus, by Theorem A,
$\operatorname{ARCdim}(\mathcal{J}[\pi,\phi])<q$. Furthermore, there are
bounds on how quickly $\overline{E}{}^{q}[\pi,\phi]$ can decrease as a
function of $q$ [Thu20, Proposition 6.11], so if we know
$\overline{E}{}^{q}[\pi,\phi]<1$, we get an upper bound on $q^{*}$ that is
smaller than $q$. See Proposition 7.1.

For lower bounds, we have the following. Set
$\overline{N}[\pi,\phi]=\overline{E}{}^{1}[\pi,\phi]$. This quantity has
several interpretations. It is the asymptotic growth rate of the number of
edge-disjoint paths in $G_{n}$ covering non-trivial loops in $G_{0}$. It is
also the asymptotic growth rate of the minimum cardinality of the fibers of
$\phi_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{n}\to G_{0}$. The bounds mentioned above on how fast
$\overline{E}{}^{q}$ decreases as a function of $q$ yield the following.

######  Theorem B.

For any recurrent forward-expanding virtual graph automorphism $[\pi,\phi]$
where $\deg(\pi)=d$, we have

$\operatorname{ARCdim}(\mathcal{J}[\pi,\phi])\geq\frac{1}{1-\log_{d}\overline{N}[\pi,\phi]}.$

For the virtual endomorphism of $f$ from Figures 1 and 2, we find by hand that
$\overline{N}(f)=2$ and $\overline{E}{}^{2}[\pi,\phi]<\sqrt{10/13}$, so
$1.6309\approx\frac{1}{1-\log_{6}2}<\operatorname{ARCdim}(J_{f})<\frac{2}{1-\log_{6}(10/13)}\approx
1.7445$; see §7.2 for details.

If $f$ is a hyperbolic rational map and $[\pi,\phi]$ an associated virtual
graph endomorphism, the quantity $\overline{N}(f)$ seems to be closely related
to the topological and metric structure of the Julia set. For example, we show
the following.

######  Theorem C.

Suppose $f$ is a critically finite hyperbolic rational map. If $J_{f}$ is a
Sierpiński carpet, then $\overline{N}{}(f)>1$.

Examples show that the converse need not hold; see §7.1.

####  1.6.2. When the conformal dimension equals $1$

M. Carrasco [Car14, Theorem 1.2] gives a metric condition, _uniformly well-
spread cut points_ (UWSCP; Definition 7.15 below), on a compact doubling
metric space $X$ which guarantees $\operatorname{ARCdim}(X)=1$. All hyperbolic
polynomials and rational maps with “gasket-type” Julia sets satisfy the UWSCP
condition. Combining his observation with our Theorem B, we obtain the
following.

######  Theorem D.

Suppose $f$ is a hyperbolic rational map. If $J_{f}$ satisfies the UWSCP
condition, then $\overline{N}(f)=1$.

We also show that Carrasco’s criterion for $\operatorname{ARCdim}=1$ is not
necessary.

######  Proposition 1.2.

Let $R$ be the rational map obtained by mating the Douady rabbit quadratic
polynomial with the basilica polynomial $z^{2}-1$. Then
$\operatorname{ARCdim}(J_{R})=1$ but $J_{R}$ does not satisfy UWSCP.

This example is shown in Figure 3. More generally, applying our methods,
InSung Park [Par21] has proved the following generalization: _A hyperbolic
critically finite rational map $f$ satisfies $\operatorname{ARCdim}(J_{f})=1$
if and only if $f$ is a crochet map, if and only if $\overline{N}{}(f)=1$._ A
crochet map is one in which any pair of points in the Fatou set is joined by a
path which meets the Julia set in a countable set of points.

Figure 3. The mating of the rabbit (yellow) and the basilica (pink)
polynomials.

####  1.6.3. Variation in a family

R. Devaney _et al._ studied the family $f_{\lambda}(z)=z^{2}+\lambda/z^{2}$
for $\lambda\in\mathbb{C}-\\{0\\}$ [DLU05]. Figure 4 shows the bifurcation
locus in the parameter plane for this family. (The four critical points at
fourth roots of $\lambda$ end up in the same orbit for this family, so
analysis is simpler; see Eq. (7.7).)

Figure 4. Parameter plane for $f_{\lambda}(z)=z^{2}+\lambda/z^{2}$. The white
regions are where the $\lambda$ critical orbit escapes to the attracting fixed
point at $\infty$; the small black Mandelbrot sets correspond to a
renormalization, where that orbit is attracted to a different attracting
cycle.

Parameters taken from the prominent “holes” along the real axis have carpet
Julia sets. In §§7.3 and 7.4, we present two one-parameter families
$\lambda_{n}^{\mathrm{skinny}},\lambda_{n}^{\mathrm{fat}}$, $n\in\mathbb{N}$
whose values converge to the left-most real parameter (at
$\lambda=-\frac{3}{16}-\frac{\sqrt{2}}{8}$) and the origin, respectively.
Figure 5 illustrates two examples.

Figure 5. Two Sierpiński carpet Julia sets in the Devaney family corresponding
to parameters $\lambda_{2}^{\mathrm{skinny}}\approx-.35891$ (left) and
$\lambda_{2}^{\mathrm{fat}}\approx-1.4996769\times 10^{-5}$ (right). The red
points are the common second iterates $x_{\lambda}$ of the finite nonzero
critical points; the white points show the orbit of $x_{\lambda}$. The insets
at lower left are local pictures in the parameter plane. Figures created with
FractalStream [Noo].

Note the difference in the apparent “thickness” of the carpets. Though each
Julia set is a Sierpiński carpet, that of $\lambda_{n}^{\mathrm{skinny}}$
visually becomes “skinnier” as $n\to\infty$, while that of
$\lambda_{n}^{\mathrm{fat}}$ visually becomes “fatter” as $n\to\infty$. The
former rate seems to be rather gradual, while the latter rate seems to be very
fast.

Recently, M. Bonk, M. Lyubich, and S. Merenkov showed that a quasi-symmetric
map between hyperbolic carpet Julia sets extends to a Möbius transformation
[BLM16]. This easily implies that the three carpets shown in Figures 1 and 5
are pairwise quasi-symmetrically inequivalent. Our techniques allow us to
quantify this distinction.

######  Theorem E.

We have

$1<\operatorname{ARCdim}(J_{\lambda_{n}^{\mathrm{skinny}}})<1+\frac{1}{\log_{2}(2n+3)},$

and

$\frac{2}{1+2^{-n}}\leq\operatorname{ARCdim}(J_{\lambda_{n}^{\mathrm{fat}}})<2.$

In particular, within this family of fixed degree, there are hyperbolic carpet
maps with conformal dimension tending to 1 and to 2. This latter result
answers a question of the first author and P. Haïssinsky [HP12].

###  1.7. Related work

Kwapisz [Kwa20] uses a similar approach to estimating the conformal dimension
for Sierpiński carpets. In his paper, he considers only the standard square
carpet, but uses very similar notions, with the slight variation that his
$q$-resistance $r(e)$ is related to our $q$-length $\alpha(e)$ by

(1.3) $r(e)=\alpha(e)^{q-1}.$

(In particular, as in this paper, there are quantities associated to the edges
rather than the vertices, as contrasted with the more common notions of
combinatorial $q$-modulus in the literature.) With this correspondence,
Kwapisz’ formulas match with ours; for instance, his $P(\mathcal{J})$ [Kwa20,
Eq. (1.2)] agrees with our $\bigl{(}E^{1}_{q}(\phi)\bigr{)}^{q}$ in Eq. (4.4).
One difference between our approaches is that he deals with signed flows as in
electrical networks, while we deal with more general unsigned tensions related
to elasticity; see the discussion in [Thu19, Appendix B].

A more substantive difference is that our $q$-conformal energies enjoy exact
sub-multiplicativity [Thu19, Prop. A.12], while Kwapisz only proves weak sub-
multiplicativity, up to a constant [Kwa20, Theorem 1.3]. On the other hand, by
only asking for weak sub-multiplicativity, he is able to get bounds on
$q$-resistance for both the graph analogous to the one we consider (his
$\mathcal{G}_{n}$) and its dual. Correspondingly, he gets numerical lower
bounds, in addition to numerical upper bounds analogous to the ones we get.
(Our numerical lower bounds rely on Theorem B, which is unlikely to be sharp
in general.)

In another direction, in the non-dynamical setting of Gromov hyperbolic
complexes, Bourdon and Kleiner [BK15] consider families of analytic invariants
such as $\ell^{p}$ cohomology and separation properties of certain associated
function spaces. It would be interesting to know if there is a connection to
this work in our setting.

### Acknowledgements

We thank Caroline Davis, InSung Park, and Giulio Tiozzo for useful
conversations. The first author was supported by the Simons Foundation under
Grant Numbers 245269 and 615022. The second author was supported by the
National Science Foundation under Grant Numbers DMS-1507244 and DMS-2110143.

##  2\. Topological dynamics

The main result of this section is the following theorem, giving good
dynamical properties of the action on a limit space.

######  Theorem F.

Suppose $X_{0},X_{1}$ are finite CW complexes equipped with complete length
metrics. Suppose $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is a $\lambda$-backward-contracting and
recurrent virtual endomorphism, and let $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ be the induced dynamics on its limit space.
Then

  1. (1)

The space $\mathcal{J}$ is connected and locally connected.

  2. (2)

The map $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is a positively expansive covering map of
degree $\deg(\pi)$.

  3. (3)

The dynamical system $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is topologically coarse expanding conformal
(cxc), in the sense of Haïssinsky and Pilgrim [HP09].

The terminology is defined in §2.1 and §2.5. One main result of [HP09] is that
topologically cxc systems have a canonically associated nonempty set of
special Ahlfors-regular metrics, called _visual metrics_. In §3, we will
describe the geometry of $\mathcal{J}$ when equipped with such metrics.

Theorem F could be deduced as follows. Nekrashevych [Nek14, Theorem 5.10]
associates to such a virtual endomorphism a self-similar recurrent contracting
group action. Such an object has also an associated limit space homeomorphic
to $\mathcal{J}$ that is connected and locally connected [Nek05, Theorem
3.5.1], establishing (1). Moreover, there is an induced dynamics on this limit
space naturally conjugate to $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$. Conclusions (2) and (3) then follow from
[HP11, Theorem 6.15]. To keep this work self-contained, and because we have
later need of certain other related technical facts, we give more direct
arguments.

To prove Theorem F, we present the limit space as a subspace of the infinite
product $(X_{1})^{\infty}$, following Ishii and Smillie [IS10]. We generalize
the theory of homotopy pseudo-orbits developed there to _families_ of homotopy
pseudo-orbits parameterized by a space $A$, and show, by generalizing their
results on homotopy shadowing, that such a family determines a map
$A\to\mathcal{J}$ (Theorem G). We prove that the limit space is locally path-
connected by applying this generalization to the case when $A=I$ is an
interval (§2.6). To obtain the needed family of homotopy pseudo-orbits in this
setting, we develop in §2.4 a general notion of approximate path-lifting.

For technical reasons, we need to control the geometry of non-rectifiable
paths. To this end, we introduce the _size_ of a path, defined to be the
diameter of the lift to a universal cover; see §2.3. We do this mainly since
we apply our development to limits of paths that need not preserve
rectifiability.

We also present the associated limit space $\mathcal{J}$ in an equivalent, but
more convenient, way as an inverse limit of spaces with increasingly large
diameter; see Theorem G′. To find path families, we also need to find
“homotopy sections” of projections from $\mathcal{J}$ We therefore need
results on homotopy shadowing and homotopy sections in those settings; see and
§2.7, respectively.

###  2.1. Topologically cxc systems

We first recall the notion of topologically cxc systems. To streamline our
presentation, we specialize the setup of [HP09] to the case of self-covers. We
next define positively expansive systems. Finally, we show that conclusions
(1) and (2) of Theorem F imply conclusion (3).

Suppose $\mathcal{J}$ is a compact, connected, and locally-connected
topological space, and $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is a self-covering. A finite open cover
$\mathcal{U}_{0}$ of $\mathcal{J}$ by connected sets generates inductively a
sequence of coverings $\mathcal{U}_{n},n\in\mathbb{N}$, via the recipe

(2.1) $\mathcal{U}_{n+1}\coloneqq\\{\widetilde{U}\mid
U\in\mathcal{U}_{n},\widetilde{U}\text{ is a component of }f^{-1}(U)\\}.$

Also set

$\mathbf{U}\coloneqq\bigcup_{n}\mathcal{U}_{n}.$

The _mesh_ of a covering of a metric space is the supremum of the diameters of
its elements.

