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# Fast Basins and Branched Fractal Manifolds of Attractors of Iterated
Function Systems

M. Barnsley and A. Vince

###### Abstract.

The fast basin of an attractor of an iterated function system (IFS) is the set
of points in the domain of the IFS whose orbits under the associated semigroup
intersect the attractor. Fast basins can have non-integer dimension and
comprise a class of deterministic fractal sets. The relationship between the
basin and the fast basin of a point-fibred attractor is analyzed. For example,
it is shown that there is a point in the interior of an attractor that is the
image, under the coding map, of a disjunctive point in code space if and only
if the fast basin contains the basin, that the projection of the usual code
space onto an attractor can be extended to a continuous map from an extended
code space onto the fast basin, and that the action of an IFS on any of its
fast basins is semi-conjugate to a symbolic IFS of partial shifts on an
extended code space.

To better understand the topology and geometry of fast basins, and because of
analogies with analytic continuation, branched fractal manifolds are
introduced. A branched fractal manifold is a metric space constructed from the
extended code space of a point-fibred attractor, by identifying some
addresses. Typically, a branched fractal manifold is a union of a
nondenumerable collection of nonhomeomorphic objects, isometric copies of
generalized fractal blowups of the attractor. A branched fractal manifold may
not be locally compact, but it always consists of a disjoint union of a
countable collection of precompact sets, each of which is homeomorphic to one
of N+1 subsets of the attractor, where N is the number of functions in the
IFS.

##  1\. Introduction

This paper on iterated function systems concerns related concepts: the fast
basin, fractal continuation, an extended code space, and the branched fractal
manifold.

The fast basin of an attractor of an iterated function system (IFS) is the set
of points, each of which possesses a chaos game orbit [2] that reaches the
attractor in finitely many steps. More precisely, let $A$ be an attractor of
an IFS, $\mathcal{F}$, here defined to be a finite set of homeomorphisms
$f_{i}:\mathbb{X}\rightarrow\mathbb{X},i=1,2,\dots,N,$ mapping a complete
metric space $\mathbb{X}$ onto itself. The fast basin of $A$ is the set of
points $x\in\mathbb{X}$ such that there exists a finite sequence of digits
$\\{\iota_{n}\\}_{n=1}^{k}\subset\\{1,2,...,N\\}$ with $f_{\iota_{1}}\circ
f_{\iota_{2}}\circ\cdots\circ f_{\iota_{k}}(x)\in A$. Equivalently, the fast
basin is the complement of the set of points, all of whose orbits do not
intersect the attractor. Some examples of fast basins, corresponding to
geometrically simple IFSs, are presented in Section 3, illustrating that they
can have interesting geometrical structure and that they are not easy to
understand.

Fast basins of affine IFSs may have non-integer Hausdorff dimension and thus
comprise a class of deterministic fractals which may be suitable for
geometrical modelling of real-world objects. But our interest here is in their
mathematical structure, which is relevant to fractal transformations [1],
tiling theory [6], and non-commutative geometry [9].

A branched fractal manifold is a certain metric space built from the code
space of a point-fibred attractor of an IFS. Its projection onto $\mathbb{X}$
is the fast basin of the attractor. The initial motivation for defining and
analyzing branched fractal manifolds was a desire to understand the
topological structures of fast basins, at least in the case of simple IFSs
consisting of a few affine maps.

Another motivation for interest in fast basins and branched fractal manifolds
is a new view of analytic continuation, introduced in [4]. According to this
view, fractal continuation provides an extension of the notion of analytic
continuation. As a simple example, consider the IFS defined by the pair of
Möbius transformations,

$\begin{pmatrix}-31+4i&8+22i\\\ 2+11i&2-4i\end{pmatrix}\text{ and
}\begin{pmatrix}-25-13i&-17+14i\\\ -11+7i&-4+13i\end{pmatrix}$

acting on the Riemann sphere. This IFS has a unique attractor, an arc of a
circle (of radius one half, centered at $3i/2$). The fast basin is the whole
circle, i.e. the analytic continuation of the arc. This illustrates a more
general situation. In [4] it is proved, under conditions that are not stated
here, that the graph of a real analytic function over an interval, say
$[0,1]$, is the attractor of an IFS whose maps are analytic on a neighborhood
of the graph, and generalized blowups of the graph, generated using the IFS,
coincide with the graph of any analytic continuation of the original function,
where both are defined. In [4] and here, we refer to generalized fractal
blowups of the graphs of fractal interpolation functions as (fractal)
continuations. In [4] it is further established that all except two fractal
continuations of the graphs of any affine fractal interpolation function
$g:[0,1]\rightarrow\mathbb{R}$ are defined for all real $x$. Although a
fractal continuation coincides with the real analytic continuation of $g$ when
$g$ is real analytic on a neighborhood of $[0,1]$, in general the graphs of
different continuations of an affine fractal interpolation function do not
coincide, except over $[0,1]$. Instead, their union, which is an example of a
fast basin, may be interpreted as an analog or generalization of analytic
continuation applied to a fractal function $g$. In this interpretation,
different continuations of $g$ belong to different branches, thus the need to
understand how these branches fit together and the motivation for the fast
basin and branched fractal manifold.

The organization of the paper is as follows. In Section 2 we define point-
fibred IFSs, their attractors, duals, basins, fast basins and fractal
continuations. A fractal continuation is a generalization of a fractal blowup
as defined by Strichartz [11]. Proposition 2.7 states that the fast basin of
an attractor is the union of its fractal continuations. Section 3 provides
examples of fast basins.

The relationship between the basin and the fast basin of an attractor is the
subject of Section 4. Theorem 4.2 shows that the image of a disjunctive point
in code space under the coding map (see below) belongs to the boundary of an
attractor if and only if the interior of the attractor is empty. Theorem 4.5
relates addresses of points in attractors to whether or not associated
continuations contain the basin, using the concept of full or reversible
addresses, defined here and in [6]. As a consequence, Corollary 4.6 states
that the fast basin contains the basin if and only if the attractor has
nonempty interior.

In Section 5, two symbolic IFSs $Z$ and $\widehat{Z}$ are introduced. The
first, $Z=\\{\mathbb{I}_{+};\sigma_{n},n=1,2,...,N\\},$ involves shift maps on
$\mathbb{I}_{+}=\\{1,2,...,N\\}^{\infty}$ and is well known, when
$\mathcal{F}$ is contractive, to be related to the attractor $A$ by the
following commutative diagram.

(1.1) $\begin{array}[c]{ccccc}&&\sigma_{n}&&\\\
&\mathbb{I}_{+}&\rightarrow&\mathbb{I}_{+}&\\\
\pi&\downarrow&&\downarrow&\pi\\\ &A&\rightarrow&A&\\\ &&f_{n}&&\end{array}$

This involves the coding map or addressing function, a continuous surjection
$\pi:\mathbb{I}_{+}\rightarrow A$. (Theorem 5.4 extends (1.1) to the case
$\mathcal{F}$ is point-fibred.) However, $Z$ is not an IFS, as defined in this
paper, as the maps of $Z$ are not invertible and its fast basin is undefined.
This is remedied by the introduction of
$\widehat{Z}=\\{\mathbb{I};\sigma_{n},n=1,2,...,N\\}$ whose maps are
invertible. Here $\mathbb{I}$ is a certain shift invariant subspace of
$\mathbb{I}_{0}=\\{-N,-(N-1),...,-1,1,2,...,N\\}^{\infty}$. The attractor of
$\widehat{Z}$ is $\mathbb{I}_{+}$ and its fast basin,
$\widehat{\mathbb{I}}\subset\mathbb{I}$, provides a symbolic representation
for any attractor of $\mathcal{F}$ and its fast basin. It is shown that the
coding map $\pi:\mathbb{I}_{+}\rightarrow A$ can be extended to various
subspaces of $\mathbb{I}$, including $\widehat{\mathbb{I}}$, and that the
action of shift maps on these subspaces is (semi-) conjugate to the action of
the $f_{n}$s (and/or their inverses) on corresponding subspaces of
$\mathbb{X}$, including the fast basin of $A$. Dual relationships also hold in
certain cases. These results are summarized by the commutative diagrams in
Theorem 5.6 and Corollaries 5.8 and 5.9.

The branched fractal manifold $\mathbb{L}$ associated with a point-fibred
attractor of an invertible IFS is defined in Section 6. It is a metric space
constructed from the symbolic fast basin $\widehat{\mathbb{I}}$ by identifying
some addresses. It is a disjoint union of certain leaves, each leaf being
homeomorphic to one of at most $N+1$ subsets of $A$, but $\mathbb{L}$ itself
may not be locally compact. Also, $\mathbb{L}$ contains isometric copies,
called sheets, of all the fractal continuations of the original attractor. The
branched fractal manifold contains, in a concise way, information on the
global addressing discussed in Section 5. A description of $\mathbb{L}$,
providing its main properties, is the content of Theorem 6.4 and Theorem 6.5.
Examples are provided.

Notions related to but distinct from fast basins and branched fractal
manifolds occur in the literature, mainly in the context of IFSs of
similitudes with the open set condition (OSC). In particular, some fast basins
are fractafolds as defined by Stritchartz [13, Section 5.4], and some are
examples of fractafold bundles as defined by Ionescu and Kumjian [9]. We
discuss these related notions in more detail in the following paragraphs.

In [11] Strichartz defines and establishes some properties of fractal blowups
(called fractal continuations here) for a similitude IFS that obeys the OSC.
Fractal blowups (continuations) for the classical Cantor set, the Sierpinski
triangle, and the Koch curve are investigated, and it is noted that various
different IFSs, with the Koch curve as the attractor, yield different blowups.
It is pointed out that in general a given IFS may yield a nondenumerable set
of non-homeomorphic blowups, where the relative topology is assumed.
Strichartz is focused on post critically finite (pcf) fractals because energy
forms and laplacians can be defined on their blowups. In [13, Section 5.4]
Strichartz defines a fractafold to be a countable union of pcf fractals, glued
together on their boundaries in such a way that every point in the fractafold
has a neighborhood that is conjugate to a neighborhood of a point on the
attractor; this notion subsumes fractal blowups. In his context, Strichartz
considers a fractafold to be a generalization of the notion of a manifold,
with the role of Euclidean space (or half-space) being played by the
attractor. In various works, see for example [12], he has shown how energy
forms and laplacians on p.c.f. fractals can be extended to fractafolds.

Some branched fractal manifolds are related to fractafold bundles, defined by
Ionescu and Kumjian [9]. They consider IFSs of the form of $\mathcal{F}$ with
$0<r_{i}\leq R_{i}<1$ and $r_{i}d(x,y)\leq d(f_{i}(x),f_{i}(y))\leq
R_{i}d(x,y)$ for all $x,y\in\mathbb{X}$ and all $i=1,2,...,N$, so that
$\mathcal{F}$ is invertible and has a unique attractor $A$. They define the
associated fractafold bundle $L$ to be $\\{\omega\times
B_{\omega}:\omega\in\mathbb{I}_{+}\\},$ but with a certain inductive limit
topology in place of the relative topology. They use the projection of $L$
onto $\mathbb{I}_{+}$ in order to build a Renault–Ionescu groupoid which
exploits the fact that if tails of two strings $\alpha,\beta\in\mathbb{I}_{+}$
are the same, and $\mathcal{F}$ obeys the OSC, then $B_{\alpha}$ and
$B_{\beta}$ are homeomorphic. This enables the establishment of properties of
an associated $C^{\ast}$ algebra on $L$. The point of view in [9] is very
different from that of the present work because it concerns the projection of
$L$ onto $\mathbb{I}_{+}$ rather than into $\mathbb{X}$.

Some notions and results presented here are anticipated in [3, 4, 6], but the
scope of the present work is much wider, most particularly with regard to the
definition and topology of fast basins and branched fractal manifolds.

##  2\.  IFSs Attractors, their Basins, Fast Basins, and Continuations

Let $\mathbb{N=\\{}1,2,3,...\\}$ and $\mathbb{N}_{0}=\\{0,1,2,...\\}$.
Throughout this paper an iterated function system (IFS) is a complete metric
space $\mathbb{X}$ together with a finite set of homeomorphisms
$f_{n}:\mathbb{X}\rightarrow\mathbb{X},n=1,2,\dots,N$, and is denoted by

$\mathcal{F}=\mathcal{F}_{\mathbb{X}}=\mathcal{\\{}\mathbb{X};f_{1},f_{2},...,f_{N}\\}.$

We use the same symbol $\mathcal{F}$ for the IFS and for the set of functions
$\\{f_{1},f_{2},...,f_{N}\\}$.

Let $H=H_{\mathbb{X}}$ be the collection of nonempty compact subsets of
$\mathbb{X}$ and define $F:H\rightarrow H$ by

$F(C)=\bigcup\limits_{f\in\mathcal{F}}f(C)$

for all $C\in H$. We also treat $F$ as a map $F:2^{\mathbb{X}}\rightarrow
2^{\mathbb{X}}$, where $2^{\mathbb{X}}$ is the collection of all subsets of
$\mathbb{X}$. For $S\subset\mathbb{X}$, define $F^{0}(S)=S$ and let $F^{k}(S)$
denote the $k$-fold composition of $F$ applied to $S$, namely, the union of
$\\{\iota_{n}\\}_{n=1}^{k}\subset\\{1,2,...,N\\}$ with $f_{\iota_{1}}\circ
f_{\iota_{2}}\circ\cdots\circ f_{\iota_{k}}(S)$ over all finite words
$\iota_{1}\iota_{2}\cdots\iota_{k}$ of length $k$.

Let $d=d_{\mathbb{X}}$ be the metric on $\mathbb{X}$, and let
$d_{H}=d_{H_{\mathbb{X}}}$ be the corresponding Hausdorff metric on
$H_{{\mathbb{X}}}$. Throughout, the topology on $H_{\mathbb{X}}$ is the one
induced by $d_{H_{\mathbb{X}}}$. It is well known that $(H,d_{H})$ is a
complete metric space because $(\mathbb{X},d)$ is complete, and that if
$(\mathbb{X},d)$ is compact then $(H,d_{H})$ is compact.

######  Definition 2.1.

An attractor of the IFS $\mathcal{F}$ is a set $A\in H_{\mathbb{X}}$ such that

  1. (1)

$F(A)=A$, and

  2. (2)

there is an open set $U\subset\mathbb{X}$ such that $A\subset U$ and
$\lim_{k\rightarrow\infty}F^{k}(S)=A,$ for all $S\in H_{\mathbb{X}}$ with
$S\subset U$, where the limit is w.r.t. the metric $d_{H}$.

The union of all open sets $U$, such that statement 2 of Definition 2.1 is
true, is called the basin of the attractor $A$ (w.r.t. $\mathcal{F}$). If
$B=B(A)$ denotes the basin of $A$, then it can be proved that statement 2 of
Definition 2.1 holds with $U$ replaced by $B$. That is, the basin of the
attractor $A$ is the largest open set $U$ such that statement 2 of Definition
2.1 holds. An IFS may not possess an attractor, or it may possess multiple
attractors. Examples and further discussion can be found in [3] and in
references therein.

The IFS $\mathcal{F}$ is said to be contractive if there is a
$\lambda\in[0,1)$ such that

$d_{\mathbb{X}}(f(x),f(y))\leq\lambda d_{\mathbb{X}}(x,y)$

for all $f\in\mathcal{F}$ and all $x,y\in\mathbb{X}$. A basic result of [8] is
that a contractive IFS possesses a unique attractor $A$, with
$B(A)=\mathbb{X}$.

######  Definition 2.2.

The dual IFS is

$\mathcal{F}^{\ast}:=\\{\mathbb{X};f_{n}^{-1},n=1,2,...,N\\}\text{.}$

If $A$ is an attractor of the IFS $\mathcal{F},$ then the set
$A^{\ast}:=\mathbb{X}\backslash B(A)$ is called the dual repeller of $A$
(w.r.t. $\mathcal{F}$). The dual repeller $A^{\ast}$, under certain conditions
[5], is an attractor of the dual IFS $\mathcal{F}^{\ast}$. This can occur, for
example, when $\mathcal{F}_{\widehat{\mathbb{C}}}$ is a Möbius IFS on the
Riemann sphere $\widehat{\mathbb{C}}$ [14].

If $f,g\in\mathcal{F},$ then $f\circ g$ denotes the composition of $f$ with
$g$. For an infinite word
$\iota=\iota_{1}\iota_{2}\iota_{3}\cdots,\;\iota_{i}\in\\{1,2,\dots,N\\}$, let

(2.1) $\displaystyle\iota|k$
$\displaystyle=\iota_{1}\iota_{2}\iota_{3}\cdots\iota_{k},$ $\displaystyle
f_{\iota|k}$
$\displaystyle=f_{\iota_{1}\iota_{2}\iota_{3}\cdots\iota_{k}}=f_{\iota_{1}}\circ
f_{\iota_{2}}\circ\cdots\circ f_{\iota_{k}},$

and $f_{\emptyset}=f_{\iota|0}={\text{id}}_{\mathbb{X}}$, the identity map on
$\mathbb{X}$.

######  Definition 2.3.

An attractor $A$ of $\mathcal{F}$ is point-fibred (w.r.t. $\mathcal{F}$) if

$\lim_{k\rightarrow\infty}f_{\iota|k}(C)\subset A,$

is a singleton subset of $\mathbb{X}$, for all
$\iota\in\\{1,2,\dots,N\\}^{\infty}$ independent of $C\subset B(A)$ with $C\in
H_{\mathbb{X}}$, where convergence is with respect to the Hausdorff metric.

All attractors in this paper are assumed to be point-fibred. It follows from
results in [8] that if the IFS $\mathcal{F}$ is contractive, then its
attractor is point-fibred. The point-fibred property allows for an addressing
scheme for points of the attractor (Definition 2.4). The following notation
will be used throughout this paper. Let $N\in\mathbb{N}$ and let

$\begin{aligned} I_{+}&=\\{+1,+2,\dots,+N\\},\\\
I_{-}&=\\{-1,-2,\dots,-N\\},\\\ I&=I_{+}\cup
I_{-},\end{aligned}\begin{aligned} \qquad\mathbb{I}_{+}&=I_{+}^{\infty},\\\
\mathbb{I}_{-}&=I_{-}^{\infty},\\\ \mathbb{I}_{0}&=I^{\infty}.\end{aligned}$
(2.2) $\mathbb{I=\\{\iota\in I}_{0}:\iota_{i}\neq-\iota_{i+1}\text{ for all
}i\in\mathbb{N\\}}.$

Let $d_{\mathbb{I}}$ be the metric on $\mathbb{I}_{0}$ defined by

$d_{\mathbb{I}}(\iota,\omega)=\begin{cases}\max\\{2^{-k}:k\in\mathbb{N}_{0},\iota_{k}\neq\omega_{k}\\}\;\;\,\text{
if }\iota\neq\omega,\\\ 0\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{if
}\iota=\omega,\end{cases}$

for all $\iota,\omega\in\mathbb{I}_{0}$. The metric $d_{\mathbb{I}}$ induces
the product topology on $\mathbb{I}_{0}$, and the metric spaces
$(\mathbb{I},d_{\mathbb{I}})$,
$(\mathbb{I}_{+},d_{\mathbb{I}}),(\mathbb{I}_{-},d_{\mathbb{I}})$ are compact
subspaces of the compact metric space $(\mathbb{I}_{0},d_{\mathbb{I}}).$

The space $(\mathbb{I}_{+},d_{\mathbb{I}})$ is sometimes called the code space
or address space for an attractor of $\mathcal{F}$ because it provides
addresses for the points of $A$, as given in the following definition.

######  Definition 2.4.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$. According to
Definition 2.3, there is a map $\pi:\mathbb{I}_{+}\rightarrow
A\subset\mathbb{X}$, called the coding map, that is well defined by

(2.3) $\pi(\iota)=\lim_{k\rightarrow\infty}f_{\iota|k}(b),$

for all $\iota\in\mathbb{I}_{+}$, and the limit is independent of $b\in B(A)$.
It is readily proved that $\pi$ is continuous and that
$\pi(\mathbb{I}_{+})=A.$ The set-valued inverse $\pi^{-1}(a)$ comprises the
set of addresses of the point $a\in A$, (w.r.t. $\mathcal{F}$).

######  Definition 2.5.

The fast basin $\widehat{B}=\widehat{B}(A)$ of an attractor $A$ of an IFS
$\mathcal{F}$ is

$\widehat{B}=\\{x\in\mathbb{X}\,:\,F^{K}(\\{x\\})\cap
A\neq\emptyset\;\;\text{for some}\;\;K\in\mathbb{N}\\}.$

The relationship between the basin $B$ and the fast basin $\widehat{B}$ is
complicated because neither $B\subset$ $\widehat{B}$ nor $\widehat{B}\supset
B$ holds in general. This topic is discussed in Section 4.

Fractal continuations were introduced in [4] in the context of fractal
interpolation, and in [3] in the context of fractal tiling. (See also the last
part of Section 1 for a discussion of related notions.) These situations are
generalized in Definition 2.6. Extend the notation of Equation (2.1) as
follows. For $\iota\in\mathbb{I}_{0}$ define $-\iota$ by

$(-\iota)_{k}:=-\iota_{k}$

for all $k\geq 1$. Define

(2.4) $\displaystyle\iota|k$
$\displaystyle=\iota_{1}\iota_{2}\iota_{3}\cdots\iota_{k}$ $\displaystyle
f_{-j}$ $\displaystyle=(f_{j})^{-1}$ $\displaystyle f_{\iota|k}$
$\displaystyle=f_{\iota_{1}}\circ f_{\iota_{2}}\circ\cdots\circ
f_{\iota_{k}},$

for all $\iota\in{\mathbb{I}}_{0},\;j=1,2,\dots,N,$ where it is understood
that $\iota|0=\emptyset$, and $f_{\emptyset}$ is the identity map on
$\mathbb{X}$.

######  Definition 2.6.

Let $A$ be an attractor of an IFS $\mathcal{F}=\\{\mathbb{X};f_{n},n\in
I_{+}\\}$. For $\theta\in\mathbb{I}_{+}$, the continuation ${B}_{\theta}$ of
$A$ (w.r.t. $\mathcal{F}$) is defined to be

$B_{\theta}:=\bigcup\limits_{k\in\mathbb{N}}f_{-\theta|k}(A)=\bigcup\limits_{k\in\mathbb{N}_{0}}f_{\theta_{1}}^{-1}\circ
f_{\theta_{2}}^{-1}\circ\dots\circ f_{\theta_{k}}^{-1}(A)\text{.}$

Note that this is a nested union in the sense that $f_{-\theta|k}(A)\subset
f_{-\theta|k+1}(A)$.