######  Definition 2.2.

The pair of the dynamical system $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ and cover $\mathcal{U}_{0}$ is said to satisfy
Axiom [Expansion] if, for some (equivalently, any) metric on $\mathcal{J}$
compatible with the topology, the mesh of $\mathcal{U}_{n}$ from Eq. (2.1)
tends to zero as $n\to\infty$. The map $f$ satisfies [Expansion] if there is
some $\mathcal{U}_{0}$ satisfying this condition.

######  Lemma 2.3.

Suppose $\mathcal{J}$ is compact, connected, and locally connected,
$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\mathcal{J}\to\mathcal{J}$ is a covering map, and
$\mathcal{U}_{0}$ is a covering so $(f,\mathcal{U}_{0})$ satisfies Axiom
[Expansion]. Then this dynamical system satisfies the following additional two
axioms:

  * •

[Degree] For fixed $n$ there is a uniform upper bound on the cardinality of
fibers of restrictions $f^{\circ n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{U}_{n}\to U_{0}$, for all $U_{0}\in\mathcal{U}_{0}$ and
$\widetilde{U}_{n}\in\mathcal{U}_{n}$.

  * •

[Irreducibility] For any nonempty open set $W\subset\mathcal{J}$, there is an
integer $N$ for which $f^{\circ N}(W)=\mathcal{J}$.

On the conditions of this lemma, $f$ is topologically cxc. The general
definition of topologically cxc is that there exists a finite open cover
$\mathcal{U}_{0}$ such that all three axioms [Expansion], [Degree], and
[Irreducibility] hold.

###### Proof.

For an self-cover satisfying axiom [Expansion], axiom [Degree] clearly holds,
since for large enough $n$ the elements of $\mathcal{U}_{n}$ will be evenly
covered.

To show [Irreducibility], fix arbitrarily a metric on $\mathcal{J}$ compatible
with its topology. For $x\in\mathcal{J}$, define

$\displaystyle Y(x)$ $\displaystyle\coloneqq\\{\,y\in X\mid\text{the backward
orbit of $y$ accumulates at $x$}\,\\}$ $\displaystyle=\\{\,y\in
X\mid\text{there is a sequence }\tilde{y}_{n_{k}}\in f^{-n_{k}}(y)\text{ s.t.\
}\lim_{k\to\infty}\tilde{y}_{n_{k}}=x\,\\}.$

We first claim $Y(x)$ is nonempty for each $x$. To see this, fix
$x\in\mathcal{J}$, and let $y$ be any accumulation point of the forward orbit
of $x$, i.e., $\lim_{k\to\infty}f^{\circ n_{k}}(x)=y$. Let
$U\in\mathcal{U}_{0}$ contain $y$. For each sufficiently large $k$, let
$\widetilde{U}_{n_{k}}$ be the unique component of $f^{-n_{k}}(U)$ containing
$x$, and pick $\tilde{y}_{n_{k}}\in\widetilde{U}_{n_{k}}\cap f^{-n_{k}}(y)$.
Then $\tilde{y}_{n_{k}}\to x$ since
$\operatorname{diam}\widetilde{U}_{n_{k}}\to 0$. Hence $y\in Y(x)$.

Now suppose $y\in Y(x)$ is arbitrary and choose $U\in\mathcal{U}_{0}$ with
$y\in U$. The same reasoning with $\operatorname{diam}\widetilde{U}_{n_{k}}$
shows that $U\subset Y(x)$. Since $X$ is connected and covered by
$\mathcal{U}_{0}$, we conclude $Y(x)=X$. Since $x$ is arbitrary, we conclude
the set of backward orbits under $f$ of each point is dense in $X$.

Now fix an arbitrary nonempty open set $W$ as in the statement of Axiom
[Irreducible]. By [Expansion], there exists $U\in\mathbf{U}$ with $U\subset
W$. Pick $y\in U$. By the previous paragraph, there exists a backward orbit of
$y$ accumulating at $y$. Pulling back $U$ along this backward orbit,
[Expansion] implies that there exists some $\widetilde{U}\in\mathbf{U}$ so
that $\widetilde{U}\subset U$ and $f^{\ocirc m}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{U}\to U$ is a covering map for some $m>0$. The previous
paragraph implies $\bigcup_{n=0}^{\infty}f^{nm}(\widetilde{U})=X$. By our
choice of $m$ and $\widetilde{U}$, this is an increasing union. Since $X$ is
compact, $f^{nm}(\widetilde{U})=X$ for some $n$. Then $N=nm$ suffices for the
statement, since $\widetilde{U}\subset W$. ∎

######  Definition 2.4 (Positively expansive).

Let $(\mathcal{J},d)$ be a metric space. A continuous surjection
$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\mathcal{J}\to\mathcal{J}$ is _positively expansive_ if there
is a constant $e>0$ so that for any $x\neq y\in\mathcal{J}$ there is an $n>0$
so that $d(f^{n}(x),f^{n}(y))>e$.

Note that a positively expansive map is locally injective.

In the case $\mathcal{J}$ is compact, positively expansive is equivalent to
the following condition: there exists a neighborhood
$N\supset\\{(x,x)\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mux\in\mathcal{J}\\}$ of the diagonal such that if $x,y\in\mathcal{J}$
satisfy $(f^{i}x,f^{i}y)\in N$ for all $i\geq 0$, then $x=y$. (In particular,
positively expansive is independent of the metric.) In addition, by a theorem
of Reddy [AH94, Theorem 2.2.10] there exists a compatible metric $D$ on
$\mathcal{J}$, called an _adapted metric_ , and _expansion constants_
$\delta>0$ and $0<\lambda<1$ such that for any $x,y\in\mathcal{J}$,

$D(x,y)\leq\delta\implies D(f(x),f(y))\geq\lambda^{-1}D(x,y).$

Note that this implies $f$ is a homeomorphism on $\delta$-balls in the
$D$-metric.

######  Proposition 2.5 (Positively expansive implies [Expansion]).

Suppose $\mathcal{J}$ is compact and locally connected, and $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is a positively expansive covering map. Then
$f$ satisfies Axiom [Expansion].

To prove this, we will need the following result.

######  Theorem 2.6 (Eilenberg constants [AH94, Theorem 2.1.1]).

Let $X$ be compact and $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ a continuous surjective local homeomorphism. Then there exist two
positive numbers $\tau$ and $\mu$ such that each subset $U$ of $Y$ with
diameter less than $\tau$ determines a decomposition of the set
$f^{-1}(U)=U_{1}\cup\dots\cup U_{d}$ with the following properties:

  1. (1)

$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muU_{i}\to U$ is a homeomorphism;

  2. (2)

for $i\neq j$ no point of $U_{i}$ is closer than $2\mu$ to a point of $U_{j}$;
and

  3. (3)

for each $\eta>0$ there exists $0<\varepsilon<\tau$ such that
$\operatorname{diam}U<\varepsilon\implies\operatorname{diam}U_{j}<\eta$ for
all $j$.

###### Proof of Proposition 2.5.

Equip $\mathcal{J}$ with an adapted metric. We apply Theorem 2.6 with
$X=Y=\mathcal{J}$ and obtain the constant $\tau$. Let $\delta$ be as in the
definition of adapted metric, and take the constant $\eta$ in Theorem 2.6 so
that $\eta<\delta$; we obtain a constant $\varepsilon$. In summary: any open
connected set $U$ of diameter at most $\varepsilon$ is evenly covered by $f$
and has preimages $\widetilde{U}_{1},\ldots,\widetilde{U}_{d}$ of diameter at
most $\delta$. The definition of adapted metric then implies that
$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\widetilde{U}_{j}\to U$ expands distances by at least the
factor $\lambda^{-1}$ and so the inverse branches $f_{j}^{-1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muU\to\widetilde{U}_{j}$ contract distances by at least the factor
$\lambda$.

Since $\mathcal{J}$ is locally connected and compact, there is a finite open
cover $\mathcal{U}_{0}$ by connected sets such that each element has diameter
at most $\varepsilon$. The previous paragraph implies that the covering
$\mathcal{U}_{1}$ has mesh at most $\lambda\varepsilon<\varepsilon$. Induction
shows $\mathrm{mesh}(\mathcal{U}_{n})<\lambda^{n}\varepsilon\to 0$ as required
in Axiom [Expansion]. ∎

###  2.2. Length spaces

In this subsection, we prepare for the proof of local connectivity by
collecting some technical results related to covering maps and length spaces.
Here, $X$ denotes a finite, hence compact, connected CW complex, equipped with
a compatible length metric. The Hopf-Rinow theorem implies that $X$ is a
geodesic metric space. While balls might not be simply-connected, they are
path-connected. The _systole_ is the length of the shortest essential loop,
i.e.,

$\operatorname{systole}(X)\coloneqq\inf\\{\ell(\gamma)\mid[\gamma]\neq
1,[\gamma]\in\pi_{1}(X)\\}.$

This is positive, since $X$ is compact. Since $X$ is a length space, any cover
$\widetilde{X}$ inherits a lifted metric by lengths of paths. Balls of radius
less than $\tfrac{1}{2}\operatorname{systole}(X)$ are simply-connected and
thus evenly covered.

######  Lemma 2.7.

Let $X$ be a finite connected CW complex with a length metric. Suppose
$p\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\widetilde{X}\to X$ is a covering map, $\widetilde{X}$ is
equipped with the lifted metric, and
$r<\tfrac{1}{4}\operatorname{systole}(X)$. Then for any
$\tilde{x}\in\widetilde{X}$ with $p(\tilde{x})=x$, the restriction
$p\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muB(\tilde{x},r)\to B(x,r)$ is an isometry.

This is standard, but we provide a proof for completeness.

###### Proof.

The definition of the metric on $\widetilde{X}$ implies that $p$ preserves the
length of paths, and is therefore 1-Lipschitz. Thus $p(B(\tilde{x},r))\subset
B(x,r)$. We have $p(B(\tilde{x},r))=B(x,r)$ since a geodesic joining $x$ to
$y\in B(x,r)$ lifts to a path of the same length joining $\tilde{x}$ to some
point $\tilde{y}$ which therefore lies in $B(\tilde{x},r)$. We now claim that
$p\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muB(\tilde{x},r)\to B(x,r)$ is an isometry. Suppose
$\tilde{a},\tilde{b}\in B(\tilde{x},r)$, and put $a=p(\tilde{a})$ and
$b=p(\tilde{b})$. Then $a,b\in B(x,r)$. Consider the piecewise geodesic path
$\gamma$ comprised of 3 length-minimizing segments which runs from $x$ to $a$,
then from $a$ to $b$, then from $b$ to $x$. This loop may not lie in $B(x,r)$.
However, $\ell(\gamma)<4r$ and both endpoints are at $x$, so $\gamma\subset
B(x,2r)$, which is evenly covered by the choice of $r$. It follows that the
middle segment lifts to a segment joining $\tilde{a}$ to $\tilde{b}$ of length
equal to $d(a,b)$. Hence $d(\tilde{a},\tilde{b})\leq d(a,b)$ and the result is
proved. ∎

If $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is a map between metric spaces, we say a non-decreasing function
$\omega_{\phi}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,\operatorname{diam}(X)]\to[0,\operatorname{diam}(Y)]$ is a _modulus of
continuity_ if, for all $E\subset X$,
$\operatorname{diam}\phi(E)\leq\omega_{\phi}(\operatorname{diam}E)$.

######  Lemma 2.8.

Suppose $X,Y$ are compact, connected, CW complexes equipped with length
metrics, and $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is a continuous map, with modulus of continuity $\omega_{\phi}$.
Let $p_{Y}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{Y}\to Y$ be any covering map, and equip $\widetilde{Y}$ with
the lifted metric. Let $p_{X}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{X}\to X$ and $\widetilde{\phi}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{X}\to\widetilde{Y}$ be the maps induced by pullback, and equip
$\widetilde{X}$ with the lifted metric from $X$. Then $\widetilde{\phi}$ is
uniformly continuous, with modulus of continuity $\omega_{\widetilde{\phi}}$
independent of the cover $p_{Y}$. Indeed, there exists $r_{0}>0$ depending
only on $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ so that we can take

$\omega_{\widetilde{\phi}}=\begin{cases}\omega_{\phi}(\delta)&\delta\leq
r_{0}\\\
\omega_{\phi}(r_{0})+\omega_{\phi}(r_{0})\delta/r_{0}&\delta>r_{0}.\end{cases}$

So $\widetilde{\phi}$ behaves just like $\phi$ at small scales, and is
Lipschitz at large scales.