The set ${B}_{\theta}$ is referred to as a continuation of $A$ or a fractal
continuation of $A$. The family $\\{{B}_{\theta}:\theta\in\mathbb{I}_{+}\\}$
is referred to as the set of continuations of $A$. We also write
${B}_{\theta|k}=f_{-\theta|k}(A)$, which is referred to as a finite
continuation of $A$. Clearly

${B}_{\theta}=\bigcup_{k\in\mathbb{N}_{0}}{B}_{\theta|k}.$

The paper [4] concerns the special case of fractal continuations of fractal
functions. In that case it is proved, under special conditions, the most
important of which is that the IFS consists of analytic functions, that the
set of continuations is uniquely determined by the attractor, independent of
the analytic IFS used to generate the attractor.

The following result, stating that the fast basin is the union of all the
continuations, is readily verified.

######  Proposition 2.7.

Let $A$ be an attractor of an IFS $\mathcal{F}=\\{\mathbb{X};f_{n},n\in
I_{+}\\}$. If $\widehat{B}$ is the fast basin of $A$ and
$\\{{B}_{\theta}:\theta\in\mathbb{I}_{+}\\}$ is the set of continuations of
$A$, then

$\widehat{B}=\bigcup\limits_{\theta\in{\mathbb{I}}_{+}}{B}_{\theta}\text{.}$

##  3\.  Fast Basin Examples

######  Example 3.1.

(Cantor Set) The fast basin of the standard Cantor set
$\mathcal{C\subset}\mathbb{R}$, w.r.t. the IFS
$\left\\{\mathbb{R};x/3,x/3+2/3\right\\},$ is

$\widehat{B}=\bigcup_{k\in\mathbb{Z}}(\mathcal{C+}2k)\text{.}$

We use the notation for Minkowski sum,
$(\mathcal{C+}2k):=\\{x+2k:x\in\mathcal{C}\\}$.

######  Example 3.2.

(Sierpinski Gasket) The fast basin of a Sierpinski triangle
$\mathcal{\triangle\subset}\mathbb{R}^{2}$, with vertices at
$a,b,c\,\mathbb{\in R}^{2},$ w.r.t. the IFS
$\left\\{\mathbb{R}^{2};(x+a)/2,(x+b)/2,(x+c)/2\right\\}$, is
$\widehat{B}=\mathcal{\bigcup}_{t\in\mathcal{G}}(\mathcal{\triangle+}t)$ where
$\mathcal{G}$ is the group generated by the set of translations by
$\\{a-b,b-c,c-a\\}$. Figure 1 shows $\widehat{B}$ when $a=(0,0),$ $b=(1,0)$,
and $c=(0,1)$.

Figure 1. This illustrates part of the fast basin of the attractor of the
iterated function system
$\mathcal{F=\\{}\mathbb{R}^{2};(x/2,y/2),((x+1)/2,y/2),(x/2,(y+1)/2)\\}.$ The
attractor is rendered in red at the lower left.

######  Example 3.3.

(Other Affine IFSs) Figure 2 illustrates part of the fast-basin of a
contractive affine IFS $({\mathbb{R}}^{2};\,f_{1},f_{2})$, where

$\displaystyle f_{1}(x,y)$
$\displaystyle=\left(\frac{x}{2}+\frac{y}{2\sqrt{3}}-1,\,\frac{x}{2\sqrt{3}}-\frac{y}{2})\right),$
$\displaystyle f_{2}(x,y)$
$\displaystyle=\left(\frac{x}{2}-\frac{y}{2\sqrt{3}}\
+1,\,-\frac{x}{2\sqrt{3}}-\frac{y}{2})\right).$

The attractor is the segment of the Koch snowflake curve near the center of
the figure. The fast basin is the union of the boundaries of the tiles of a
tiling of the plane by Koch snowflakes and other related tiles.

Figure 2. See Example 3.3. This shows the fast basin for the Koch curve using
an IFS of similitudes. The Koch curve is located near the center, in various
colours.

Figure 3 illustrates part of the fast basin for the contractive affine IFS
$({\mathbb{R}}^{2};\,f_{1},f_{2})$, where

$\displaystyle f_{1}(x,y)$
$\displaystyle=\left(\frac{x}{2}+\frac{1}{2},\,\frac{x}{2}+\frac{2y}{5}+\frac{1}{4}\right),$
$\displaystyle f_{2}(x,y)$
$\displaystyle=\left(\frac{x}{2}-\frac{1}{2},\,-\frac{x}{2}+\frac{2y}{5}+\frac{1}{4}\right),$

whose attractor, at the center of the image, is the graph of a fractal
interpolation function. At each branch point, there is a countable infinity of
distinct branches.

Figure 3. See Example 3.3.

######  Example 3.4.

(Topological Dimension of the Fast Basin) We use the following definition: an
orbit of a point  $x\in\mathbb{X}$, under the IFS
$\mathcal{F}{=}\\{\mathbb{X};\,f_{1},..,f_{N}\\}$, is
$\left\\{f_{\iota_{k}}\circ f_{\iota_{k-1}}\circ
f_{\iota_{1}}(x)\right\\}_{k=0}^{\infty}\subset\mathbb{X}$ for some
$\iota\in\left\\{1,2,...,N\right\\}^{\infty}$. If $\mathcal{F}$ is an IFS on
${\mathbb{R}}^{M}$ consisting of affine functions and whose attractor $A$ is
contained in a proper linear subspace $V$ of $\mathbb{R}^{M}$, then it is easy
to show that the fast basin of $A$ is also contained in $V$. Moreover, any
orbit of any point not in $V$ has empty intersection with $A$. It follows that
the topological dimension of the fast basin of $A$ is strictly less than that
of the underlying space. For instance, if $M=N=2,$ and

$\text{ }f_{1}(x,y)=(x/2,y/2),\qquad\qquad f_{2}(x,y)=((x+1)/2,y/2)$

for all $(x,y)\in\mathbb{R}^{2},$ then $A=[0,1]\subset\mathbb{R}$,
$B=\mathbb{R}^{2},$ and $\widehat{B}\subset\mathbb{R}$. Any orbit of any point
in $\mathbb{R}^{2}\setminus\\{(x,0)\,:\,x\in{\mathbb{R}}\\}$ does not meet the
attractor (although the closure of the orbit does meet the attractor) while,
for any point in $\mathbb{R}$, there exists an orbit that reaches $A$ in
finitely many steps, i.e. the orbit intersects the attractor.

######  Example 3.5.

An example, illustrating a relationship between a fast basin and analytic
continuation, is provided by the contractive IFS

$\mathcal{F=}\\{\mathbb{C\times C};f_{+1+i},f_{-1+i},f_{+1-i},f_{-1-i}\\},$

where

$\displaystyle f_{\pm 1-i}\begin{pmatrix}z\\\ w\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}&0\\\ \frac{\pm
1-i}{2}&\frac{1}{4}\end{pmatrix}\begin{pmatrix}z\\\
w\end{pmatrix}+\begin{pmatrix}\frac{\pm 1-i}{2}\\\ \frac{\mp
i}{2}\end{pmatrix};$ $\displaystyle f_{\pm 1+i}\begin{pmatrix}z\\\
w\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}&0\\\ \frac{\pm
1+i}{2}&\frac{1}{4}\end{pmatrix}\begin{pmatrix}z\\\
w\end{pmatrix}+\begin{pmatrix}\frac{\pm 1+i}{2}\\\ \frac{\pm
i}{2}\end{pmatrix}.$

The unique attractor $A$ of $\mathcal{F}$ is the graph of $z\mapsto z^{2}$
over the square $-1\leq\operatorname{Re}z,\operatorname{Im}z\leq+1$. The fast
basin $\widehat{B}$ of $A$ (w.r.t. $\mathcal{F}$) is the manifold

$\left\\{(z,z^{2})\,:\,z\in\mathbb{C}\right\\}.$

######  Example 3.6.

An example that is related to a Schottky group is provided by the loxodromic
Möbius IFS

$\mathcal{F}=\left\\{\widehat{\mathbb{C}};\,f_{1}(z)=\frac{az-i(2+\sqrt{3})}{z-i(2+\sqrt{3})+a-1},\,f_{2}(z)=\frac{az+i(2+\sqrt{3})}{-z-i(2+\sqrt{3})+a-1}\right\\}$

where $a=\frac{-i(2+\sqrt{3})C+1}{1-C}$, $C\in\mathbb{C}$ with
$\left|C\right|\notin\\{0,1\\},$ $f_{1}(1)=1$ and $f_{2}(-1)=-1$. We find

$f_{1}^{\prime}(1)=f_{2}^{{}^{\prime}}(-1)=C\text{,}$

so if $\left|C\right|\ll 1$, then $\mathcal{F}$ has a totally disconnected
attractor $A$, located within two small circles, one centered at $+1$ and one
centered at $-1$. In this case the dual repeller $A^{\ast}$ is also totally
disconnected and is located within two small circles, one centered at
$(2+\sqrt{3})i$ and the other at $-(2+\sqrt{3})i$. The maps $f_{1},f_{2}$ are
the generators of a Schottky group whose limit set $\Lambda$ is also totally
disconnected, and we have $A\subset\Lambda,$ $A^{\ast}\subset\Lambda,$ and
$\widehat{B}\subset\Lambda$. It follows that $\widehat{B}$ is totally
disconnected.

##  4\. When Does the Fast Basin Contain the Basin?

Consider the following examples showing that, in general, there is no
containment relationship between the basin and the fast basin. Since the IFS

$\left\\{{\mathbb{R}};\,\frac{1}{2}\,x,\;\frac{1}{2}\;x+\frac{1}{2}\right\\}$

is contractive, the basin $B={\mathbb{R}}$. It is not hard to show, for
example via Proposition 2.7, that the fast basin $\widehat{B}={\mathbb{R}}$.
Therefore, $\widehat{B}=B$ for this IFS. Likewise, the IFS

$\left\\{\mathbb{R};\;\frac{1}{3}\,x,\;\frac{1}{3}\,x+\frac{2}{3}\right\\}$

is contractive, but, as in Example (3.1), the fast basin is not all of
${\mathbb{R}}$. Therefore $\widehat{B}\subsetneq B$ for this IFS. Finally
consider the IFS $\mathcal{F}=\\{\mathbb{P}^{1};\,f_{1},f_{2}\\}$, where
$\mathbb{P}^{1}$ is $1$-dimensional projective space, homeomorphic to the
circle, and which we denote by ${\mathbb{R}}\cup\\{\infty\\}$ (the reals with
an additional point at infinity). The functions are

$f_{1}(x)=\frac{9x}{20-2x},\qquad\qquad f_{2}(x)=\frac{11x+9}{2x+18}.$

The attractor $A$ of $\mathcal{F}$ is the interval $[0,1]$; the attractor
$A^{*}$ of the dual IFS is ${\mathbb{R}}\setminus(-9/2,11/2)$. Therefore the
basin of $A$ is $B={\mathbb{R}}\setminus A^{*}=(-9/2,11/2)$. It is
straightforward to check that the fast basin is $\widehat{B}={\mathbb{R}}$.
Therefore, in this example, $B\subsetneq\widehat{B}$.

From the above examples we see that all three possibilities
$\widehat{B}=B,\,\widehat{B}\subsetneq B,\,B\subsetneq\widehat{B}$ can occur.
However, if the attractor $A$ has nonempty interior (in the topology of
$\mathbb{X}$), then $B\subseteq\widehat{B}$ (see Theorem 4.6).

Define the shift map $S:\mathbb{I}_{0}\rightarrow\mathbb{I}_{0}$ by

$S(\iota_{1}\iota_{2}\dots)=\iota_{2}\iota_{3}\dots$

for all $\iota\in\mathbb{I}_{0}$. Note that $S$ is continuous and maps
$\mathbb{I}$ onto $\mathbb{I}$, $\mathbb{I}_{+}$ onto $\mathbb{I}_{+}$, and
$\mathbb{I}_{-}$ onto $\mathbb{I}_{-}$.

######  Definition 4.1.

A word $\theta\in\mathbb{I}_{+}$ is disjunctive if every finite word is a
subword of $\theta$. In fact, if $\theta$ is disjunctive, then every finite
word (in the alphabet $I_{+}$) appears as a subword in $\iota$ infinitely many
times. Moreover, an equivalent definition of a disjunctive word is that its
orbit under the shift map is dense in the code space.

######  Theorem 4.2.

Let $A$ be an attractor of an IFS $\mathcal{F}$, and let
$\theta\in\mathbb{I}_{+}$ be disjunctive. Then $\pi(\theta)\in A^{\circ}$ if
and only if $A^{\circ}\neq\emptyset$.

###### Proof.

Since $\theta\in\mathbb{I}_{+}$ is disjunctive, the orbit,
$\left\\{S^{k}(\theta)\right\\}_{k=0}^{\infty}$, of $\theta$ under the shift
map $S:$ $\mathbb{I}_{+}\mathbb{\rightarrow I}_{+}$ is dense in
$\mathbb{I}_{+}$. Since $\pi$ is continuous,
$\left\\{\pi_{\mathcal{F}}(S^{k}(\theta))\right\\}_{k=0}^{\infty}$ is dense in
$A$. By way of contradiction assume that $\pi_{\mathcal{F}}(\theta)\in\partial
A$, the boundary of $A$. Since each $f\in\mathcal{F}$ is a homeomorphism,
$f(A^{o})\subset A^{o}$ or, equivalently, $\left(f|_{A}\right)^{-1}(\partial
A)\subset\partial A$. Therefore $\pi_{\mathcal{F}}(S^{k}(\theta))\in\partial
A$ for all $k\in\mathbb{N}_{0}$, which would mean that
$A=\overline{\left\\{\pi_{\mathcal{F}}(S^{k}(\theta))\right\\}_{k=0}^{\infty}}\subset\overline{\partial
A}=\partial A$ which implies $A^{\circ}=\emptyset,$ which contradicts
$A^{o}\neq\emptyset$. ∎

Theorem 4.5 below provides sufficient conditions for a fractal continuation of
an attractor to contain the basin of the attractor. It involves the equivalent
notions of a full and a reversible word in $\mathbb{I}_{+}$. These concepts
were introduced in [6]; the equivalence of full and reversible is part of
Theorem 4.4.

######  Definition 4.3.

If $A$ is an attractor of an IFS $\mathcal{F}$, call
$\theta\in{\mathbb{I}}_{+}$ full if there exists a nonempty compact set
$A^{\prime}\subset A^{o}$ such that, for any positive integer $M$, there exist
$n>m\geq M$ such that

$f_{\theta_{n}}\circ f_{\theta_{n-1}}\circ\cdots\circ
f_{\theta_{m+1}}(A)\subset A^{\prime}.$

Call $\theta\in\mathbb{I}_{+}$ reversible w.r.t. an attractor $A$ and IFS
$\mathcal{F}$ if there exists an
$\omega=\omega_{1}\omega_{2}\cdots\in\mathbb{I}_{+}$ such that $\omega$ is the
address of some point in $A^{o}$ and, for every pair of positive integers $M$
and $L$, there is an integer $m\geq M$ such that

$\omega_{1}\omega_{2}\cdots\omega_{L}=\theta_{m+L}\theta_{m+L-1}\cdots\theta_{m+1}.$

In some cases it is easy to check if a word is reversible. For example, if
$\theta=\theta_{1}\theta_{2}\cdots\theta_{k}\theta_{1}\theta_{2}\cdots\theta_{k}\theta_{1}\cdots$
is periodic and
$\pi(\theta_{k}\theta_{k-1}\cdots\theta_{1}\theta_{k}\theta_{k-1}\cdots\theta_{1}\theta_{k}\cdots)$
lies in the interior of $A$, then $\theta$ is full/reversible.

######  Theorem 4.4.

([6]) For an IFS $\mathcal{F}$, let $A$ be an attractor. With respect to $A$
and $\mathcal{F}$:

  1. (1)

There are infinitely many disjunctive words in $\mathbb{I}_{+}$ for $N\geq 2$.

  2. (2)

If $A^{o}\neq\emptyset$, then every disjunctive word is reversible.

  3. (3)

A word is reversible if and only if it is full.

######  Theorem 4.5.

Let $A$ be an attractor of an IFS $\mathcal{F}$ with basin $B$. If
$\theta\in\mathbb{I}_{+}$ is full/reversible, then

$B\subset{B}_{\theta}.$

###### Proof.

Let $x\in B$. It sufices to show that $x\in
f_{\theta_{1}}^{-1}f_{\theta_{2}}^{-1}f_{\theta_{3}}^{-1}\cdots
f_{\theta_{n}}^{-1}(A)$ for some $n$, or equivalently

$f_{\theta_{n}}f_{\theta_{n-1}}\cdots f_{\theta_{1}}(x)\in A.$

By the definition of attractor, for any $\epsilon>0$ there is an
$M_{\epsilon}$ such that if $m\geq M_{\epsilon}$, then

$F^{m}(x)\subset A_{\epsilon},$

where $A_{\epsilon}$ is the open $\epsilon$-neighborhood of $A$. Because
$\theta$ is assumed to be full, there exists a compact $A^{\prime}$ with
$A^{\prime}\subset A^{o}$ with the property that, for any $M$ there exist
$n>m\geq M$ such that

$f_{\theta_{n}}\circ\cdots\circ f_{\theta_{m+1}}(A^{\prime})\subset A^{o}.$

This implies that there exists an $\epsilon_{0}$-neighborhood
$A_{\epsilon_{0}}$ of $A$ such that

$f_{\theta_{n}}\circ\cdots\circ f_{\theta_{m+1}}(A_{\epsilon_{0}})\subset
A^{o},$

for some $\epsilon_{0}>0$. If $M\geq M_{\epsilon_{0}}$, then

$f_{\theta_{n}}\circ\cdots\circ f_{\theta_{m+1}}\circ
f_{\theta_{m}}\circ\cdots\circ f_{\theta_{1}}(x)\in
f_{\theta_{n}}\circ\cdots\circ f_{\theta_{m+1}}(A_{\epsilon_{0}})\subset A,$

as required. ∎

######  Corollary 4.6.

Let $A$ be an attractor of an IFS $\mathcal{F}$ on a complete metric space
$\mathbb{X}$, with basin $B$ and fast basin $\widehat{B}$. Then
$B\subseteq\widehat{B}$ if and only if $A^{o}\neq\emptyset$. Moreover, if
$\mathcal{F}$ is contractive and $A^{o}\neq\emptyset$, then
$B=\widehat{B}=\mathbb{X}$.

###### Proof.

Assume that If $A^{o}\neq\emptyset$. Let $\theta$ be a disjunctive sequence;
by Theorem 4.4 there are infinitely many. By the same theorem $\theta$ is
full/reversible. By Theorem 4.5 and Proposition 2.7 we have $B\subseteq
B_{\theta}\subseteq\widehat{B}$. If $\mathcal{F}$ is contractive, then, as
mentioned in the introduction, it is well known that $B=\mathbb{X}$.

Conversely, assume that $A^{o}=\emptyset$. If $A$ is an attractor, the fast
basin of $A$ is

$\widehat{B}=\\{(f_{\iota})^{-1}(A)\,:\,\iota\in{\mathbb{I}}_{+}^{*}\\},$

where ${\mathbb{I}}_{+}^{*}$ is the set of all finite sequences in the
alphabet $I_{+}$. Since $(f_{\iota})^{-1}(A)$ has empty interior,
$\widehat{B}$ is nowhere dense by the Baire Category Theorem. Since $B$ is
open, clearly $B\nsubseteq\widehat{B}$. ∎

##  5\.  Symbolic IFSs and The Extended Coding Map

The goal of this section is to extend the classic coding map given in
Definition 2.4. We begin with two examples of “symbolic IFSs”. For $n\in I$,
define the inverse shift maps
$\sigma_{n}:\mathbb{I}_{0}\rightarrow\mathbb{I}_{0},$ by

(5.1) $\sigma_{n}(\iota_{1}\iota_{2}\cdots)=n\iota_{1}\iota_{2}\cdots,$

and define $\sigma_{n}:\mathbb{I}\rightarrow\mathbb{I}$ by

(5.2)
$\sigma_{n}(\iota_{1}\iota_{2}\cdots)=\left\\{\begin{array}[]{c}n\iota_{1}\iota_{2}\cdots\\\
\iota_{2}\iota_{3}\cdots\end{array}\right.\begin{array}[]{c}\text{if
}\iota_{1}\neq-n,\\\ \text{if }\iota_{1}=-n.\end{array}$

######  Example 5.1 (First Symbolic IFS).

If

$Z=\\{\mathbb{I}_{+};\,s_{n},\;n\in I_{+}\\},$

then $Z$ is an IFS each of whose functions is a contraction with scaling
factor $1/2$. The unique attractor is $\mathbb{I}_{+}$.

######  Example 5.2 (Second Symbolic IFS).

If

$\widehat{Z}=\\{\mathbb{I};\;\sigma_{n},n\in I_{+}\\},$

then $\widehat{Z}$ is an IFS, for which it is easily verified that each
function $\sigma_{n}\,:\,\mathbb{I}\rightarrow\mathbb{I}$ is bi-Lipshitz with

$\frac{1}{2}\,d_{\mathbb{I}}(\iota,\omega)\leq
d_{\mathbb{I}}\left(\sigma_{n}(\iota),\sigma_{n}(\omega)\right)\leq
2\,d_{\mathbb{I}}(\iota,\omega)$

for all $\iota,\omega\in\mathbb{I}$. In particular, each $\sigma_{n}$ is a
homeomorphism with inverse $\sigma_{-n}:\mathbb{I\rightarrow}\mathbb{I}$. The
dual IFS is

$\widehat{Z}^{*}=\\{\mathbb{I};\,s_{n},n\in I_{-}\\}.$

Although neither $\widehat{Z}$ nor $\widehat{Z}^{*}$ is contractive, both have
attractors as proved in Theorem 5.3 below.