###### Proof.

Put $s_{0}\coloneqq\frac{1}{4}\operatorname{systole}(Y)$. By the uniform
continuity of $\phi$, there exists
$0<r_{0}<\frac{1}{4}\operatorname{systole}(X)$ such that for each $x\in X$ and
$y=\phi(x)$, we have $\phi(B(x,r_{0}))\subset B(y,s_{0})$. Fix now
$\tilde{x}\in\widetilde{X}$ and put $x=p_{X}(\tilde{x})$,
$\tilde{y}=\widetilde{\phi}(\tilde{x})$, and $y=p_{Y}(\tilde{y})=\phi(x)$. By
Lemma 2.7,

$p_{Y}^{-1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muB(y,s_{0})\to B(\tilde{y},s_{0})$

and

$p_{X}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muB(\tilde{x},r_{0})\to B(x,r_{0})$

are isometric homeomorphisms. Now fix $0<r\leq r_{0}$. Then

$\widetilde{\phi}(B(\tilde{x},r))=(p_{Y}^{-1}\circ\phi\circ
p_{X})(B(\tilde{x},r))=p_{Y}^{-1}(\phi(B(x,r))\subset
p_{Y}^{-1}(B(y,\omega_{\phi}(r)))\subset B(\tilde{y},\omega_{\phi}(r)),$

establishing the estimate in the case $\delta\leq r_{0}$. For the other case,
suppose $a,b\in\widetilde{Y}$ are at distance $R\geq r_{0}$, and let $\gamma$
be a geodesic joining $a$ to $b$. Divide $\gamma$ into sub-segments
$\gamma=\gamma_{0}*\gamma_{1}*\ldots*\gamma_{n}$ with $\ell(\gamma_{0})\leq
r_{0}$ and $\ell(\gamma_{i})=r_{0}$ for $i=1,\ldots,n$, so that
$nr_{0}\leq\ell(\gamma)\leq(n+1)r_{0}$. Then

$\displaystyle d(\widetilde{\phi}(a),\widetilde{\phi}(b))$
$\displaystyle\leq\sum_{i=0}^{n}\operatorname{diam}\phi(\gamma_{i})\leq\omega_{\phi}(r_{0})+\omega_{\phi}(r_{0})d(a,b)/r_{0}.\qed$

###  2.3. Sizes of paths and traces of homotopies

It would be nice to always work with the length of paths, but it turns out
that not all the paths we consider are rectifiable. (In particular, we
consider paths in the Julia set $\mathcal{J}$ and their projections to the
finite approximations $G_{n}$; these projections are usually not rectifiable.)
We could consider the diameter of paths, but we also need to lift paths to
covers. We work instead with a hybrid.

######  Convention 2.9.

For paths $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$, the path $\overline{\gamma}$ is the reversed path, and
$\gamma_{1}*\gamma_{2}$ denotes composition of paths, defined when
$\gamma_{1}(1)=\gamma_{2}(0)$. For homotopies $H\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X$, we will more generally use the same notations
$\overline{H}$ and $H_{1}*H_{2}$, always operating on the first input (which
is an interval).

######  Definition 2.10.

For $X$ a locally simply-connected length space, $A$ a simply-connected
auxiliary space (usually the interval), and $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X$ a continuous map, there are lifts
$\widetilde{\gamma}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to\widetilde{X}$ of $\gamma$ to the universal cover of $X$. The _size_
of $\gamma$ is the diameter of $\widetilde{\gamma}$ with respect to the lifted
metric on $\widetilde{X}$:

$\operatorname{size}(\gamma)\coloneqq\max_{s,t\in
I}d_{\widetilde{X}}(\widetilde{\gamma}(s),\widetilde{\gamma}(t)).$

The proof of the following lemma is straightforward.

######  Lemma 2.11 (Properties of size for paths).

The notion of size for paths (with $A$ the interval) satisfies the following
properties.

  1. (0)

Well-defined: $\operatorname{size}(\gamma)$ is independent of which lift of
$\gamma$ to $\widetilde{X}$ you take.

  2. (1)

Bounded by length: when $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$ is rectifiable, $\operatorname{size}(\gamma)\leq\ell(\gamma)$.

  3. (2)

Invariance under lifts: if $p\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is a covering map, $Y$ is a length space, $X$ is equipped with
the lifted metric, $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to Y$ is a path, and $\widetilde{\gamma}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$ is a lift of $\gamma$ under $p$, then
$\operatorname{size}(\widetilde{\gamma})=\operatorname{size}(\gamma)$.

  4. (3)

Sub-additive under path composition:
$\operatorname{size}(\gamma_{1}*\gamma_{2})\leq\operatorname{size}(\gamma_{1})+\operatorname{size}(\gamma_{2})$.

  5. (4)

Shortening: If $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is $\lambda$-Lipschitz and $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$ is a path in $X$, then
$\operatorname{size}(f\circ\gamma)\leq\lambda\cdot\operatorname{size}(\gamma)$.

As a result of point (1), we will prefer to give statements with hypotheses on
the _size_ of paths and construct paths with bounds on _length_ , even if we
don’t necessarily need the length bounds for our applications.

Another central feature of our development is the following.

######  Definition 2.12.

For $X$ a locally simply-connected length space, $A$ an auxiliary space, and
$H\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muI\times A\to X$ a homotopy of maps from $A$ to $X$, a _trace_
of $H$ is a path of the form $t\mapsto H(t,a)$ for fixed $a\in A$. The _trace
size_ of $H$ is the maximum size of a trace:

$\operatorname{tracesize}(H)\coloneqq\sup_{a\in
A}\,\operatorname{size}(H(\cdot,a)).$

If two maps $f,g\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X$ are homotopic by a homotopy of trace size at most $K$, then we
write $f\sim_{K}g$.

For a homotopy $H\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times I\to X$ between two paths $\gamma_{0}$ and $\gamma_{1}$, be
careful to distinguish between its _size_ and _trace size_. For instance, if
$H$ has bounded size, then the $\gamma_{i}$ must also have bounded size, while
two paths that are very long can still have a homotopy of bounded trace size.

The trace size of a homotopy between two paths is sensitive to the
parameterization of the domain of the two paths, which in turn is sensitive to
details like exactly how one defines the concatenation operation on paths. We
will specify the parameterization when necessary.

One thing to note now is that, if $\beta$ is a path of size $R$ and any
$\varepsilon>0$, then, for any concatenatable $\gamma_{1},\gamma_{2}$ and
suitable parameterization of the domain,

$\gamma_{1}*\beta*\overline{\beta}*\gamma_{2}\sim_{R+\varepsilon}\gamma_{1}*\gamma_{2}.$

(The parameterization to make this work uses a very small interval in the
domain for $\beta*\overline{\beta}$ on the left hand side.)

######  Lemma 2.13.

Suppose $X,Y,Z$ are length spaces. If $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ and $g_{0},g_{1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muY\to Z$ are maps with $g_{0}\sim_{K}g_{1}$ via the homotopy
$\alpha\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times Y\to Z$, then $g_{0}\circ f\sim_{K}g_{1}\circ f$ via the homotopy
$f^{*}\alpha\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times X\to Z$ given by $(t,x)\mapsto\alpha(t,f(x))$.

###### Proof.

The only non-trivial point is the bound on the trace size, which follows since
every trace of $f^{*}\alpha$ is a trace of $\alpha$. ∎

###  2.4. Approximate path-lifting

Suppose $X$, $Y$, and $A$ are topological spaces, and suppose we are given
maps $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ and $g\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to Y$. An _approximate lift_ of $g$ under $\phi$ with constant $K$ is a
map $g^{\prime}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X$ such that $\phi\circ g^{\prime}\sim_{K}g$.

Now suppose further that $X,Y$ are finite connected CW complexes, equipped
with complete length metrics, and $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is continuous and surjective on fundamental group. We consider
the problem of approximately lifting paths in $Y$ under $\phi$ to paths in
$X$.

In this subsection, the constants appearing in the conclusions depend on
$\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ and on the other constants appearing in the statements. We
suppress their dependence on $\phi$.

######  Proposition 2.14 (Controlled approximate path-lifting).

Suppose $X$ and $Y$ are finite connected CW complexes, $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is continuous, and $\phi_{*}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\pi_{1}(X)\to\pi_{1}(Y)$ is surjective. Then there exist a positive
constant $K$ so that for any path $\gamma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to Y$ joining endpoints $y_{0}$ to $y_{1}$, and any preimages
$x_{i}\in\phi^{-1}(y_{i})$ for $i=0,1$ of these endpoints, there exists an
approximate lift $\gamma^{\prime}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$ with $\gamma^{\prime}(i)=x_{i}$ for $i=0,1$ with homotopy
$H\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0muI\times I\to Y$ of trace size $K$. Concretely,

  * •

$H(0,t)=\gamma(t)$ and $H(1,t)=\phi\circ\gamma^{\prime}(t)$ for $0\leq t\leq
1$;

  * •

$H(s,i)=y_{i}$ for $i=0,1$ and $0\leq s\leq 1$; and

  * •

$\operatorname{size}(s\mapsto H(s,t))\leq K$ for each $t\in[0,1]$.

There exist constants $C_{0}$ and $C_{1}$ so that if $\gamma$ is rectifiable,
then so is $\gamma^{\prime}$, and
$\ell(\gamma^{\prime})<C_{0}+C_{1}\ell(\gamma)$.

$\scriptstyle{\phi}$ Figure 6. Approximate path lifting. The traces of the
homotopies are the red lines on the right; these must be of bounded size.

In other words, approximate lifting increases lengths by controlled amounts,
and the failure of a path to lift is measured by a homotopy whose traces are
uniformly bounded in size, independent of the path $\gamma$; see Figure 6. (We
do not use the fact that lengths are increased by controlled amounts in this
paper, but it helps add motivation.) We will also say that $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ satisfies the $K$-APL condition.

Before proving the general statement, we first prove it for loops $\gamma$ of
bounded length.

######  Lemma 2.15 (Controlled approximate loop-lifting).

In the setup of Proposition 2.14, fix $C>0$. Then there exist constants $L$
and $K$ with the following property. For any $x\in X$, $y=\phi(x)$, and loop
$\lambda\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to Y$ loop based at $y$ with $\operatorname{size}(\lambda)\leq C$,
there is a rectifiable loop $\lambda^{\prime}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X$ based at $x$ with $\ell(\lambda^{\prime})<L$ so that
$\phi\circ\lambda^{\prime}\sim_{K}\lambda$.

###### Proof.

First fix $x\in X$ and $y=\phi(x)$. The length metrics on $X$ and $Y$ induce
norms on the fundamental groups $\pi_{1}(X,x)$ and $\pi_{1}(Y,y)$. Since
$\operatorname{size}(\lambda)\leq C$, the norm of $[\lambda]$ is also bounded
by $C$. Since $\phi_{*}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\pi_{1}(X,x)\to\pi_{1}(Y,y)$ is surjective, there exists $L(x)$ such that
the image of the ball of radius $L(x)$ in $\pi_{1}(X,x)$ contains the ball of
radius $C$ in $\pi_{1}(Y,y)$. Now vary $x\in X$. By compactness, $\sup_{x\in
X}L(x)$ is finite; call this $L$. Thus there exists a loop $\lambda^{\prime}$
based at $x$ of length at most $L$ for which
$\phi\circ\lambda^{\prime}\sim\lambda$.