######  Theorem 5.3.

  1. (1)

The IFS $\widehat{Z}$ has a point-fibred attractor $\mathbb{I}_{+}$ with basin
$\mathbb{I}\backslash\mathbb{I}_{-}$ and fast basin

(5.3) $\widehat{\mathbb{I}}:=\left\\{\iota\in\mathbb{I}\,:\,\text{there
exists}\;k\in\mathbb{N}_{0}\,\text{such
that}\;\iota_{j}\in\begin{cases}I_{-}\;\;&\text{if}\;\;j\leq k\\\
I_{+}\;\;&\text{if}\;\;j>k\end{cases}\right\\}.$

  2. (2)

The dual IFS $\widehat{Z}^{*}$ has point-fibred attractor $\mathbb{I}_{-}$
with basin $\mathbb{I}\backslash\mathbb{I}_{+}$ and fast basin

${\widehat{\mathbb{I}}}^{*}:=\left\\{\iota\in\mathbb{I}\,:\,\text{there
exists}\;k\in\mathbb{N}_{0}\,\text{such
that}\;\iota_{j}\in\begin{cases}I_{+}\;\;&\text{if}\;\;j\leq k\\\
I_{-}\;\;&\text{if}\;\;j>k\end{cases}\right\\}.$

###### Proof.

We will show that the IFS $\widehat{Z}$ has point-fibred attractor
$\mathbb{I}_{+}$ with basin $\mathbb{I}\backslash\mathbb{I}_{-}$ and dual
repeller $\mathbb{I}_{-}$. The rest of the proof follows immediately, and is
omitted. The compact nonempty set $\mathbb{I}_{+}$ is contained in
$\mathbb{I}$, and $\mathbb{I}_{+}=\widehat{Z}(\mathbb{I}_{+}):=\cup_{n\in
I_{+}}\sigma_{n}(\mathbb{I}_{+})$. An open neighborhood of $\mathbb{I}_{+}$ is
$\mathbb{I}\backslash\mathbb{I}_{-}$. If
$\iota\in\mathbb{I}\backslash\mathbb{I}_{-}$, then there is
$K=K(\iota)\in\mathbb{N}$ such that $\iota_{K}\in I_{+},$ from which it
follows that $\alpha_{1}\in I_{+}$ for all $\alpha\in$
$\widehat{Z}^{K-1}(\\{\iota\\})$. In turn, this implies

$d_{H}(\widehat{Z}^{K+j}(\\{\iota\\}),\mathbb{I}_{+})\leq 2^{-j-1}\text{,}$

for all $j\in\mathbb{N}$. This proves that
$\left\\{\widehat{Z}^{k}(\\{\iota\\})\right\\}_{k\in\mathbb{N}}$ converges to
$\mathbb{I}_{+}$ for all $\iota\in$ $\mathbb{I}\backslash\mathbb{I}_{-}$.

Now suppose that $C\in H(\mathbb{I})$ and
$C\subset\mathbb{I}\backslash\mathbb{I}_{-}$. Let $\iota\in C$, and let
$K=K(\iota)$ be as above. Then for all $\omega\in\mathcal{B}(\iota,2^{-K-1})$
we have $\omega_{K}\in I_{+}$ and hence

$d_{H}(\widehat{Z}^{K(\iota)+j}(\\{\omega\\}),\mathbb{I}_{+})\leq
2^{-j-1}\text{,}$

which shows that
$\left\\{\widehat{Z}^{k}(\overline{\mathcal{B}(\iota,2^{-K(\iota)-1})})\right\\}_{k\in\mathbb{N}}$
converges to $\mathbb{I}_{+}$. Using its compactness, $C$ can be covered by a
finite set of such balls, and we conclude that
$\left\\{\widehat{Z}^{k}(C)\right\\}_{k\in\mathbb{N}}$ converges to
$\mathbb{I}_{+}$. On the other hand,
$\mathbb{I}_{-}\subset\widehat{Z}(\mathbb{I}_{-})$ so
$\mathbb{I}_{-}\subset\widehat{Z}^{k}(\mathbb{I}_{-})$ for all
$k\in\mathbb{N}$, so
$\left\\{\widehat{Z}^{k}(\mathbb{I}_{-})\right\\}_{k\in\mathbb{N}}$ does not
converge to $\mathbb{I}_{+}$, so we conclude that the basin of the attractor
$\mathbb{I}_{+}$ (w.r.t. $\widehat{Z}$) is
$\mathbb{I}\backslash\mathbb{I}_{-}$. Hence, by definition, the dual repeller
is $\mathbb{I}_{-}$. ∎

Proposition 5.4 below is a generalization, along the lines of Kieninger [10,
Section 4.2] and Hata [7, Theorem 3.2], to point-fibred attractors, of ideas
in Hutchinson’s work [8, 2.1.(8), p.716; 3.1.(3), p.724]. It asserts, in
particular, a semiconjugacy between $\mathcal{F}$ acting on $A$ and the
contractive IFS $Z=\\{\mathbb{I}_{+};\sigma_{n},n\in I_{+}\\}$ of Example 5.1
acting on its attractor, $\mathbb{I}_{+}$.

######  Theorem 5.4.

Let $A$ be a point-fibred attractor of the IFS
$\mathcal{F}=\\{\mathbb{X};f_{n},n\in I_{+}\\}$. The map
$\pi:\mathbb{I}_{+}\rightarrow A$ from the attractor of $Z$ to the attractor
of $\mathcal{F}$ given in Equation (2.3) is uniformly continuous, and the
diagram

$\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\mathbb{I}_{+}&\rightarrow&\mathbb{I}_{+}&\\\
\pi&\downarrow&&\downarrow&\pi\\\ &A&\rightarrow&A&\\\ &&f_{n}&&\end{array}$

is commutes for all $n\in I_{+}$.

Theorem 5.4 describes the behaviour of the functions of the IFS on points of
the attractor, not outside the attractor. The results below describe the
behaviour of the functions of the IFS on the fast basin of the attractor and,
in the case that the IFS is contractive with attractor with nonempty interior,
on all of the space $\mathbb{X}$. These new results, Theorem 5.6 and its
corollaries, together with Theorem 5.4, provide a more complete symbolic
description than the one provided by Theorem 5.4 alone, of the structure and
dynamics of point-fibred attractors of IFSs.

Extend the notation of Equation (2.2) as follows

$\displaystyle\mathbb{J}_{+}$ $\displaystyle\mathbb{=}\mathbb{\\{\iota\in
I}:\exists K\in\mathbb{N}_{0}\ni S^{K}(\mathbb{\iota)}\text{
}\in\mathbb{I}_{+}\mathbb{\\}}\text{, }$ $\displaystyle\mathbb{J}_{-}$
$\displaystyle\mathbb{=}\mathbb{\\{\iota\in I}:\exists K\in\mathbb{N}_{0}\ni
S^{K}(\mathbb{\iota)}\text{ }\in\mathbb{I}_{-}\mathbb{\\}},$

and note that

$\displaystyle\mathbb{I}_{+}$
$\displaystyle\subset\widehat{\mathbb{I}}\subset\mathbb{J}_{+}\subset\overline{\mathbb{J}_{+}}=\mathbb{I\subset
I}_{0},\qquad\text{and }$ $\displaystyle\mathbb{I}_{-}$
$\displaystyle\subset{\widehat{\mathbb{I}}}^{*}\subset\mathbb{J}_{-}\subset\overline{\mathbb{J}_{-}}=\mathbb{I\subset
I}_{0}.$

In particular, the spaces $\mathbb{J}_{\pm}$ are dense in $\mathbb{I}$.
Moreover, $\sigma_{n}(\mathbb{J}_{+})=\mathbb{J}_{+}$ and
$\sigma_{n}(\mathbb{J}_{-})=\mathbb{J}_{-}$ for all $n\in I$. In addition,
$\sigma_{n}|_{\mathbb{J}_{+}}:$ $\mathbb{J}_{+}\rightarrow\mathbb{J}_{+}$ and
$\sigma_{n}|_{\mathbb{J}_{-}}:$ $\mathbb{J}_{-}\rightarrow\mathbb{J}_{-}$ are
homeomorphisms.

######  Definition 5.5.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$. The extended
coding map $\pi\,:\,\mathbb{J}_{+}\rightarrow\mathbb{X}$ is given by

(5.4) $\pi(\iota)=\lim_{k\rightarrow\infty}f_{\iota|k}(b),$

for all $\iota\in\mathbb{J}_{+}$, where $b\in B(A)$. The limit in (5.4) exists
and is independent of $b$ because, by the definition of $\mathbb{J}_{+}$,
there is $K\in\mathbb{N}_{0}$ such that $S^{K}(\iota)\in\mathbb{I}_{+},$ which
means that we can write

$\pi(\iota)=f_{\iota|K}(\pi(S^{K}(\iota))).$

Clearly, the extended coding map agrees with the standard coding map
(Definition 2.4) on ${\mathbb{I}}_{+}$. We will use the same notation for both
maps. When the dual repeller $A^{*}$ is a point-fibred attractor of the dual
IFS ${\mathcal{F}}^{*}$, we denote the associated extended coding map by
${\pi}^{\ast}\,:\,{\mathbb{J}}_{-}\rightarrow\mathbb{X}$.

Although $\mathbb{I}$ does not serve as a code space - in particular, a result
analogous to Theorem 5.4 does not hold - the sets ${\mathbb{J}}_{\pm}$, that
are dense in $\mathbb{I}$, do.

######  Theorem 5.6.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$. The extended
coding map $\pi\,:\,\mathbb{J}_{+}\rightarrow\mathbb{X}$ defined by (5.4) is
continuous and agrees with the standard coding map on $\mathbb{I}_{+}$. The
left diagram below is commutative for all $n\in I=I_{+}\cup I_{-}$.

If $A^{\ast}$, the dual repeller of $A$, is a point-fibred attractor of the
dual IFS $\mathcal{F}^{\ast}$, then the right diagram below is commutative for
all $n\in I$.

$\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\mathbb{J}_{+}&\rightarrow&\mathbb{J}_{+}&\\\
{\pi}&\downarrow&&\downarrow&{\pi}\\\ &\mathbb{X}&\rightarrow&\mathbb{X}&\\\
&&f_{n}&&\end{array}\qquad\qquad\qquad\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\mathbb{J}_{-}&\rightarrow&\mathbb{J}_{-}&\\\
{\pi}^{\ast}&\downarrow&&\downarrow&{\pi}^{\ast}\\\
&\mathbb{X}&\rightarrow&\mathbb{X}&\\\ &&f_{n}&&\end{array}$

######  Example 5.7.

If $\mathcal{F}$ is as in Example 3.6, then the ranges of both $\pi$ and
$\pi^{\ast}$ are contained in $\Lambda$.

######  Corollary 5.8.

If $\mathcal{F}$ is an IFS with point-fibred attractor $A$ and fast basin
$\widehat{B}$, then $\pi:\widehat{\mathbb{I}}\rightarrow\widehat{B}$ is
surjective and the left diagram below is commutative for all $n\in I_{-}$.

If $A^{\ast}=\mathbb{X}\backslash B(A)$ is a point-fibred attractor of the
dual IFS $\mathcal{F}^{\ast}$, then
${\pi}^{*}:{\widehat{\mathbb{I}}}^{*}\rightarrow{\widehat{B}}_{{\mathcal{F}}^{*}}$
is surjective and the right diagram below is commutative for all $n\in I_{+}$.

$\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\widehat{\mathbb{I}}&\rightarrow&\widehat{\mathbb{I}}&\\\
{\pi}&\downarrow&&\downarrow&{\pi}\\\ &\widehat{B}&\rightarrow&\widehat{B}&\\\
&&f_{n}&&\end{array}\qquad\qquad\qquad\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\widehat{\mathbb{I}}^{\ast}&\rightarrow&\widehat{\mathbb{I}}^{\ast}&\\\
{\pi}^{\ast}&\downarrow&&\downarrow&{\pi}^{\ast}\\\
&{\widehat{B}}_{\mathcal{F}^{\ast}}&\rightarrow&{\widehat{B}}_{\mathcal{F}^{\ast}}&\\\
&&f_{n}&&\end{array}.$

###### Proof.

This is a corollary of Theorem 5.6. We simply note that
$\sigma_{-n}(\widehat{\mathbb{I}})\subset\widehat{\mathbb{I}}$ and
$\sigma_{n}(\widehat{\mathbb{I}}^{\ast})\subset\widehat{\mathbb{I}}^{\ast}$
for all $n\in I_{+}$. ∎

The following corollary provides a model for attractor-repeller pairs, for
example the loxodromic Möbius case discussed in [14].

######  Corollary 5.9.

Let $\mathcal{F}$ be an IFS with point-fibred attractor $A$ on a complete
metric space $X$ such that $A^{\circ}\neq\emptyset$. If $\mathcal{F}$ is
contractive, then ${\pi}(\widehat{\mathbb{I}})=\mathbb{X}$ and the left
diagram below commutes for all $n\in I_{-}$. If $\mathbb{X}$ is compact, let
the dual repeller $A^{\ast}$ be a point-fibred attractor of
$\mathcal{F}^{\ast}$ with $(A^{\ast})^{\circ}\neq\emptyset$. Then
${\pi}(\widehat{\mathbb{I}})={\pi}^{*}(\widehat{\mathbb{I}}^{*})=\mathbb{X}$;
the left diagram commutes for all $n\in I_{-}$, and the right diagram commutes
for all $n\in I_{+}$.

$\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\widehat{\mathbb{I}}&\rightarrow&\widehat{\mathbb{I}}&\\\
{\pi}&\downarrow&&\downarrow&{\pi}\\\ &\mathbb{X}&\rightarrow&\mathbb{X}&\\\
&&f_{n}&&\end{array}\qquad\qquad\qquad\begin{array}[]{ccccc}&&\sigma_{n}&&\\\
&\widehat{\mathbb{I}}^{*}&\rightarrow&\widehat{\mathbb{I}}^{*}&\\\
{\pi}^{\ast}&\downarrow&&\downarrow&{\pi}^{\ast}\\\
&\mathbb{X}&\rightarrow&\mathbb{X}&\\\ &&f_{n}&&\end{array}$

###### Proof.

The result follows from Corollary 5.8 once it is shown that the fast basin of
$A$ is ${\mathbb{X}}$ and the fast basin of $A^{*}$ is also ${\mathbb{X}}$. In
the case of a contractive IFS with $A^{o}\neq\emptyset$, the fast basin and
the basin coincide and both equal $\mathbb{X}$; see Section 2. The dual
repeller in this case is empty.

In the $\mathbb{X}$ compact case, we will show that the fast basin of $A^{*}$
is ${\mathbb{X}}$; a similar argument shows that the fast basin of $A$ is
${\mathbb{X}}$. It is known [5, Theorem 5.2] that the basin
$B(A^{*})={\mathbb{X}}\setminus A$. So by Proposition 2.7 and Theorems 4.5 and
4.4, if $x\notin A$, then $x$ is in the fast basin of $A^{*}$. Lastly, if
$x\in A$, let $y\neq x$ be any other point of $A$. If $\iota$ is any address
of $y$, then $\pi(\iota)=\lim_{k\rightarrow\infty}f_{\iota|k}(A)$. Therefore
there is a $k$ such that $x\notin f_{\iota|k}(A)$, which implies that

$z:=f_{\iota_{k}}^{-1}\circ f_{\iota_{k-1}}^{-1}\circ\cdots\circ
f_{\iota_{1}}^{-1}(y)\notin A.$

Since $z$ is in the basin of $A^{*}$ and $(A^{*})^{o}\neq\emptyset$, there is
$\omega_{1}\omega_{2}\cdots\omega_{j}$ for some $J$ such that
$f_{\omega_{1}}^{-1}\circ f_{\omega_{2}}^{-1}\circ\cdots\circ
f_{\iota_{j}}^{-1}(z)\in A$. Hence $x$ lies in the fast basin of $A^{*}$. ∎

Theorem 5.6 and its corollaries have many consequences including the
following: (i) the feasibility of continuous assignment of addresses to points
in $\widehat{B}(A)$; (ii) symbolic dynamics on $\widehat{\mathbb{I}}$ are
semiconjugate to corresponding dynamics on $\widehat{B}(A)$; (iii) if $A$ has
nonempty interior as a subset of $\mathbb{X}$, then we obtain addresses and
dynamics on the basin $B(A)$; (iv) description of the relationship between
attractors and dual repellers; (v) extension of fractal homeomorphisms,
generalizing results in [1] and [6]; (vi) address structures for
continuations, branched fractal manifolds and tilings, thereby clarifying and
extending ideas in [6].

##  6\.  Fractal Manifold

Throughout this section, $A$ is a point-fibred attractor of an IFS
$\mathcal{F}$ on a complete metric space $\mathbb{X}$. The branched fractal
manifold generated by $(\mathcal{F},A)$ is constructed in this section.
Properties of the branched fractal manifold are the subject of Theorem 6.4.

######  Definition 6.1.

For $\iota\in\widehat{\mathbb{I}}$, let

$k_{\iota}=\min\\{k:{\pi}(S^{k}(\iota))\in A\\}\in\mathbb{N}_{0}\text{.}$

Define

$[\iota]=\iota|k_{\iota},\text{ and
}\left\langle\iota\right\rangle=S^{k_{\iota}}(\iota),$

to be the integer part and the fractional part of the address
$\iota=\iota_{1}\iota_{2}\cdots\iota_{k_{\iota}}\cdots$ (w.r.t. $\mathcal{F}$
and $A$) respectively.

######  Example 6.2 (Integral and fractional parts).

Consider the case $\mathcal{F}=\\{\mathbb{R};f_{0},f_{1}\\},$ where
$f_{0}(x)=\frac{1}{2}x$ and $f_{2}(x)=\frac{1}{2}x+\frac{1}{2}.$ The unique
attractor is $A=[0,1]$. We will determine the integer and fractal parts of the
two elements of $\widehat{\mathbb{I}}$, namely $-0\,-0\,-0\,\overline{1}$ and
$-0\,-0\,-1\,0\,\overline{1}$. Noting that $\pi(\overline{1})=1$ and
$\pi(-0\,\overline{1})=2$ we have

$\displaystyle\,[-0\,-0\,-0\,\overline{1}]$ $\displaystyle=-0\,-0\,-0,$
$\displaystyle\langle-0\,-0\,-0\,\overline{1}\rangle$
$\displaystyle=\overline{1}$

Noting that $\widehat{\pi}(-0\,-0\,-1\,0\,\overline{1})=0$, we have

$\displaystyle\left[-0\,-0\,-1\,0\,\overline{1}\right]$
$\displaystyle=\emptyset,$
$\displaystyle\langle-0\,-0\,-1\,0\,\overline{1}\rangle$
$\displaystyle=-0\,-0\,-1\,0\,\overline{1}.$

Define an equivalence relation $\sim$ on $\widehat{\mathbb{I}}$ by
$\iota\sim\omega$ if and only if

  1. (1)

$[\iota]=[\omega]$, and

  2. (2)

$\pi(\iota)=\pi(\omega)$,

and let

$\mathbb{L}=\widehat{\mathbb{I}}/\sim.$

For $\iota\in\widehat{\mathbb{I}}$, let $\widetilde{\iota}$ denote the
equivalence class to which $\iota$ belongs. Maps
$\widetilde{\pi}:\widehat{\mathbb{I}}\rightarrow\mathbb{L}$ and
$\widehat{\pi}:\mathbb{L\rightarrow}\mathbb{X}$ are well defined by
$\widetilde{\pi}(\iota)=\widetilde{\iota}$ and
$\widehat{\pi}(\widetilde{\iota})={\pi}(\iota)$, respectively. The maps
$\widehat{\pi}$ and $\widetilde{\pi}$ will be called projection maps.

For $\alpha:=\widetilde{\iota}\in\mathbb{L}$, let

$k_{\alpha}:=k_{\iota}\text{,
}\qquad\qquad[\alpha]:=[\iota],\qquad\qquad\left\langle\alpha\right\rangle:=\widetilde{\left\langle\iota\right\rangle}.$

These entities do not depend on the representative $\iota$ of the equivalence
class $\alpha$. It is useful to think of $\mathbb{L}$ in terms of leaves,
panicles, and sheets.

######  Definition 6.3.

If $\theta$ is a finite word in the alphabet $I_{-}$, then

$l(\theta)=\\{\alpha\in\mathbb{L}\,:\,[\alpha]=\theta\\}$

will be called a leaf of $\mathbb{L}$, and

$p(\theta)=l(\emptyset)\cup l(\theta_{1})\cup
l(\theta_{1}\theta_{2})\cup\cdots\cup l(\theta)$

will be called a panicle of $\mathbb{L}$. Because of the special importance of
the case $\theta=\emptyset$, define

$\mathbb{K}:=l(\emptyset)=p(\emptyset)\text{.}$

If $\theta\in{\mathbb{I}}_{-}$, we will refer to the “infinite panicle”

$\mathbb{L}_{\theta}:=\bigcup_{k=0}^{\infty}l(\theta|k)$

as a sheet of $\mathbb{L}$.

Define a metric on $\mathbb{L}$ as follows. That it is a metric is part of
Theorem 6.4 below. For $\alpha,\beta\in\mathbb{L}$ let

$\displaystyle K$
$\displaystyle=\max\left\\{k\,:[\alpha]|k=[\beta]|k\right\\},$
$\displaystyle[\alpha,\beta]$ $\displaystyle=[\alpha]|K=[\beta]|K,$
$\displaystyle p(\alpha,\beta)$ $\displaystyle=p([\alpha,\beta]),$

and

(6.1)
$d_{\mathbb{L}}(\alpha,\beta)=\min\,\left\\{d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\iota))+d_{\mathbb{X}}(\widehat{\pi}(\iota),\widehat{\pi}(\beta)):\iota\in
p(\alpha,\beta)\right\\}.$

Because $p(\alpha,\beta)=f_{[\alpha,\beta]}(A)$, where the notation of
Equation 2.4 is used, it is equivalent to define

(6.2)
$d_{\mathbb{L}}(\alpha,\beta)=\min\left\\{d_{\mathbb{X}}\left(\widehat{\pi}(\alpha),x\right)+d_{\mathbb{X}}\left(x,\widehat{\pi}(\beta)\right)\,:\,x\in
f_{[\alpha,\beta]}(A)\right\\}.$

The metric space $({\mathbb{L}},\widetilde{d})$ will be called the branched
fractal manifold or just the f-manifold generated by $\mathcal{(F},A)$. A
description of the f-manifold is the intent of the following theorem. Shift
maps $\widetilde{\sigma}_{n}\,:\,\mathbb{L}\rightarrow\mathbb{L}$ are well-
defined by

$\widetilde{\sigma}_{n}(\widetilde{\iota})=\widetilde{\sigma_{n}(\iota)}.$

These maps occur in statement (9).