We must bound the trace size of the homotopy; in fact we bound its size. For
any $\lambda$ and $\lambda^{\prime}$ as above, we can lift
$\phi\circ\lambda^{\prime}$ and $\lambda$ to paths in the universal cover
$\widetilde{Y}$. Fix lifts $\widetilde{\phi\circ\lambda^{\prime}}$ and
$\widetilde{\lambda}$, respectively, joining common endpoints. The
concatenation
$\widetilde{\alpha}\coloneqq\widetilde{\phi\circ\lambda^{\prime}}*\overline{\widetilde{\lambda}}$
is a loop in the simply-connected length CW complex $\widetilde{Y}$. By Lemma
2.8 we have

$\operatorname{diam}(\widetilde{\alpha})\leq\operatorname{diam}(\widetilde{\phi\circ\lambda^{\prime}})+\operatorname{diam}(\widetilde{\lambda})\leq\omega_{\widetilde{\phi}}(L)+C\eqqcolon
D_{1}.$

By [DK18, Lemma 9.51], $\widetilde{Y}$ is uniformly simply-connected. This
implies that $\widetilde{\alpha}$ is homotopic to a constant map via a
homotopy whose image has diameter $K\coloneqq K(D_{1})$, as desired. ∎

###### Proof of Proposition 2.14.

Pick (arbitrarily) a basepoint $x_{*}\in X$, set $y_{*}\coloneqq\phi(x_{*})$,
and pick an arbitrary constant $C>0$. Divide $\gamma$ into sub-paths

$\gamma=\gamma_{0}*\dots*\gamma_{n-1}$

with $\operatorname{size}(\gamma_{i})\leq C$. Let the endpoints of
$\gamma_{i}$ be $z_{i},z_{i+1}\in Y$, and for each $i$ pick a path $\rho_{i}$
from $y_{*}$ to $z_{i}$, of length less than $\operatorname{diam}(Y)$. Pick
also paths $\rho_{0}^{\prime},\rho_{n}^{\prime}$ from $x_{*}$ to
$x_{0},x_{1}$, respectively, of length less than $\operatorname{diam}(X)$.
Then $\lambda_{i}\coloneqq\rho_{i}*\gamma_{i}*\overline{\rho_{i+1}}$ is a loop
based at $y_{*}$ of size less then $2\operatorname{diam}(X)+C$. By Lemma 2.15,
there are constants $L_{1},K_{1}$ so that, for each $i$, there is a loop
$\lambda_{i}^{\prime}$ in $X$ based at $x_{*}$ with
$\ell(\lambda_{i}^{\prime})<L_{1}$ and
$\phi\circ\lambda_{i}^{\prime}\sim_{K_{1}}\lambda_{i}$. In addition,
$\sigma_{0}\coloneqq(\phi\circ\rho_{0}^{\prime})*\overline{\rho_{0}}$ is a
loop of size less than
$\omega_{\widetilde{\phi}}(\operatorname{diam}Y)+\operatorname{diam}X$, so
there are constants $L_{2},K_{2}$ and a loop $\sigma_{0}^{\prime}$ based at
$x_{*}$ of length less than $L_{2}$ so that
$\phi\circ\sigma_{0}^{\prime}\sim_{K_{2}}\sigma_{0}$. Similarly pick an
approximate lift $\sigma_{n}^{\prime}$ of
$\sigma_{n}\coloneqq\rho_{n}*\overline{(\phi\circ\rho_{n}^{\prime})}$. Now set

$\displaystyle K$ $\displaystyle\coloneqq\max(K_{1},K_{2})$ $\displaystyle D$
$\displaystyle\coloneqq\max(\operatorname{diam}(X),\omega_{\widetilde{\phi}}(\operatorname{diam}(Y))$
$\displaystyle\gamma^{\prime}$
$\displaystyle\coloneqq\overline{\rho^{\prime}_{0}}*\sigma_{0}^{\prime}*\lambda_{1}^{\prime}*\lambda_{2}^{\prime}*\dots*\lambda_{n-1}^{\prime}*\sigma_{n}^{\prime}*\rho^{\prime}_{n}$

(with suitable parameterization of the domain for $\gamma^{\prime}$), so that

$\displaystyle\phi\circ\gamma^{\prime}$
$\displaystyle\sim_{K}\overline{(\phi\circ\rho_{0}^{\prime})}*\sigma_{0}*\lambda_{1}*\lambda_{2}*\dots*\lambda_{n-1}*\sigma_{n}*(\phi\circ\rho_{n}^{\prime})$
$\displaystyle\sim_{D+\varepsilon}\gamma_{0}*\gamma_{1}*\dots*\gamma_{n-1}=\gamma,$

as desired.

To get the bounds on length in the case that $\gamma$ is rectifiable, choose
the initial decomposition of $\gamma$ into sub-paths $\gamma_{i}$ so that, for
$i>0$, $\ell(\gamma_{i})\geq C$; then we have $n\leq 1+\ell(\gamma)/C$. We can
choose the paths $\lambda_{i}^{\prime}$, $\rho_{i}^{\prime}$, and
$\sigma_{i}^{\prime}$ all to have length bounded by a constant, which then
gives the desired bound on $\ell(\gamma)$. ∎

######  Remark 2.16.

Lemma 2.15 gives a bound on the overall size of the homotopy, but Proposition
2.14 only gives a bound on the trace sizes.

###  2.5. Multi-valued dynamical systems

We turn our attention back to dynamics. We think of two spaces and a pair of
maps between them, $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$, as a _multi-valued dynamical system_. We
introduce an associated limit space and describe it in two different ways, as
in [IS10], but adopting slightly different notation. Here is how to translate
between their (IS) and our (PT) notation:

$\displaystyle X^{i}_{IS}$ $\displaystyle=X_{i,PT};$
$\displaystyle(\iota,\sigma)_{IS}=(\iota,f)_{IS}$
$\displaystyle=(\phi,\pi)_{PT};$ $\displaystyle X^{+\infty}_{IS}$
$\displaystyle=\mathcal{J}_{PT}.$

(There are also minor differences in the indexing.)

Suppose $X_{1}$ and $X_{0}$ are compact topological spaces and
$\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ are two continuous maps. An _orbit_ is a
sequence of points $x_{i}\in X_{1}$ with $\pi(x_{i})=\phi(x_{i+1})$ when both
sides are defined. If $x_{i}$ is defined for $1\leq i\leq n$ for some $n\geq
1$, we get the space $X_{n}$ of orbits of length $n$. If $x_{i}$ is defined
for all $1\leq i$, we get the _limit space_ $\mathcal{J}\subset
X_{1}^{\mathbb{N}}$ of one-sided infinite orbits $(x_{0},x_{1},\ldots)$. Note
the typographical distinction between the abstract limit space $\mathcal{J}$
of a virtual endomorphism and the concrete limit space
$J_{R}\subset\widehat{\mathbb{C}}$ of a rational map $R$.

With this setup, there are two families of canonical maps

$\displaystyle\phi^{n+1}_{n},\pi^{n+1}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n+1}$ $\displaystyle\to X_{n},$
$\displaystyle\phi^{n+1}_{n}(x_{1},x_{2},\ldots,x_{n},x_{n+1})$
$\displaystyle\coloneqq(x_{1},x_{2},\ldots,x_{n})$
$\displaystyle\pi^{n+1}_{n}(x_{1},x_{2},\ldots,x_{n},x_{n+1})$
$\displaystyle\coloneqq(x_{2},\ldots,x_{n},x_{n+1}).$

We also set $\phi^{1}_{0}\coloneqq\phi$ and $\pi^{1}_{0}\coloneqq\pi$. We can
compose these to get maps $\phi^{n}_{k},\pi^{n}_{k}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to X_{k}$ for $0\leq k\leq n$.

######  Convention 2.17.

Our indexing convention is such that the index of the domain appears as a
superscript and index of the codomain appears as a subscript. This way,
composition corresponds to “contraction” of indices, as is conventional in
tensor notation.

We can present the space $\mathcal{J}$ as an inverse limit of the sequence
$\phi^{n+1}_{n}$:

$\cdots$$X_{n}$$X_{n-1}$$\cdots$$X_{1}$$X_{0}$$\mathcal{J}$$\scriptstyle{\phi^{n+1}_{n}}$$\scriptstyle{\phi^{n}_{n-1}}$$\scriptstyle{\phi^{n-1}_{n-2}}$$\scriptstyle{\phi^{2}_{1}}$$\scriptstyle{\phi^{1}_{0}}$$\scriptstyle{\phi^{\infty}_{n}}$$\scriptstyle{\phi^{\infty}_{n-1}}$$\scriptstyle{\phi^{\infty}_{1}}$$\scriptstyle{\phi^{\infty}_{0}}$

where the $\phi^{\infty}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to X_{n}$ are analogues of $\phi^{n}_{k}$:

$\phi^{\infty}_{n}(x_{1},x_{2},\dots)\coloneqq(x_{1},x_{2},\dots,x_{n}).$

There is also a canonical map $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ induced by the one-sided shift, a kind of
analogue of $\pi^{n+1}_{n}$:

$f(x_{1},x_{2},\ldots)\coloneqq(x_{2},\ldots).$

There are two natural modifications of a pair of maps $(\pi,\phi)$: we can
_iterate_ it, replacing the pair by

$\pi^{n}_{0},\phi^{n}_{0}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\rightrightarrows X_{0},$

or we can _reindex_ , replacing the pair by

$\pi^{n}_{n-1},\phi^{n}_{n-1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\rightrightarrows X_{n-1}.$

Neither of these operations changes the limit space $\mathcal{J}$, but
iterating replaces the dynamics of $f$ on $\mathcal{J}$ by $f^{n}$, while
reindexing does not change $f$.

We now restrict attention to expanding systems, as in [IS10], but adopting
terminology of Nekrashevych [Nek14] and the second author [Thu20]. We continue
with some definitions.

######  Definition 2.18.

Virtual endomorphisms. A pair of continuous maps $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ between topological spaces is a _virtual
endomorphism_ if $\pi$ is a covering map of finite degree.

######  Convention 2.19.

In this section we are exclusively concerned with virtual endomorphisms where
$X_{0},X_{1}$ are finite connected CW complexes, $X_{0}$ is equipped with a
complete length metric $d_{0}$, and $X_{1}$ is equipped with the length metric
$d_{1}$ induced by the covering $\pi$, i.e., the length metric on $X_{1}$ so
that $\pi$ is a local isometry.

######  Definition 2.20.

Suppose $0<\lambda<1$. The virtual endomorphism $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\to X_{0}$ is _$\lambda$ -backward-contracting_ if
$d_{0}(\phi(a),\phi(b))\leq\lambda d_{1}(a,b)$ for all $a,b\in X_{1}$, i.e.,
$\phi$ is a uniform contraction. Equivalently, we say the virtual endomorphism
is $\lambda^{-1}$-_forward-expanding_. A virtual endomorphism is backward
contracting (equivalently, forward expanding) if it is $\lambda$-backward-
contracting for some $0<\lambda<1$.

######  Definition 2.21.

The virtual endomorphism $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is _recurrent_ if $X_{0}$ and $X_{1}$ are
connected and $\phi_{*}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\pi_{1}(X_{1})\to\pi_{1}(X_{0})$ is surjective.

If $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is recurrent, then $X_{n}$ is connected for
each $n$.

In the setting of a virtual endomorphism, the map $\pi^{n+1}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n+1}\to X_{n}$ defined above is also a covering map. We will use the
fact that $X_{n+1}$ is a pullback, which concretely gives the following lemma,
among other pullback diagrams.

######  Lemma 2.22.

Given maps $F\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{1}$ and $G\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n}$ for which $F\circ\phi=\pi^{n}_{0}\circ G$, there exists a
unique map $A\to X_{n+1}$ such that the following diagram commutes:

$X_{1}$$X_{0}$$X_{n+1}$$X_{n}$$A$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}G}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}F}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi^{n}_{0}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\phi^{n+1}_{n}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi^{n+1}_{1}}$

Via pullback of length metrics, each space $X_{n}$ inherits a length metric
$d_{n}$. The degree of $\pi^{n}_{0}$ is $d^{n}$, and the valence of $X_{n}$ is
constant in $n$, so $\operatorname{diam}X_{n}\to\infty$ as $n\to\infty$.

######  Remark 2.23.

From a virtual endomorphism $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$, we will not see every cover of $X_{0}$
among the covers $\pi^{n}_{0}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to X_{0}$ (or their normal closures). Studying the exact covers
that appear leads to a very interesting group, the _iterated monodromy group_
$\operatorname{IMG}(\pi,\phi)\trianglelefteq\pi_{1}(X_{0})$. This subgroup
defines a different Galois cover $\widetilde{X}_{0}^{\prime}$ of $X_{0}$ than
the universal cover. Instead of measuring the size by lifting paths to the
universal cover as in Definition 2.10, we could get a different (smaller)
notion by lifting to $\widetilde{X}_{0}^{\prime}$ instead. The difference is
inessential for this paper.

######  Definition 2.24.

A _homotopy pseudo-orbit_ is a sequence, for $i\geq 1$, of points $x_{i}\in
X_{1}$ and paths $\alpha_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,1]\to X_{0}$ such that

  * •

$\alpha_{i}(0)=\pi(x_{i})$, $\alpha_{i}(1)=\phi(x_{i+1})$

  * •

$\ell(\alpha_{i})\leq C$ for some $C\geq 0$ independent of $i$.

Two homotopy pseudo-orbits $(x_{i},\alpha_{i})$ and
$(x^{\prime}_{i},\alpha^{\prime}_{i})$ are _homotopic_ if there is a sequence
of paths $\beta_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,1]\to X_{1}$ with

  * •

$\beta_{i}(0)=x_{i}$ and $\beta_{i}(1)=x_{i}^{\prime}$;

  * •

for $i\geq 1$,
$\alpha_{i}*(\phi\circ\beta_{i+1})\sim(\pi\circ\beta_{i})*\alpha_{i}^{\prime}$;
and

  * •

$\ell(\beta_{i})<B$ for some $B>0$ independent of $i$.

See Figure 7 and [IS10, Definition 6.4; Figure 5a]. Following Ishii and
Smillie, we use lengths in the condition on $\alpha_{i}$ and $\beta_{i}$, but
size would work equally well, since these are paths in a CW complex.

$x_{i}$$x_{i}^{\prime}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\beta_{i}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi(\beta_{i})}$$\sim$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\phi(\beta_{i+1})}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\alpha^{\prime}_{i}}$$x_{i+1}$$x_{i+1}^{\prime}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color<EMAIL_ADDRESS>
Figure 7. In homotopic pseudo-orbits,
$\alpha_{i}*\phi(\beta_{i+1})\sim\pi(\beta_{i})\cdot\alpha_{i}^{\prime}$.

The following result appears in [IS10, §7, 8].

######  Theorem 2.25 (Homotopy shadowing).

Suppose $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is forward-expanding. Then every homotopy
pseudo-orbit is homotopic to an orbit, and this orbit is unique.

We will need a generalization of the homotopy shadowing theorem to a setting
where the orbit depends on a parameter $a$ lying in a space $A$. We develop
this notion in close parallel to the above notions and Ishii and Smillie.

######  Definition 2.26.

For $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ a virtual endomorphism between locally
simply-connected length spaces and $A$ an auxiliary space, a _family of
homotopy pseudo-orbits parameterized by $A$ of trace size $K$_ is a sequence
of maps $x\coloneqq(x_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{1})_{i\geq 1}$ and a sequence of homotopies
$\alpha\coloneqq(\alpha_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{0})_{i\geq 1}$, so that

  1. (1)

$\alpha_{i}$ is a homotopy from $\pi\circ x_{i}$ to $\phi\circ x_{i+1}$, in
the sense that $\alpha_{i}(0,\cdot)=\pi\circ x_{i}$ and
$\alpha_{i}(1,\cdot)=\phi\circ x_{i+1}$, and

  2. (2)

there exists a constant $K<\infty$ so that for each $i\geq 1$, we have

$\operatorname{tracesize}(\alpha_{i})=\sup_{a\in
A}\operatorname{size}(\alpha_{i}(\cdot,a))\leq K.$

See Figure 8.

######  Definition 2.27.

Two families $(x,\alpha)$ and $(x^{\prime},\alpha^{\prime})$ of homotopy
pseudo-orbits parameterized by $A$ are _homotopic_ if there exists a constant
$B$ and a sequence $\beta=(\beta_{i})_{i\geq 1}$ of homotopies
$\beta_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{1}$ such that

  1. (1)

$\beta_{i}$ is a homotopy from $x_{i}$ to $x_{i}^{\prime}$, in the sense that
$\beta_{i}(0,\cdot)=x_{i}$ and $\beta_{i}(1,\cdot)=x_{i}^{\prime}$;

  2. (2)

for each $i\geq 1$, we have $\operatorname{tracesize}(\beta_{i})\leq B$; and

  3. (3)

for each $i\geq 1$, the map $\alpha_{i}*(\phi\circ\beta_{i+1})$ is homotopic
to $(\pi\circ\beta_{i})*\alpha_{i}^{\prime}$ in the following sense. There is
a map $H_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times I\times A\to X_{0}$ such that
$H_{i}(0,\cdot,\cdot)=\alpha_{i}*(\phi\circ\beta_{i+1})$,
$H_{i}(1,\cdot,\cdot)=(\pi\circ\beta_{i})*\alpha_{i}^{\prime}$, and for all
$0\leq t\leq 1$, $H_{i}(t,0,\cdot)=\pi\circ x_{i}$ and
$H_{i}(t,1,\cdot)=\phi\circ x_{i+1}^{\prime}$.

These conditions in (3) guarantee that the homotopic squares appearing in
Figure 7 remain squares throughout the interpolating maps
$H_{i}(\cdot,\cdot,a)$. We do not require a size bound on the homotopies
$H_{i}$ in condition (3).

Here is our generalization of Ishii-Smillie’s homotopy shadowing result.

$X_{0}$$X_{0}$$X_{0}$$X_{0}$$\cdots$$X_{1}$$X_{1}$$X_{1}$$X_{1}$$\cdots$$A$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color<EMAIL_ADDRESS>
Figure 8. A homotopy pseudo-orbit of maps of $A$

######  Theorem G.

Suppose $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is a $\lambda$-backward-contracting virtual
endomorphism of CW complexes with induced dynamics $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ on its limit space. Let
$(x_{i},\alpha_{i})_{i\geq 1}$ be a family of homotopy pseudo-orbits
parameterized by $A$.

  1. (1)

There exists

    1. (a)

a map $x^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to\mathcal{J}$, i.e., a family of orbits of $f$ parameterized by $A$,
and

    2. (b)

a homotopy $(\beta_{i}^{\infty})_{i\geq 1}$ from $(x,\alpha)$ to $x^{\infty}$.

  2. (2)

If $K$ is the bound on $\operatorname{tracesize}(\alpha_{i})$, then
$\operatorname{tracesize}(\beta^{\infty}_{i})\leq K^{\prime}\coloneqq
K/(1-\lambda)$.

To make sense of part 1(b) of the statement, we regard an orbit as a family of
homotopy pseudo-orbits parameterized by $A$, namely
$(x^{\infty}_{i},\alpha^{\infty}_{i})_{i\geq 1}$, where each
$\alpha^{\infty}_{i}$ is a constant homotopy.

###### Proof.

This is a straightforward modification of the proof of [IS10, Theorem 7.1].
Since we will need the notation later, we shamelessly copy their proof, more
or less word for word, with slight adjustments to indexing. We let the pseudo-
orbit $x$ now depend on a parameter $a\in A$, so that
$x\coloneqq(x_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{1})_{i\geq 1}$, and we denote the collection of homotopies by
$\alpha\coloneqq(\alpha_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{0})_{i\geq 1}$. To ease notation, we think of our
homotopies $\alpha_{i}$ and $\beta_{j}$ below as paths in the space of
continuous maps from $A$ to $X_{0}$ and $X_{1}$, respectively.

We inductively define a sequence of families of homotopy pseudo-orbits as
follows. Set $x_{i}^{0}\coloneqq x_{i}$ and
$\alpha_{i}^{0}\coloneqq\alpha_{i}$. Suppose that a family of homotopy pseudo-
orbits $(x_{i}^{n},\alpha_{i}^{n})_{i\geq 1}$ is defined. Then, since $\pi$ is
a covering and $\alpha_{i}^{n}(0,\cdot)=\pi\circ x_{i}^{n}$, there exists a
unique lift $\beta_{i}^{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X^{1}$ of $\alpha_{i}^{n}$ by $\pi$ so that
$\beta_{i}^{n}(0,\cdot)=x_{i}^{n}$, by the homotopy lifting property of
covering maps. Put $\alpha_{i}^{n+1}\coloneqq\phi\circ\beta_{i+1}^{n}$ and
$x_{i}^{n+1}\coloneqq\beta_{i}^{n}(1)$. Then, we have $\pi\circ
x_{i}^{n+1}=\pi\circ\beta_{i}^{n}(1)=\alpha_{i}^{n}(1)=\phi(x_{i+1}^{n})=\phi(\beta_{i+1}^{n}(0))=\alpha_{i}^{n+1}(0)$
and $\phi\circ x_{i+1}^{n+1}=\phi\circ\beta_{i+1}^{n}(1)=\alpha_{i}^{n+1}(1)$.
This means that, once we verify a trace size bound,
$((x_{i}^{n+1}),(\alpha_{i})^{n+1}))_{n+1}\eqqcolon(x^{n+1},\alpha^{n+1})$ is
a family of homotopy pseudo-orbits.

Contraction implies that the length of the traces of the homotopies
$\alpha_{i}^{n}$ are bounded by $K\lambda^{n}$ for $n\geq 1$; this is [IS10,
Lemma 7.2]. Concatenating the homotopies $\alpha_{i}^{n}$ for $n=1,2,\ldots$
and scaling the time parameters in the homotopy to consecutive intervals in
$[0,1)$ as in their proof, we obtain a sequence of maps
$\alpha^{\infty}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,1)\times A\to X_{0}$ for $i\geq 1$.

To get a map defined on $[0,1]\times A$, we need to say a little more. First,
as in their proof, for fixed $a\in A$, the path $\alpha^{\infty}_{i}(t,a)$ is
Cauchy as $t\to 1$ in the sense that, for any $\varepsilon>0$, there is
$\delta<1$ so that for $t_{0},t_{1}>\delta$, we have
$d_{X_{0}}(\alpha^{\infty}_{i}(t_{0},a),\alpha^{\infty}_{i}(t_{1},a))<\varepsilon$.
Furthermore these paths are uniformly Cauchy as $a$ varies. There is therefore
a well-defined limit $\alpha_{i}^{\infty}(1,a)$, and the continuous functions
$\alpha_{i}^{\infty}(t,\cdot)\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{0}$ converge uniformly to $\alpha_{i}^{\infty}(1,\cdot)$. By the
Uniform Limit Theorem, the limiting function $\alpha_{i}^{\infty}$ restricted
to $\\{1\\}\times A$ is therefore continuous. The standard proof of the
Uniform Limit Theorem shows that, in fact, we get a continuous function
$\alpha^{\infty}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,1]\times A\to X_{0}$, as desired.

We also have sequences of maps $\beta_{i}^{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{1}$ and concatenating the $\beta_{i}^{n}$’s for
$n=1,2,3,\ldots$ and extending by the Uniform Limit Theorem yields
$\beta_{i}^{\infty}$, a lift of $\alpha_{i}^{\infty}$ under $\pi$. We put
$x^{\infty}_{i}\coloneqq\beta_{i}^{\infty}(\cdot,1)$ and note that
$x^{\infty}$ defines a family of orbits, i.e. a map $x^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to\mathcal{J}$; in the associated family of homotopies, the homotopies
are constant. By construction, $\beta^{\infty}$ gives a homotopy between the
family of homotopy pseudo-orbits $(x,\alpha)$ and the family of orbits
$x^{\infty}$. Since $\pi$ is an isometry, the trace sizes of the
$\beta_{i}^{\infty}$ are bounded by $K^{\prime}\coloneqq K/(1-\lambda)$, as
required. ∎

In our later applications, it will be useful to restate Theorem G in terms of
maps into the $X_{n}$. For its proof we will need the following lemma.

######  Lemma 2.28.

Suppose $(x,\alpha)$ is a family of homotopy pseudo-orbits parameterized by
$A$, with trace size $K$. Then for each $n\geq 1$ there are unique lifted maps
$\widetilde{x}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n}$ and homotopies $\widetilde{\alpha}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{n}$ so that $\pi^{n}_{1}\circ\widetilde{x}_{n}=x_{n}$,
$\pi^{n}_{0}\circ\widetilde{\alpha}_{n}=\alpha_{n}$, and
$\widetilde{\alpha}_{n}$ is a homotopy from $\widetilde{x}_{n}$ to
$\phi^{n+1}_{n}\circ\widetilde{x}_{n+1}$ of trace size $K$.

These lifted maps are shown in Figure 9.

$X_{0}$$X_{0}$$X_{0}$$X_{0}$$X_{0}$$X_{1}$$X_{1}$$X_{1}$$X_{1}$$X_{2}$$X_{2}$$X_{2}$$X_{3}$$X_{3}$$X_{4}$$A$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{3}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{3}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.3252,0.3252,0.3252}\color[rgb]{0.3252,0.3252,0.3252}\definecolor[named]{pgfstrokecolor}{rgb}{0.3252,0.3252,0.3252}\pgfsys@color@gray@stroke{0.3252}\pgfsys@color@gray@fill{0.3252}\phi^{3}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi^{2}_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi^{3}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color@gray@fill{0.21}\pi^{3}_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21,0.21,0.21}\color[rgb]{0.21,0.21,0.21}\definecolor[named]{pgfstrokecolor}{rgb}{0.21,0.21,0.21}\pgfsys@color@gray@stroke{0.21}\pgfsys@color<EMAIL_ADDRESS>Figure 9. Lifting a family of homotopy pseudo-orbits parameterized by $A$ to
covering spaces

###### Proof.

Proceed by induction on $n$, first constructing $\widetilde{\alpha}_{n}$ and
then $\widetilde{x}_{n+1}$. Recall that $\alpha_{n}(0)=\pi\circ x_{n}$ and
$\alpha_{n}(1)=\phi\circ x_{n+1}$. We start by setting
$\widetilde{x}_{1}\coloneqq x_{1}$. If we have constructed
$\widetilde{x}_{n}$, then by unique homotopy lifting applied to the covering
map $\pi^{n}_{0}$, there is a unique function $\widetilde{\alpha}_{n}$ that is
a lift of $\alpha_{n}$ with starting point $\widetilde{x}_{n}$; since
$\widetilde{\alpha}_{n}$ is a lift, the trace size is still $K$, as desired.
We next construct $\widetilde{x}_{n+1}$. Let
$\widetilde{x}_{n+1,n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n}$ be $\widetilde{\alpha}_{n}(1)$. Then, by the defining
property of $\alpha_{n}$, we have
$\pi^{n}_{0}\circ\widetilde{x}_{n+1,n}=\phi\circ x_{n+1}$. Since we have a
pullback square (Lemma 2.22), there is a unique map
$\widetilde{x}_{n+1}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n+1}$ compatible with these two projections. ∎

######  Theorem G′.

Suppose $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is a $\lambda$-backward-contracting virtual
endomorphism of CW complexes with limit space $\mathcal{J}$, and
$(x_{i},\alpha_{i})_{i\geq 1}$ is a family of homotopy pseudo-orbits. Let
$x^{\infty}$ and $\beta$ be the families and homotopies given by Theorem G,
and let $\widetilde{x}\coloneqq(\widetilde{x}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n})_{n\geq 1}$ and
$\widetilde{x}^{\infty}=(\widetilde{x}^{\infty}_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{n})_{n\geq 1}$ be the families of maps given by Lemma 2.28 from
$x$ and $x^{\infty}$. Then for each $n\geq 0$, we have
$\widetilde{x}^{\infty}_{n}=\phi^{n+1}_{n}\circ\widetilde{x}^{\infty}_{n+1}$
and $\widetilde{x}_{n}\sim_{K^{\prime}}\widetilde{x}^{\infty}_{n}$, where
$K^{\prime}\coloneqq K/(1-\lambda)$.