######  Theorem 6.4.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$ with fast basin
$\widehat{B}\subset\mathbb{X}$. Let $(\mathbb{L},d_{\mathbb{L}})$ be the
f-manifold generated by $(\mathcal{F},A)$. Let
$\widehat{\pi}:\mathbb{L\rightarrow}\widehat{B}$ and
$\widetilde{\pi}:\widehat{\mathbb{I}}\mathbb{\rightarrow L}$ be the associated
projection maps. The following statements hold.

  1. (1)

The minimum in Equation (6.1) is achieved, and $d_{\mathbb{L}}$ is a metric on
$\mathbb{L}$.

  2. (2)

The set of all leaves form a partition of $\mathbb{L}$, and each leaf is
nonempty.

  3. (3)

The image of $\mathbb{L}$ under $\widehat{\pi}$ is the fast basin, i.e.,
$\widehat{\pi}(\mathbb{L})=\widehat{B}$.

  4. (4)

For any $\theta\in\mathbb{I}_{-}$, the projection $\widehat{\pi}$ maps the
sheet $\mathbb{L}_{\theta}$ isometrically onto the continuation
$B_{-\theta}\subseteq\widehat{B}\subseteq\mathbb{X}$:

$d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\beta))=d_{\mathbb{L}}(\alpha,\beta)\text{.}$

for all $\alpha$,$\beta\in\mathbb{L_{\theta}}$.

  5. (5)

Each panicle is homeomorphic to $A$.

  6. (6)

The projection maps $\widetilde{\pi}:\widehat{\mathbb{I}}\mathbb{\rightarrow
L}$ and $\widehat{\pi}:\mathbb{L\rightarrow}\widehat{B},$ are continuous; in
particular

$d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\beta))\leq
d_{\mathbb{L}}(\alpha,\beta)\text{.}$

for all $\alpha$,$\beta\in\mathbb{L}$.

  7. (7)

If $A$ is pathwise connected, then $\mathbb{L}$ is pathwise connected.

  8. (8)

If the metric spaces $(\mathbb{X},d_{\mathbb{X}})$ and
$(\mathbb{X},d_{\mathbb{X}}^{\prime})$ have the same topology, then
$\left(\mathbb{L},d_{\mathbb{L}}\right)$ and
$(\mathbb{L},d_{\mathbb{L}}^{\prime})$ have the same topology, where the
metric $d_{\mathbb{L}}^{\prime}$ is defined by Equation (6.1) with
$d_{\mathbb{X}}$ replaced by $d_{\mathbb{X}}^{\prime}$.

  9. (9)

With $\sigma_{n}:\widehat{\mathbb{I}}\rightarrow\widehat{\mathbb{I}}$ as
defined in Section 5 and $\widetilde{\sigma}_{n}:\mathbb{L\rightarrow L}$ as
defined above, both $\sigma_{n}$ and $\widetilde{\sigma}_{n}$ are continuous
and injective, for all $n\in I_{-}$, and we have the commutative diagram,

$\begin{array}[]{ccccc}&\widehat{\mathbb{I}}&\overset{\sigma_{n}}{\rightarrow}&\widehat{\mathbb{I}}&\\\
\widetilde{\pi}&\downarrow&&\downarrow&\widetilde{\pi}\\\
&\mathbb{L}&\underset{\widetilde{\sigma}_{n}}{\rightarrow}&\mathbb{L}&\\\
\widehat{\pi}&\downarrow&&\downarrow&\widehat{\pi}\\\
&\widehat{B}&\underset{f_{n}}{\rightarrow}&\widehat{B}.&\end{array}$

###### Proof.

The proofs of (2,3,9) are straightfoward.

(1) The set $f_{[\alpha,\beta]}(A)$ in (6.2) is compact because $A$ is compact
and $f_{[\alpha,\beta]}$ is continuous. For fixed $\alpha,\beta$, the function
$d_{\mathbb{X}}(\widehat{\pi}(\alpha),x)+d_{\mathbb{X}}(x,\widehat{\pi}(\beta))$
is continuous in $x\in f_{[\alpha,\beta]}(A)$. Hence the minimum is achieved
at some $x^{\ast}\in f_{[\alpha,\beta]}(A).$

Concerning the metric, we establish only the triangle inequality:
$d_{\mathbb{L}}(\alpha,\gamma)+d_{\mathbb{L}}(\gamma,\beta)=d_{\mathbb{X}}(\widehat{\pi}(\alpha),x_{1})+d_{\mathbb{X}}(x_{1},\widehat{\pi}(\gamma))+d_{\mathbb{X}}(\widehat{\pi}(\gamma),x_{2})+d_{\mathbb{X}}(x_{2},\widehat{\pi}(\beta))$,
where $x_{1}\in f_{[\alpha,\gamma]}(A)$ minimizes
$d_{\mathbb{X}}(\widehat{\pi}(\alpha),x_{1})+d_{\mathbb{X}}(x_{1},\widehat{\pi}(\gamma))$
and $x_{2}\in f_{[\beta,\gamma]}(A)$ minimizes
$d_{\mathbb{X}}(\widehat{\pi}(\gamma),x_{2})+d_{\mathbb{X}}(x_{2},\widehat{\pi}(\beta)).$
We have either $f_{[\alpha,\gamma]}(A)\subset f_{[\alpha,\beta]}(A)$ or
$f_{[\beta,\gamma]}(A)\subset f_{[\alpha,\beta]}(A).$ Without loss of
generality assume that $f_{[\beta,\gamma]}(A)\subset f_{[\alpha,\beta]}(A)$.
Then

$\displaystyle d_{\mathbb{L}}(\alpha,\gamma)+d_{\mathbb{L}}(\gamma,\beta)$
$\displaystyle=d_{\mathbb{X}}(\widehat{\pi}(\alpha),x_{1})+d_{\mathbb{X}}(x_{1},\widehat{\pi}(\gamma))+d_{\mathbb{X}}(\widehat{\pi}(\gamma),x_{2})+d_{\mathbb{X}}(x_{2},\widehat{\pi}(\beta))$
$\displaystyle\geq
d_{\mathbb{X}}(\widehat{\pi}(\alpha),x_{1})+d_{\mathbb{X}}(x_{1},x_{2})+d_{\mathbb{X}}(x_{2},\widehat{\pi}(\beta))$
$\displaystyle\geq
d_{\mathbb{X}}(\widehat{\pi}(\alpha),x_{2})+d_{\mathbb{X}}(x_{2},\widehat{\pi}(\beta))\text{
(where }x_{2}\in f_{[\beta,\gamma]}(A)\text{)}$
$\displaystyle\geq\min\\{d_{\mathbb{X}}(\widehat{\pi}(\alpha),x)+d_{\mathbb{X}}(x,\widehat{\pi}(\beta)):x\in
f_{[\alpha,\beta]}(A)\\}$ $\displaystyle=d_{\mathbb{L}}(\alpha,\beta)\text{.}$

(4) Assume that $\alpha,\beta\in\mathbb{L}$ lie in the same sheet, and
therefore lie in a common panicle, say $p([\beta])$. Then
$[\alpha,\beta]=[\alpha]$ and $f_{[\alpha,\beta]}(A)=f_{[\alpha]}(A)$, which
implies that $\widehat{\pi}(\alpha)\in f_{[\alpha]}(A)$. By the triangle
inequality we have
$d_{\mathbb{L}}(\alpha,\beta)=\min\\{d_{\mathbb{X}}(\widehat{\pi}(\alpha),x)+d_{\mathbb{X}}(x,\widehat{\pi}(\beta)):x\in
f_{[\alpha,\beta]}(A)\\}=d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\beta))$.
That the image of $\mathbb{L}_{\theta}$ under $\widehat{\pi}$ is $B_{-\theta}$
follows from the definitions of $\widehat{\pi}$ and of the continuation
$B_{-\theta}$.

(5) It follows from the definitions that $\widehat{\pi}$ maps the panicle
$p(\theta)$ onto $f_{\theta}(A)$. The result now follows from that facts that
$f_{\theta}$ is a homeomorphism and that, by part (4), the maping
$\widehat{\pi}$ restricted to $p(\theta)$ is an isometry.

(6) The continuity of $\widehat{\pi}:\mathbb{L}\rightarrow\mathbb{X}$ follows
from the inequality
$d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\beta))\leq
d_{\mathbb{L}}(\alpha$,$\beta)$, which follows from the triangle inequality.

Concerning the continuity of
$\widetilde{\pi}:\widehat{\mathbb{I}}\rightarrow\mathbb{L}$, let
$\iota\in\widehat{\mathbb{I}}$ be given. If $\omega\in\widehat{\mathbb{I}}$
and $\omega|K=\iota|K$ for $K$ sufficiently large, that is, if
$d_{\mathbb{I}}(\iota,\omega)$ is sufficiently small, then
$\widetilde{\omega}$ and $\widetilde{\iota}$ lie the same leaf. By part (4)
$d_{\mathbb{L}}(\widetilde{\pi}(\iota),\widetilde{\pi}(\omega))=d_{\mathbb{X}}(\widehat{\pi}(\widetilde{\iota}),\widehat{\pi}(\widetilde{\omega}))=d_{\mathbb{X}}(\widehat{\pi}(\iota),\widehat{\pi}(\omega)).$
The continuity of $\widetilde{\pi}:\widehat{\mathbb{I}}\rightarrow\mathbb{L}$
then follows from the continuity of
$\widehat{\pi}:\widehat{\mathbb{I}}\rightarrow\mathbb{X}$ by Theorem 5.6.

(7) Assume that $A$ is pathwise connected. Given any two points
$\alpha,\beta\in\mathbb{L}$, the leaf $\mathbb{K}$ is contained in the
intersection of the sheets containing $\alpha$ and $\beta$. It is therefore
sufficient to prove that any sheet in pathwise connected; so assume that
$\alpha$ and $\beta$ lie in the same sheet, which implies that they lie in the
same panicle $p(\theta)$ for some finite word $\theta$. But by (5), this
panicle is homeomorphic to $A$.

(8) In the metric space $(\mathbb{X}$,$d_{\mathbb{X}})$, let
$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(x,r)$ denote a ball of radius $r>0$
centered at $x$. Two metrics $d_{\mathbb{X}}$ and $d_{\mathbb{X}}^{\prime}$ on
$\mathbb{X}$ generate the same topology if, given any ball
$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(x,r)$, there is a positive radius
$r^{\prime}$ such that
$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}}^{\prime})}(x,r^{\prime})\subset\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(x,r)$
and, given any
$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}}^{\prime})}(x,r^{\prime}),$ there is a
positive radius $r$ such that
$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(x,r)\subset\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}}^{\prime})}(x,r^{\prime})$.
In the present situation, let the metrics on $\mathbb{L}$ defined using
$d_{\mathbb{X}}$ and $d_{\mathbb{X}}^{\prime}$ be denoted by $d_{\mathbb{L}}$
and $d_{\mathbb{L}}^{\prime}$. Let $r>0$ be given and $\alpha\in\mathbb{L}$ be
given. We will show that there is $r^{\prime}>0$ so that
$\mathcal{B}_{(\mathbb{L},d_{\mathbb{L}}^{\prime})}(\alpha,r^{\prime})\subset\mathcal{B}_{(\mathbb{L},d_{\mathbb{L}})}(\alpha,r).$
The rest of the argument is then obtained by switching the roles of the key
players.