###### Proof.

Immediate from Theorem G and Lemma 2.28. ∎

###  2.6. Proof of Theorem F

An inverse limit of connected spaces is connected, so $\mathcal{J}$ is
connected.

We now show that $f$ is a covering map. Let
$x^{\infty}=(x_{0},x_{1},\ldots)\in X_{0}\times X_{1}\times\ldots$ represent
an element of $\mathcal{J}$. A finite $CW$ complex is locally contractible
[Hat02, Prop. A.4], so there is a connected neighborhood $U$ of $x_{0}$ which
is contained in a contractible set. The set $U$ is then evenly covered by
$\pi$; let $\widetilde{U}_{i}$, for $i=1,\ldots,\deg(\pi)$, be the components
of its preimages in $X_{1}$. The induced map $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is the pullback the covering map $\pi$ under
$\phi^{\infty}_{0}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to X_{0}$. This immediately implies that the neighborhood
$(\phi_{0}^{\infty})^{-1}(U)$ of $x^{\infty}$ is evenly covered by the
neighborhoods $(\phi^{\infty}_{1})^{-1}(\widetilde{U}_{i})$ under $f$.

We now show that $\mathcal{J}$ is locally path-connected and hence locally
connected. In fact we will show that $\mathcal{J}$ is _weakly locally path-
connected_ at every point: for all $x\in\mathcal{J}$ and every open
neighborhood $U\ni x$, there is a smaller open neighborhood $x\in V\subset U$
so that any two points in $V$ can be connected by a path in $U$. This is
enough, since weak local path-connectivity implies that path components of
open sets are open.

We switch to thinking of $\mathcal{J}$ as a subset of $(X_{1})^{\infty}$. Fix
$p^{\infty}=(p_{1},p_{2},\ldots)\in\mathcal{J}$. Let $d_{1}$ denote the length
metric on $X_{1}$. The definition of the product topology says that, for any
fixed $\varepsilon_{0}>0$, a neighborhood basis of $p^{\infty}$ is given by
those neighborhoods $U_{m,\varepsilon}$ of $p^{\infty}$ for $m\geq 1$ and
$\varepsilon<\varepsilon_{0}$, consisting of points
$q^{\infty}=(q_{1},q_{2},\ldots)\in\mathcal{J}$ for which
$d_{1}(p_{i},q_{i})<\varepsilon$ for $i=1,\ldots,m$. Using local
contractibility and Lemma 2.7, choose $\varepsilon_{0}$ so that balls in
$X_{0}$ of radius $\varepsilon_{0}$ or smaller are contained in a contractible
neighborhood and hence evenly covered by $\pi$, and so that $\pi$ is an
isometry on these balls. Fix $m$ and $\varepsilon<\varepsilon_{0}$ and put
$U\coloneqq U_{m,\varepsilon}$.

Let $K$ be the constant for approximate path-lifting given by Proposition 2.14
for the map $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\to X_{0}$, and choose $n\geq m$ large enough that
$\lambda^{n-m}K/(1-\lambda)<\varepsilon/3$. Let $V\subset U$ be
$U_{n,\varepsilon/3}$, and fix $q^{\infty}\in V$. Using Theorem G, we are
going to show $\mathcal{J}$ is weakly locally path-connected by constructing a
path $x^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to\mathcal{J}$ from $p^{\infty}$ to $q^{\infty}$ which is contained in
$U$.

To construct $x^{\infty}$, we will first construct a family
$x=(x_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\to X_{1})_{i\geq 1}$ of homotopy pseudo-orbits of paths joining
$p^{\infty}$ to $q^{\infty}$ parameterized by $A=I$, the unit interval; as a
path, each $x_{i}$ joins $p_{i}$ to $q_{i}$. We first construct the $x_{i}$
for $1\leq i\leq n$, where $n$ is the integer from the previous paragraph; in
this range $x_{i}$ will be an actual orbit (i.e., for $i<n$ the homotopies
$\alpha_{i}$ are constant). We begin by choosing the path $x_{n}$. By
construction, $d_{1}(p_{n},q_{n})<\varepsilon/3$; let $x_{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{1}$ be a path exhibiting this. By decreasing induction, for
$1\leq i<n$ define $x_{i}$ to be the lift of $\phi\circ x_{i+1}$ under $\pi$
starting at $p_{i}$. Since $\phi$ is a contraction and $\pi$ is an isometry on
balls of radius $\varepsilon/3$, the image of $x_{i}$ is contained in the
$\varepsilon/3$-ball about $p_{i}$. In particular, $q_{i}$ is the only element
of $\pi^{-1}(\phi(q_{i+1}))$ in this ball, so $x_{i}$ ends at $q_{i}$.

We complete the construction of $x$ by constructing $x_{i}$ for $i>n$, by
increasing induction starting with $x_{n}$. For $i\geq n$, supposing we have
defined $x_{i}$, let $x_{i+1}$ be an approximate lift under $\phi$ of
$\pi\circ x_{i}$ with trace size bounded by $K$ joining $p_{i}$ to $q_{i}$
(using Proposition 2.14). Let $\alpha_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{0}$ for $i\geq n$ be the corresponding homotopies between
$\pi\circ x_{i}$ and $\phi\circ x_{i+1}$. We have completed the definition of
our family of homotopy pseudo-orbits with trace size $K$.

By Theorem G, the family of homotopy pseudo-orbits $(x_{i})_{i\geq
1},(\alpha_{i})_{i\geq 1}$ defines

  1. (i)

a family of orbits $(x_{i}^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to X_{1})_{i\geq 1}$ yielding a map $x^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\to\mathcal{J}$, and

  2. (ii)

a family of homotopies $(\beta_{i}^{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times A\to X_{1})_{i\geq 1}$ with $\beta_{i}^{\infty}$ joining $x_{i}$
to $x_{i}^{\infty}$.

The bounds on the trace size of $\beta_{i}^{\infty}$ from Theorem G are not
enough for our purposes: to make sure the path $x^{\infty}$ remains within
$U$, we need to make sure that $\beta_{i}^{\infty}$ has trace size less than
$\varepsilon$ for $1\leq i\leq m$, while Theorem G gives a constant trace size
$K/(1-\lambda)$. Thus we consider the proof of the Theorem, which expresses
each element $\beta_{i}^{\infty}$ as a concatenation
$\beta_{i}^{1}*\beta_{i}^{2}*\cdots*\beta_{i}^{k}*\cdots$, where each
$\beta_{i}^{k}$ is obtained from $\alpha_{i+k}$ by repeatedly lifting by $\pi$
(a total of $k+1$ times) and composing with $\phi$ (a total of $k$ times).
Thus $\beta_{i}^{k}$ is constant (trace size $0$) for $i+k<n$, and otherwise
has trace size bounded by $\lambda^{k}K$. In particular, for $i\leq m$, we
have

$\operatorname{tracesize}(\beta_{i}^{\infty})\leq\sum_{k=n-i}^{\infty}\operatorname{tracesize}(\beta_{i}^{k})\leq\frac{\lambda^{n-i}}{1-\lambda}K\leq\frac{\lambda^{n-m}}{1-\lambda}K<\varepsilon/3.$

Finally, we estimate the size of $x_{i}^{\infty}$ for $1\leq i\leq m$. Recall
that $\beta_{i}^{\infty}$ is a homotopy relative to endpoints from the path
$x_{i}$ to the path $x_{i}^{\infty}$. We have

$\operatorname{diam}(x_{i}^{\infty})\leq\operatorname{tracesize}(\beta^{\infty}_{i})+\operatorname{diam}(x_{i})+\operatorname{tracesize}(\beta^{\infty}_{i})<\varepsilon/3+\varepsilon/3+\varepsilon/3=\varepsilon.$

This concludes the proof of weak local path connectivity, and hence local
connectivity.

It remains to prove that $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is positively expansive. Let $\varepsilon_{0}$
be the parameter chosen above, so that $d_{0}$-balls of radius
$\varepsilon_{0}$ are contained in contractible sets. Consider the
neighborhood of the diagonal $N\coloneqq
U_{1,\varepsilon_{0}}=\\{(x,y)\in\mathcal{J}\times\mathcal{J}\mid
d_{1}(x_{1},y_{1})<\varepsilon_{0}\\}$. To prove that $f$ is positively
expansive, it suffices to show that, if $(f^{i}x,f^{i}y)\in N$ for each $i\geq
0$, then $x=y$; so let us suppose the iterates remain in $N$. If we think of
$\mathcal{J}$ as a subset of $X_{1}^{\mathbb{N}}$, the map $f$ is given by the
left-shift. Thus $d_{1}(x_{i},y_{i})<\varepsilon_{0}$ for each $i\geq 0$.

Since $d_{1}$ is a length metric, for each such $i$ there exists a path
$\beta_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu[0,1]\to X_{1}$ with $\ell(\beta_{i})<\varepsilon_{0}$ joining $x_{i}$ to
$y_{i}$. By construction, the paths $\pi(\beta_{i})$ and $\phi(\beta_{i+1})$
join the same endpoints for each $i\geq 1$. We have
$\operatorname{diam}(\pi(\beta_{i}))<\varepsilon_{0}$ by construction, while
$\operatorname{diam}(\phi(\beta_{i+1}))<\lambda\varepsilon_{0}$; hence the
union of these paths lies in an $\varepsilon_{0}$-ball in $X_{0}$. Since the
ball is contractible, the two paths are homotopic. The collection
$\beta\coloneqq(\beta_{i})_{i\geq 1}$ is therefore a homotopy between the
orbits $x=(x_{i})_{i}$ and $y=(y_{i})_{i}$. By Theorem 2.25, we then have
$x=y$, completing the proof that $f$ is positively expansive.∎

###  2.7. Homotopy sections

Suppose $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is a continuous map between topological spaces. A _homotopy
section_ of $\phi$ is a continuous map $\sigma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muY\to X$ such that $\phi\circ\sigma\sim_{K}\mathrm{id}_{Y}$ for some
constant $K$. The main result of this section is the existence of homotopy
sections to $\mathcal{J}$.

######  Proposition 2.29.

Suppose $\phi,\pi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{1}\rightrightarrows X_{0}$ is a $\lambda$-backward-contracting and
recurrent virtual endomorphism of finite CW complexes. Suppose
$\sigma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{0}\to X_{1}$ is a homotopy section of $\phi$, with
$\phi\circ\sigma\sim_{K}\mathrm{id}_{X_{0}}$. Then for each $n\in\mathbb{N}$,
we have the following.

  1. (1)

There is a canonically associated family

$(x^{n},\alpha^{n})=(x^{n}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to X_{1},\alpha^{n}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times X_{n}\to X_{0})_{i\geq 1}$

of homotopy pseudo-orbits parameterized by $X_{n}$ with trace sizes at most
$K$, agreeing with the identity map on $X_{n}$ on the first $n$ factors.
Concretely, we require that $x_{i}=\pi^{i}_{1}\circ\phi^{n}_{i}$ for $i\leq n$
and $\alpha_{i}$ is constant for $i<n$.

  2. (2)

There exists $\tilde{x}^{n}_{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to\mathcal{J}$ with
$\phi^{\infty}_{n}\circ\tilde{x}^{n}_{\infty}\sim_{K^{\prime}}\mathrm{id}_{X_{n}}$,
where $K^{\prime}\coloneqq K/(1-\lambda)$.

Observe that in the lifted length metrics, the diameters of the $X_{n}$ tend
to infinity exponentially fast; but conclusion (2) says that the compositions
$\phi^{\infty}_{n}\circ\tilde{x}^{n}_{\infty}$ are at uniformly bounded
distance from the identity.

###### Proof.

We will construct a family of homotopy pseudo-orbits $(x_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to X_{1})_{i\geq 1}$. Fix a homotopy $\alpha\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muI\times X_{0}\to X_{0}$ joining the identity to $\phi\circ\sigma$ with
trace size $K$.