If $\alpha$ and $\beta$ are contained in the same sheet, then by (4), we have
$d_{\mathbb{L}}(\alpha,\beta)=d_{\mathbb{X}}(\widehat{\pi}(\alpha),\widehat{\pi}(\beta))$,
so that

$\mathcal{B}_{(\mathbb{L},d_{\mathbb{L}})}(\alpha,r)=\\{\beta\in\mathbb{L}:\widehat{\pi}(\beta)\in\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(\widehat{\pi}(\alpha),r)\\}\text{.}$

Now choose $r^{\prime}>0$ so small that

$\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}}^{\prime})}(\widehat{\pi}(\alpha),r^{\prime})\subset\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(\widehat{\pi}(\alpha),r).$

Then, if $r^{\prime}$ is sufficiently small,

$\displaystyle\mathcal{B}_{(\mathbb{L},d_{\mathbb{L}})}(\alpha,r^{\prime})$
$\displaystyle=\\{\beta\in\mathbb{L}:\widehat{\pi}(\beta)\in\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(\widehat{\pi}(\alpha),r^{\prime})\\}$
$\displaystyle\subset\\{\beta\in\mathbb{L}:\widehat{\pi}(\beta)\in\mathcal{B}_{(\mathbb{X},d_{\mathbb{X}})}(\widehat{\pi}(\alpha),r)\\}=\mathcal{B}_{(\mathbb{L},d_{\mathbb{L}})}(\alpha,r).$

It now suffices to show that if $\alpha$ is fixed and $\beta$ is sufficiently
close to $\alpha$, then $\alpha$ and $\beta$ lie on the same sheet. Assume
that $\\{\beta_{n}\\}$ is a sequence of points in $\mathbb{L}$ that converge
to $\alpha$, but $\beta_{n}$ is not in the same sheet as $\alpha$ for every
$n$. By the definition of the distance on the f-manifold, this implies that
there is a sequence $\\{\gamma_{n}\\}$ such that $\gamma_{n}\in
p(\alpha,\beta_{n})$ and $\widehat{\pi}(\gamma_{n})$ converges to
$\widehat{\pi}(\alpha)$. Because $\alpha$ is fixed, the set
$\\{p(\alpha,\beta_{n})\\}$ has at most finitely many distinct elements. So
there is an infinite subsequence $\\{\gamma^{\prime}_{n}\\}$ of
$\\{\gamma_{n}\\}$ and the corresponding subsquence $\\{\beta^{\prime}_{n}\\}$
of $\\{\beta_{n}\\}$ such that

$\gamma^{\prime}_{n}\in p(\alpha,\beta^{\prime}_{n})=p(\alpha,\beta_{0})$

for some fixed $\beta_{0}$. Since $\alpha,\gamma^{\prime}_{n}\in p(\alpha)$
for all $n$ and $p(\alpha)$ is compact by statement (5), it follows from the
fact that $\widehat{\pi}(\gamma^{\prime}_{n})$ converges to
$\widehat{\pi}(\alpha)$ and statement (4) that $\\{\gamma_{n}\\}$ converges to
$\alpha$. But from the facts that $p(\alpha,\beta_{0})$ is compact and
$\gamma_{n}^{\prime}\in p(\alpha,\beta_{0})$, it follows that $\alpha\in
p(\alpha,\beta_{0})$. Therefore, $\alpha$ and $\beta^{\prime}_{n}$ are in the
same panicle, and hence the same sheet, for all $n$, a contradiction. ∎

######  Theorem 6.5.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$. For the
f-manifold of $(\mathcal{F},A)$ there are at most $\left|\mathcal{F}\right|+1$
topologically distinct leaves. Specifically, the leaf
$l(\emptyset)=\mathbb{K}$ is homeomorphic to $A$, and the leaf $l(\theta)$ for
$\theta\neq\emptyset$ is homeomorphic to $A\backslash f_{i_{\theta}}(A)$ where
$-i_{\theta}$ is the last component of $\theta$.

###### Proof.

To simplify notation, when the intent is clear, we make no distiction between
$\iota\in\widehat{\mathbb{I}}$ and its equivalence class in $\mathbb{L}$. For
$\alpha\in\mathbb{L}$, note that ${\pi}(\langle\alpha\rangle)\in A$. For a
leaf $l(\theta)$, let $\langle
l(\theta)\rangle=\\{\langle\alpha\rangle:\alpha\in l(\theta)\\}$. By
definition, $\iota\in\langle(\theta)\rangle$ if and only if $\pi(\iota)\in A$
and $f_{S^{i}(\iota)}(\pi(\iota))\notin A$ for $i=0,1,2,...,k_{\iota}-1$; the
last condition here is equivalent to $f_{i_{\theta}}(\pi(\iota))\notin A$
where $-i_{\theta}$ is the last entry in $\theta$. Therefore ${\pi}(\langle
l(\theta)\rangle)=A\backslash f_{i_{\theta}}(A)$. This implies that
$\widehat{\pi}(l(\theta))=f_{\theta}(\pi(\langle
l(\theta)\rangle))=f_{\theta}(A\backslash f_{i_{\theta}}(A))$. Since
$\widehat{\pi}$ restricted to $l(\theta)$ is a homeomorphism by part (4) of
Theorem 6.4, and $f_{\theta}$ is a homeomorphism, we have $l(\theta)$ is
homeomorphic to $A\backslash f_{i_{\theta}}(A)$. ∎

######  Corollary 6.6.

Let $A$ be a point-fibred attractor of an IFS $\mathcal{F}$. The closures of
any two distinct leaves of the f-manifold are non-overlapping.

###### Proof.

This follows from part (5) of Theorem 6.4, Theorem 6.5, and the fact that the
boundary of a compact set has empty interior. ∎

######  Example 6.7.

The attractor of the IFS $\ ({\mathbb{R}};\,f_{1},f_{2}\\}$ where
$f_{1}(x)=\frac{1}{3}x$ and $f_{2}(x)=\frac{1}{3}x+\frac{2}{3}$, is the Cantor
set $\mathcal{C}$. The f-manifold $\mathbb{L}$ is the disjoint union of its
leaves, each of which is homeomorphic to $\mathcal{C}$. The image of each leaf
under the projection map $\widehat{\pi}$ is is related by a similitude to
$\mathcal{C}$, with scaling factor equal to $3^{k}$ for some
$k\in{\mathbb{N}}_{0}$.

######  Example 6.8.

The attractor of the IFS $\ ({\mathbb{R}};\,f_{0},f_{1}\\}$ where
$f_{0}(x)=\frac{1}{2}x$ and $f_{1}(x)=\frac{1}{2}x+\frac{1}{2}$, is the unit
interval $A=[0,1]$. The f-manifold is a connected, branched $1$-manifold. The
set ${\mathbb{K}}=l(\emptyset)$ is homeomorphic to $A=[0,1]$. According to
Theorem 6.5, all other leaves are homeomorphic to half open intervals. Each
panicle, a union of leaves, is homeomorphic to a closed interval. Routine
calculation shows the following. The length of the projection of each leaf,
under $\widehat{\pi}$, is a power of $2$. Each point $\alpha\in\mathbb{L}$ is
contained in a neighborhood that is homeomorphic to the open interval $(0,1)$
or, if it is a branch point, to the union of a countably infinite number of
copies of $[0,1)$ where only $\alpha$ is common to the copies. The branch
points are isolated. More exactly, the branch points are the points of
$\mathbb{L}$ whose projection under $\widehat{\pi}$ are of the form
$2^{n},\,n\geq 0,$ or $-2^{n}+1,\,n\geq 0$. Each sheet of $\mathbb{L}$ is
mapped isometrically by $\widehat{\pi}$ to either ${\mathbb{R}}$ or the
interval $[0,\infty)$ or the interval $(-\infty,1]$. For example the sheet
${\mathbb{L}}_{\overline{-1\,-0}}$ is homeomorphic to ${\mathbb{R}}$, and
${\mathbb{L}}_{\overline{-1}}$ is homeomorphic to $(-\infty,1]$. See Figure 4.

Figure 4. A portion of the f-manifold of Example 6.8. Each leaf $l(\theta)$ is
labeled by $-\theta$. Note that there are additional leaves, with longer
labels, incident with some of the branch points in the diagram, that are not
included in the diagram.

######  Example 6.9.

While it is easy to visualize the branched fractal manifold in the above
example, it is harder to picture it in the following example, illustrated in
Figure 5. Here the IFS $\mathcal{F}=$ $\\{\mathbb{R}^{2};f_{i},\,i=1,2,3,4\\}$
is affine, with the following property. There is a triangle $ABC$ with points
$c\in AB$, $a\in BC$, $b\in CA$, where $XY$ is the line segment joining the
points $X$ and $Y$, and the triangles $abc,$ $ABC,$ $Abc$, $aBc,$ $abC$ are
non-degenerate. Moreover, $f_{1}(ABC)=Abc$, $f_{2}(ABC)=aBc,$
$f_{3}(ABC)=abC$, and $f_{4}(ABC)=abc$. IFSs of this kind are discussed in
[1]. This IFS possesses a unique attractor, the filled triangle with vertices
$ABC$. This attractor, a small triangle in multicolours, is located near the
center of each of the four pictures; directly below it is a small yellow
triangle. If $\theta\in\mathbb{I}_{+}$ is reversible, then
$\widehat{B}_{\theta}$ $=\mathbb{R}^{2}$. The branched fractal manifold
consists of non-denumerably many copies of $\mathbb{R}^{2}$ glued together
appropriately. Each region of glue is triangular. The top left panel
illustrates $B_{ijk1}$ for a fixed choice of $ijk$. It comprises 256 copies of
$A,$ namely the sets $f_{i}^{-1}\circ f_{j}^{-1}\circ f_{k}^{-1}\circ
f_{1}^{-1}\circ f_{m}\circ f_{n}\circ f_{o}\circ f_{p}(A)$ for all
$m,n,o,p\in\\{1,2,3,4\\}$, each copy in a colour that is supposed to be
different from its neighbours. The top right panel illustrates $B_{ijk1}\cup
B_{ijk2}$, where each of the subtriangles $f_{i}^{-1}\circ f_{j}^{-1}\circ
f_{k}^{-1}\circ f_{2}^{-1}\circ f_{m}\circ f_{n}\circ f_{o}\circ f_{p}(A)$ for
all $m,n,o,p\in\\{1,2,3,4\\}$ is coloured so that it is different from its
neighbours. The two images may be thought to be glued together on $B_{ijk}$.
Similarly, the bottom left panel illustrates $B_{ijk1}\cup B_{ijk2}\cup
B_{ijk3}$ and the bottom right panel illustrates $B_{ijk1}\cup B_{ijk2}\cup
B_{ijk3}\cup B_{ijk4}$.

Figure 5. Fractal manifold under construction. See Example 6.9.

ACKNOWLEGEMENTS

We thank Alan Carey for interesting discussions and helpful comments. We thank
Louisa Barnsley for technical help and suggestions.

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