For $1\leq i\leq n$, let $x^{n}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX_{n}\to X_{1}$ be the natural maps as defined in the statement;
likewise, for $1\leq i<n$, let $\alpha^{n}_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muA\times I\to X_{0}$ be the constant homotopies.

For $i\geq n$, set by induction $x_{i+1}\coloneqq\sigma\circ\pi\circ x_{i}$.
Then

$\phi\circ x_{i+1}=\phi\circ\sigma\circ\pi\circ x_{i}\sim_{K}\pi\circ x_{i}$

with homotopy $\alpha_{i}=(\pi\circ x_{i})^{*}(\alpha)$ (see Lemma 2.13). This
gives the desired family of homotopy pseudo-orbits.

The second assertion follows immediately from Theorem G′, applied to the
family $(x,\alpha)$ of homotopy pseudo-orbits parameterized by $X_{n}$
constructed above. ∎

To get started applying this result, we have the following lemma.

######  Lemma 2.30.

Suppose $X,Y$ are finite connected graphs and $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is surjective on the fundamental group. Then there exists a
homotopy section $\sigma\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muY\to X$ of $\phi$.

As a corollary of Proposition 2.29, we then have

######  Corollary 2.31.

If $\pi,\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{1}\rightrightarrows G_{0}$ is a backward-contracting recurrent virtual
endomorphism of graphs with limit space $\mathcal{J}$, then there exists a
constant $K>0$ and a family of maps $\sigma^{n}_{\infty}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG_{n}\to\mathcal{J}$ for $n\in\mathbb{N}$ such that
$\phi^{\infty}_{n}\circ\sigma^{n}_{\infty}\sim_{K}\mathrm{id}_{G_{n}}$.

###### Proof of Lemma 2.30.

The statement is invariant under homotopy equivalence, so we may assume $Y$ is
a rose of $k$ circles with basepoint $y$. Let $x\in\phi^{-1}(y)$ be a
basepoint for $G_{1}$. Fix one of the $k$ circles, say $\beta\subset Y$. The
assumption that $\phi$ induces a surjection between fundamental groups implies
there is a loop $\alpha$ based at $x$ for which
$\phi(\alpha)\sim_{K(\beta)}\beta$ relative to $y$. We set
$\sigma|_{\beta}=\alpha$. Doing this for each circle and putting
$K\coloneqq\max_{\beta}K(\beta)$ proves the claim. ∎

##  3\. The conformal gauge

We recall here from [HP09] the construction of two natural classes of metrics,
one larger than the other, associated to certain expanding dynamical systems.

###  3.1. Convention

Throughout this section, we suppose $\mathcal{J}$ is compact, connected, and
locally connected, and $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ is a positively expansive self-cover of degree
$d\geq 2$. Proposition 2.5 and Lemma 2.3 imply that the dynamics of
$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\mathcal{J}\to\mathcal{J}$ is topologically cxc. It follows
that there exists a finite open cover $\mathcal{U}_{0}$ such that, as in the
notation from §2.1, the mesh of the coverings $\mathcal{U}_{n}$ tends to zero
as $n\to\infty$, and in addition, for all $\widetilde{U}\in\mathcal{U}_{n}$
with $f^{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{U}\to U\in\mathcal{U}_{0}$, we have $\deg(f^{n}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widetilde{U}\to U)=1$, i.e., each such $U$ is evenly covered by each
iterate.

###  3.2. Visual metrics

The metrics we construct are most conveniently defined coarse-geometrically as
visual metrics on the boundary of a certain rooted Gromov hyperbolic
1-complex. Before launching into technicalities, we quickly summarize the
development. The _visual metrics_ on $\mathcal{J}$ have the properties that
there exists a constant $0<\theta<1$ such that for any $n\in\mathbb{N}$ and
any $U\in\mathcal{U}_{n}$, we have $\operatorname{diam}U\asymp\theta^{n}$, and
these $U$ are uniformly nearly round. See Theorem 3.2 below for the precise
statements. The _snowflake gauge_ is the set of metrics bilipschitz equivalent
to some power of a visual metric. The snowflake gauge is an invariant of the
topological dynamics. Visual metrics are Ahlfors regular with respect to the
Hausdorff measure in their Hausdorff dimension, and this measure is comparable
to the measure of maximal entropy. The _Ahlfors-regular gauge_ of metrics is
the larger set of all Ahlfors-regular spaces quasi-symmetrically equivalent to
a visual metric; it too is an invariant of the topological dynamics.

####  $1$-complex

From coverings as above, we define a rooted hyperbolic 1-complex $\Sigma$ to
get the visual metrics. In addition to the $\mathcal{U}_{n}$ for $n\geq 0$,
let $\mathcal{U}_{-1}$ be the trivial covering $\\{\mathcal{J}\\}$. The
vertices of $\Sigma$ are the elements of the $\mathcal{U}_{n}$ for $n\geq-1$,
with root $\mathcal{U}_{-1}$. If $U\in\mathcal{U}_{n}$, the _level_ of $U$ is
$|U|\coloneqq n$. Edges of $\Sigma$ are of two types.

  * •

_Horizontal_ edges join $U,U^{\prime}\in\mathcal{U}_{n}$ if $U\cap
U^{\prime}\neq\emptyset$.

  * •

_Vertical_ edges join $U,U^{\prime}$ if $\bigl{|}|U|-|U^{\prime}|\bigr{|}=1$
and $U\cap U^{\prime}\neq\emptyset$.

We equip $\Sigma$ with the word length metric in which each edge has length
$1$. The _level_ of an edge is the maximum of the levels of its endpoints.

#### Compactification

We compactify $\Sigma$ in the spirit of W. Floyd rather than of M. Gromov, as
follows. Let $\varepsilon>0$ be a parameter, and let $d_{\varepsilon}$ be the
length metric on $\Sigma$ obtained by scaling so edges at level $n$ have
length $\theta^{n}$ where $\theta=e^{-\varepsilon}$. The metric space
$(\Sigma,d_{\varepsilon})$ is not complete. Its completion
$\overline{\Sigma}_{\varepsilon}$, sometimes called the _Floyd completion_ ,
adjoins the corresponding boundary
$\partial_{\varepsilon}\Sigma\coloneqq\overline{\Sigma}_{\varepsilon}-\Sigma$.
The extension of $d_{\varepsilon}$ to a metric on the boundary
$\partial_{\varepsilon}\Sigma$ is called a _visual metric_ on the boundary.

#### Snapshots

To set up the statement of the next theorem, we need a definition.

######  Definition 3.1.

Suppose $X$ is a metric space and $0<\theta<1$. A sequence
$(\mathcal{S}_{n})_{n}$ of finite coverings of $X$ is called a _sequence of
snapshots of $X$ with scale parameter $\theta$_ if there exists a constant
$C>1$ such that

  1. (1)

(scale and roundness) For all $n$ and all $S\in\mathcal{S}_{n}$, there exists
$x_{S}\in S$ with

$B(x_{S},C^{-1}\theta^{n})\subset S\subset B(x_{S},C\theta^{n}),$

  2. (2)

(nearly disjoint) For all $n$, the collection of pairs $\\{(x_{S},S)\mid
S\in\mathcal{S}_{n}\\}$ may be chosen so that in addition

$B_{\varepsilon}(x_{S},C^{-1}\theta^{n})\cap
B_{\varepsilon}(x_{S^{\prime}},C^{-1}\theta^{n})\neq\emptyset$

whenever $S$ and $S^{\prime}$ are distinct elements of $\mathcal{S}_{n}$.

The elements $S$ need not be either open nor connected.

#### Properties of visual metrics

Theorem 3.2 summarizes highlights of [HP09, Chapter 3], specialized to the
conventions in §3.1.

######  Theorem 3.2 (Visual metrics).

Suppose the topological dynamical system $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ and sequence of open covers
$(\mathcal{U}_{n})_{n}$ satisfy the assumptions in §3.1. Let $\Sigma$ be the
associated 1-complex.

There exists $\varepsilon_{0}>0$ such that for all
$0<\varepsilon<\varepsilon_{0}$, the boundary $\partial_{\varepsilon}\Sigma$
equipped with the visual metric $d_{\varepsilon}$ is naturally homeomorphic to
$\mathcal{J}$. Moreover, for each $\varepsilon$ in this range there exists
$C>1$ such that the following hold. Let $\theta=e^{-\varepsilon}<1$; we denote
by $B_{\varepsilon}$ an open ball with respect to $d_{\varepsilon}$.

  1. (1)

(Snapshot property) The sequence $(\mathcal{U}_{n})_{n}$ is a sequence of
snapshots of $\mathcal{J}$ with scale parameter $\theta$.

  2. (2)

For $q\coloneqq\frac{\log d}{\varepsilon}$, for all $x\in\mathcal{J}$ and all
$r<\operatorname{diam}(\mathcal{J})$, the Hausdorff $q$-dimensional measure
satisfies

$C^{-1}r^{q}<\mathcal{H}^{q}(B_{\varepsilon}(x,r))<Cr^{q}.$

  3. (3)

There exists a unique $f$-invariant probability measure $\mu_{f}$ of maximal
entropy $\log d$ supported on $\mathcal{J}$. The support of $\mu_{f}$ is equal
to all of $\mathcal{J}$, and for all Borel sets $E$, we have
$C^{-1}<\mu_{f}(E)/\mathcal{H}^{q}(E)<C$.

  4. (4)

There exists $r_{0}<\operatorname{diam}(\mathcal{J})$ so that for all
$r<r_{0}$, we have
$f(B_{\varepsilon}(x,r))=B_{\varepsilon}(f(x),\theta^{-1}r)$, and the
restriction $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muB_{\varepsilon}(x,r)\to B_{\varepsilon}(f(x),\theta^{-1}r)$ scales
distances by the factor $\theta^{-1}$.

  5. (5)

If two different parameters $\varepsilon$, $\varepsilon^{\prime}$ and two
different open covers $\mathcal{U}_{0}$, $\mathcal{U}^{\prime}_{0}$ are
employed in the construction, the resulting metrics $d$, $d^{\prime}$ are
snowflake equivalent.

#### Snowflake equivalence

Two metrics $d,d^{\prime}$ are _snowflake equivalent_ if there exist
parameters $\beta,\beta^{\prime}>0$ such that the ratio
$d^{\beta}/(d^{\prime})^{\beta^{\prime}}$ is bounded away from zero and
infinity. Put another way, snowflake equivalent means bilipschitz, after
raising the metrics to appropriate powers. The _snowflake gauge_ of
$(f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\mathcal{J}\to\mathcal{J})$ is the snowflake equivalence class
of some (and by Theorem 3.2(5), equivalently, of any) visual metric on
$\mathcal{J}$. Thus, the snowflake gauge of the dynamical system
$f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip
6.0mu plus 1.0mu\mathcal{J}\to\mathcal{J}$ depends only on the topological
dynamics, and not on choices.

A snowflake equivalence $d\mapsto d^{\prime}$ preserves the property of being
a sequence of snapshots, though the scale parameter and constant $C$ may
change. However, if $(\mathcal{S}_{n})_{n}$ is a sequence of snapshots in a
metric space $X$, if $Y$ is another metric space, and if $h\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muX\to Y$ is only a quasi-symmetry, then the transported sequence
$(\mathcal{T}_{n})_{n}$ comprised of images of elements of $\mathcal{S}_{n}$
under $h$ need not be a sequence of snapshots: the scale condition in
definition of sequence of snapshots (Definition 3.1(1)) can fail. (See §3.3
for a definition of quasi-symmetry.)

#### Conformal elevator and naturality of gauges

Suppose now $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a hyperbolic rational
function. Its Julia set $\mathcal{J}$ can now be equipped with at least two
natural metrics: the round spherical metric, and a visual metric. The Koebe
distortion principles imply that small spherical balls can be blown up via
iterates of $f$ to balls of definite size, with uniformly bounded distortion.
The same is true for the visual metric, by Theorem 3.2(4). Combining these
observations in a technique known as the _conformal elevator_ implies the
following.

######  Proposition 3.3.

On Julia sets of hyperbolic rational functions, the spherical and visual
metrics are quasi-symmetrically equivalent.

This is true much more generally [HP09, Theorems 2.8.2 and 4.2.4]. In all but
the most restricted cases, however, the spherical and visual metrics are not
snowflake equivalent. In the visual metric, the map $f$ is locally a homothety
with a constant factor which is the same at all points. In the spherical
metric, the image of a ball under $f$ need not be a ball, and what’s more,
typically, the magnitude of the derivative $\lvert f^{\prime}(z)\rvert$ varies
as $z$ varies in $J_{f}$. The exceptions include maps such as $f(z)=z^{d}$.

###  3.3. Ahlfors-regular spaces

A metric space is _doubling_ if there is an integer $N$ such that any ball of
radius $r$ is covered by at most $N$ balls of radius $r/2$. Doubling is a
finite-dimensionality condition: the Assouad Embedding Theorem asserts that a
doubling metric space is snowflake equivalent to a subset of a finite
dimensional spherical space [Hei01].

Ahlfors regularity is a homogeneity condition that implies doubling. A space
with both a metric and a measure $(Z,d,\mu)$ is _Ahlfors regular_ of exponent
$q$ if $\mu(B(z,r))\asymp r^{q}$ for each $r<\operatorname{diam}(Z)$, where
the implicit constant is independent of $z$ and of $r$. The Hausdorff
dimension of such a space is necessarily equal to $q$, and in fact the given
measure $\mu$ is comparable to the $q$-dimensional Hausdorff measure. A metric
space is _Ahlfors regular_ if its Hausdorff measure in its Hausdorff dimension
is Ahlfors regular.

Suppose $(Z,d,\mu)$ and $(Z^{\prime},d^{\prime},\mu^{\prime})$ are connected
compact doubling metric spaces. A homeomorphism $h\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muZ\to Z^{\prime}$ is a _quasi-symmetry_ (qs) if there is a constant $H\geq
1$ such that for each $r<\operatorname{diam}(Z)$ and each $z\in Z$, there is
an $s>0$ such that

$B_{d^{\prime}}(h(z),s)\subset h(B_{d}(z,r))\subset B_{d^{\prime}}(h(z),Hs).$

In other words: the image of a round ball is nearly round.222Strictly
speaking, this is the definition of a _weakly qs_ map; this is equivalent to
the standard but less intuitive definition of a qs map in our setting; see
[Hei01]. A quasi-symmetric map does not, in general, preserve the property of
Ahlfors regularity, though it does preserve the property of being doubling.

The _Ahlfors-regular conformal gauge_ $\mathcal{G}(Z)$ of a metric space $Z$
is the set of all Ahlfors-regular metric spaces qs equivalent to $Z$; it may
be empty. Let $f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J}$ be a positively expansive self-cover as in the
conventions of §3.1 and let $d_{\varepsilon}$ be a visual metric from Theorem
3.2. Conclusion (2) of that theorem shows that visual metrics are Ahlfors
regular and so $\mathcal{G}(f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J})\coloneqq\mathcal{G}(\mathcal{J},d_{\varepsilon})$
is nonempty. For example, if $f$ is a hyperbolic rational function with Julia
set $\mathcal{J}$, the spherical metric on $\mathcal{J}$ is Ahlfors regular.
Proposition 3.3 then implies that the spherical and visual metrics on
$\mathcal{J}$ both belong to the gauge $\mathcal{G}(f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J})$.

The _Ahlfors-regular conformal dimension_
$\operatorname{ARCdim}(f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J})$ is the infimum of the Hausdorff dimensions of
the metrics in the Ahlfors regular conformal gauge. It is thus another
numerical invariant of the topological dynamics. Note that by definition, for
any Ahlfors-regular metric $d$ in $\mathcal{G}(f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J})$, we have

$\operatorname{ARCdim}(f\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0mu\mathcal{J}\to\mathcal{J})\leq\operatorname{hdim}(\mathcal{J},d).$

###  3.4. Approximately self-similar spaces

Our proof of Theorem A uses a result of Carrasco and Keith-Kleiner that says
that for certain classes of spaces, the Ahlfors-regular conformal dimension is
a critical exponent of combinatorial modulus. (See Theorem 5.11.) Here, we
introduce that class of spaces.

The following definition appears in [Kle06, §3].

######  Definition 3.4 (Approximately self-similar).

A compact metric space $(Z,d)$ is _approximately self-similar_ if there exists
a constant $L\geq 1$ such that if $B(z,r)\subset Z$ for
$0<r<\operatorname{diam}(Z)$, then there exists an open set $U\subset Z$ which
is $L$-bilipschitz equivalent to the rescaled ball $(B(z,r),\frac{1}{r}d)$.

An immediate consequence of Theorem 3.2 is the following.

######  Corollary 3.5 (Visual metrics are self-similar).

In the setting of Theorem 3.2, visual metrics are approximately self-similar.

###### Proof.

Let $\varepsilon$, $r_{0}$, and $\theta$ be as in Theorem 3.2, and choose any
$0<r<\operatorname{diam}(\mathcal{J})$. Let $B_{\varepsilon}(x,r)$ be any
ball. If $r>\theta r_{0}$ we take $U\coloneqq B_{\varepsilon}(x,r)$ and note
that the metric space $(B_{\varepsilon}(x,r),d_{\varepsilon})$ is bilipschitz
equivalent to the metric space
$(B_{\varepsilon}(x,r),\frac{1}{r}d_{\varepsilon})$. If $r<\theta r_{0}$, let
$n$ be the unique integer for which $\theta r_{0}<\theta^{-n}r\leq r_{0}$. Set
$U\coloneqq B_{\varepsilon}(f^{n}x,\theta^{-n}r)$. By Theorem 3.2(4), the open
set $U$ is isometric to the rescaled metric space
$(B_{\varepsilon}(x,r),\theta^{-n}d_{\varepsilon})=(B_{\varepsilon}(x,r),\frac{1}{r}\cdot
r\theta^{-n}d_{\varepsilon})$ which is in turn bilipschitz equivalent to
$(B_{\varepsilon}(x,r),\frac{1}{r}d_{\varepsilon})$ by our choice of $n$. ∎

##  4\. Energies of graph maps

In this section, we introduce asymptotic $q$-conformal energies associated to
virtual endomorphisms of graphs and related analytical notions of extremal
length. This section is a review of concepts from [Thu19], which contains
further motivation, especially in its Appendix A.

In this section, all graphs are assumed to be finite.

###  4.1. Weighted and $q$-conformal graphs

######  Definition 4.1.

Suppose $G$ is a graph, and $1\leq q\leq\infty$.

  1. (1)

For $1<q\leq\infty$, a _$q$ -conformal structure_ on $G$ is a positive _$q$
-length_ $\alpha(e)$ on each edge $e$, giving a length metric $|dx|$ in which
$e$ has length $\alpha(e)$. For $q=\infty$, this is also called a _length
graph_.

  2. (2)

For $q=1$, a _$1$ -conformal structure_ on $G$ is a positive weight $w(e)$ on
each edge $e$. These weights do not determine a length structure. (In cases
where we take derivatives for maps from a $1$-conformal graph, the length
structure is arbitrary.) A $1$-conformal graph is also called a _weighted
graph_.

We will write $G^{q}$ for a graph together with a $q$-conformal structure on
it.

######  Remark 4.2.

Although $q$-conformal structures for $q\in(1,\infty]$ are all formally the
same data, we distinguish the value of $q$ for two reasons. First, it helps us
keep track of which energies $E^{p}_{q}$ to consider. Secondly, from another
point of view a $q$-conformal structure is naturally thought of as an
equivalence class of pairs $(\ell,w)$ of a length structure $\ell$ and weights
$w$ under a rescaling depending on $q$ [Thu19, Def. A.17], and from that point
of view there are two natural length-like structures in the picture.

The distinctions between $q$-conformal graphs as $q$ varies arise from how
various related analytical quantities are defined and scale under changes of
weights. Imagine each edge as “thickened” with an extra $(q-1)$-dimensional
space, to obtain a metric planar “rectangle” equipped with $q$-dimensional
Hausdorff measure $\mathcal{H}^{q}$. Scaling the length of an edge by a factor
$\lambda$ changes the total $\mathcal{H}^{q}$-measure by the factor
$\lambda^{q}$ and therefore scales in the imaginary direction “orthogonal” to
the edge by the factor $\lambda^{q-1}$.

An important special case of weighted graphs (i.e., $q=1$) is when the
underlying space is a $1$-manifold $\bigcup_{i}J_{i}$, where each $J_{i}$ is
an interval or the circle. We may regard each $J_{i}$ as a graph, by adding
the endpoints of the interval, or an arbitrary vertex on the circle. Up to
isomorphism, the result is unique. A formal sum $C=\sum w_{i}J_{i}$ with
$w_{i}>0$ then determines a weighted graph in a unique way, by setting the
weight of the unique edge in $J_{i}$ to be $w_{i}$. Ignoring the graph
structure, we call $C$ also a weighted $1$-manifold.

######  Definition 4.3.

A _curve_ on a space $X$ is a connected 1-manifold $C$ (either an interval or
circle) together with a map $c\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muC\to X$. We refer to the curve by just the map $c$ to be short, but the
underlying domain $C$ is part of the data. (Note the distinction with a _path_
, where the domain is a fixed interval.) For a _multi-curve_ we drop the
restriction that $C$ be connected, giving $C=\bigcup_{i}J_{i}$ and a map $c$
determined by $c_{i}\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muJ_{i}\to X$. A _strand_ of a multi-curve is one of its component curves
$c_{i}$. A _weighted multi-curve_ is similar, but where $C=\sum w_{i}J_{i}$
has the structure of a weighted graph.

###  4.2. Energies and extremal length

For any $1\leq p\leq q\leq\infty$ and piecewise linear (PL) map
$\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG^{p}\to H^{q}$ from a $p$-conformal graph to a $q$-conformal graph,
there is an energy $E^{p}_{q}(\phi)$ with well-behaved properties. For any of
these energies and a homotopy class $[\phi]$ of maps as above, we denote by
$E^{p}_{q}[\phi]$ the infimum of the energy over maps in the class. In this
paper we only need a few special cases.

#### Case $p=1$ and $1<q<\infty$

We switch names to consider a PL map $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muW\to G^{q}$ from a weighted (or $1$-conformal) graph $W$ to a
$q$-conformal graph $G^{q}$, where $1<q<\infty$. Let $q^{\vee}=q/(q-1)$ be the
Hölder conjugate of $q$.

Define

$E^{1}_{q}(\phi)\coloneqq\lVert n_{\phi}\rVert_{q^{\vee},G^{q}}.$

We now explain the notation. For $y\in G^{q}$, the value

$n_{\phi}(y)\coloneqq\sum_{\phi(x)=y}w(e(x))$

is the weighted number of pre-images of $y$, where for $x\in W$, the quantity
$e(x)$ denotes an edge containing $x$. The edge-weights $\alpha$ on $G^{q}$,
when interpreted as lengths, define a length measure $|dy|$ on the underlying
set of $G^{q}$ such that $\int_{e}|dy|=\alpha(e)$ for each edge $e$ of
$G^{q}$. The quantity $n_{\phi}(y)$ may be infinite (if, e.g., $\phi$
collapses an edge to the point $y$) or undefined (if, e.g., $x\in\phi^{-1}(y)$
is a vertex incident to edges with different weights). However, the assumption
that $\phi$ is PL and our convention that the graphs are finite imply that the
set of such $y$ is finite, hence of $|dy|$-measure zero. The quantity $\lVert
n_{\phi}\rVert_{q^{\vee},G^{q}}$ is then the usual $L^{q^{\vee}}$ norm of the
function $n_{\phi}$ with respect to the measure $|dy|$. For instance,
$E^{1}_{\infty}(\phi)$ is the weighted total length of the image of $\phi$.

If $n_{\phi}$ is constant on edges of $G^{q}$—automatic if $\phi$ minimizes
energy in its homotopy class—the formula for the energy is concretely given by

(4.4)
$E^{1}_{q}(\phi)=\biggl{(}\sum_{e\in\operatorname{Edge}(G)}\alpha(e)n_{\phi}(e)^{q^{\vee}}\biggr{)}^{1/q^{\vee}}.$

######  Definition 4.5.

A weighted multi-curve $c\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muC\to G$ on a graph $G$ is _reduced_ if the restriction to each strand is
either constant, or has arbitrarily small perturbations that are locally
injective.

Thus the images of the strands of a reduced multi-curve have no backtracking.
Reduced curves $\phi$ minimize $E^{1}_{q}$ in their homotopy class [Thu19,
Proposition 3.8 and Lemma 3.10].

We also define the _$q$ -extremal-length_ of a homotopy class of maps
$\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muW\to G^{q}$ by

(4.6)
$\operatorname{EL}_{q}[\phi]\coloneqq\bigl{(}E^{1}_{q}[\phi]\bigr{)}^{q}.$

The terminology is justified: as mentioned in [Thu16, §5.2], the minimizer of
extremal length over a suitable homotopy class of maps exists and is realized
by a map with nice properties, and the minimum value, when formulated as an
extremal problem, mimics the usual definition of extremal length. See also
§5.5 below.

#### Case $1<p<\infty$ and $q=\infty$

The _$p$ -harmonic energy_ of a PL map $\phi\nobreak\mskip
2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\allowbreak\mskip 6.0mu plus
1.0muG^{p}\to K$ from a $p$-conformal graph to a length graph is