# Pseudo-Böttcher components of the wandering set of inner mappings

Igor Yu. Vlasenko  Algebra and Topology department, Institute of Mathematics,
Kyiv, Ukraine<EMAIL_ADDRESS>
###### Abstract.

This article explores the topology of Pseudo-Böttcher totally invariant
connected components of the wandering set in dynamical systems generated by
non-invertible inner (open surjective isolated) mappings of compact surfaces.
We describe the possible topological types of these invariant connected
subsets, which are more diverse then corresponding components of
homeomorphisms.

##  1\. Introduction

This article contributes to the understanding of the wandering set in
dynamical systems generated by non-invertible inner (open surjective isolated)
mappings of compact surfaces. It focuses on invariant connected subsets
(components) of the wandering set which can be distinguished using filtration
of the wandering set by attractor-repeller pairs. Such components are well
understood in the invertible case but has not been studied in the non-
invertible case.

In the invertible case their topological type can only be a disc or an
annulus. In the non-invertible case we introduce a class of annulus-like
totally invariant connected components of the wandering set of dynamical
systems, generated by inner mappings of compact surfaces, called Pseudo-
Böttcher components (Definition 3.7.1). We describe all possible topological
types of Pseudo-Böttcher components, including new topological types specific
to non-invertible case, in Theorem 5.0.13.

Dynamical systems generated by non-invertible maps are not mainstream, so
there is a need to explain maps, invariant sets, classes and topological
invariants used and their difference from mainstream invertible dynamical
systems in detail.

Inner mappings as a separate subclass of non-invertible continuous maps of
two-dimensional manifolds were introduced by S. Stoilov [12]. Stoilov defined
inner mappings as open zero-dimensional continuous maps, proved that inner
mappings are in fact open discrete maps, and proved his well-known theorem
that the class of inner mappings coincides with the class of perturbations of
holomorphic mappings by the right action of the group of homeomorphisms. In
the case of higher-dimensional manifolds, the class of open zero-dimensional
maps no longer coincides with the class of open discrete maps. That is why Yu.
Yu. Trokhymchuk [13] defined the class of inner mappings in general case as a
class of open discrete maps (Definition 2.2.1). Homeomorphisms,
diffeomorphisms, holomorphic maps belong to the class of inner mappings.

Classical topological theory of dynamical systems generated by invertible maps
considers them up to conjugacy with a homeomorphism and defines a lot of well-
known invariant sets such as wandering and nonwandering ones. It is usually
hard to apply those classic invariant sets to non-invertible maps. At most,
‘‘$\omega$’’ wariants of invariant sets are used, such as
$\omega$-nonwandering set. However, among all continuous maps, inner mappings
are the natural class to which the methods of topological theory of invertible
dynamical systems can be extended. Being open, inner mappings preserve base
topology at a point, and being discrete, they allow to trace locally a partial
trajectory of a point back-and-forth unambiguously.

Analogues of classic invariant sets for inner mappings are built in author’s
book [15]. We can speak about trajectories (Definition 3.1.1), wandering sets
(Definition 3.3.1) and so on, because the corresponding definitions reduce to
classic ones when applied to invertible maps.

The wandering set of non-invertible inner mappings can be just as complex as
the nonwandering set of invertible mappings. In this paper, we focus on a
specific subsets of the wandering set: totally invariant subsets (Definition
2.0.1). A new class called Pseudo-Böttcher regular components is defined
(Definition 3.7.1) which extends the class of Böttcher components from [14].
This paper answers the questions about topology of Pseudo-Böttcher regular
components. Similar questions were left unexplored in the paper [14]. Theorem
5.0.13 shows that without singular points Pseudo-Böttcher regular components
are just open annuli that is the same as in the invertible case, but with
singular points those components become quite complex topologically, as
described in Theorem 5.0.13.

##  2\. Preliminary information

Let $f\colon M\to M$ be a continuous surjective map of a compact manifold $M$
without boundary.

Due to the difference between invertible and non-invertible maps the notion of
invariant set has to be split into two notions.

######  Definition 2.0.1.

A set $X\subset M$ is called

  * •

_invariant_ set if $f(X)=X$.

  * •

_totally invariant_ set if $f^{-1}(X)=X$.

A totally invariant set is invariant because $f(X)=f(f^{-1}(X))=X$. The
converse is not true for non-invertible maps. A set $X$ can be invariant
($f(X)=X$) but not totally invariant ($f^{-1}(X)\not=X$).

For the invertible maps any invariant set is totally invariant.

###  2.1. Attractor-repeller pairs.

Filtration of the manifold $M$ by attractor-repeller pairs is part of the
theory of chain-recurrent sets and the ‘‘fundamental theorem of dynamical
systems’’ which has been developed by Conley [3] for continuous flows on a
compact metric space and has been generalized by Franks for homeomorphisms [4]
and Hurley for endomorphisms [5].

######  Definition 2.1.1.

An open set $U\subset M$ is called _strictly attracting_ if
$f(\overline{U})\subset U$.

A strictly attracting open set $U$ generates

  * •

the _attractor_ $A:=\cap_{n\geq 0}f^{n}(U)$,

  * •

the _basin of attraction_ $B_{A}:=\cup_{n\geq 0}f^{-n}(U)$ and

  * •

the _repeller_ $R:=M\setminus B_{A}$.

Figure 1.  Attractor-repeller pair $(A,R)$ and a strictly attracting
neighborhood $U$.

$A\not=\emptyset$ and $R\not=\emptyset$ because $M$ is compact. By
construction, $f(A)=A$, $f(B_{A})=B_{A}$, $f^{-1}(B_{A})=B_{A}$. Therefore,
$f(R)=R$, $f^{-1}(R)=R$. If $f$ is not invertible, $f^{-1}(A)$ can be
different from $A$.

Repeller $R$ and basin of attraction $B_{A}$ are totally invariant sets.
Attractor $A$ is invariant, but possibly not totally invariant set.

For homeomorphisms, an attractor-repeller pair $(A,R)$ is symmetric: an
attractor $A$ for the map $f$ becomes repeller for the map $f^{-1}$. For non-
invertible maps an attractor-repeller pair $(A,R)$ is not symmetric: repeller
$R$ is totally invariant, while attractor $A$ is not.

######  Definition 2.1.2.

A set $D:=B_{A}\setminus\cup_{n\geq 0}f^{-n}(A)$ is called the _region of
strict wandering_ of the attractor $A$.

######  Definition 2.1.3.

A connected component $S$ of the region of strict wandering is called a
_component of strict wandering_ of the attractor $A$.

By construction, the region of strict wandering and its components of strict
wandering are open sets.

###  2.2. Definition of inner mappings.

######  Definition 2.2.1.

A map $f$ is called

  1. (1)

_open_ if the image of any open set is open;

  2. (2)

_closed_ if the image of any closed set is closed;

  3. (3)

_discrete_ if the preimage of any point is a discrete set (a set is called
_discrete_ if any point of the set is isolated);

  4. (4)

_finite-to-one_ if the preimage of any point is finite;

  5. (5)

_inner mapping_ if $f$ is continuous open discrete surjective map of $M$.

######  Theorem 2.2.2 (S. Stoilov [12]).

Let $M$ be a two-dimensional manifold and $f\colon M\to\mathbb{C}$ be an inner
mapping. Then there exists a Riemannian surface $R$ which is a complex
structure on $M$, a homeomorphism $h\colon M\to R$, and a holomorphic function
$F\colon R\to C$ such that $f=F\circ h$.

######  Definition 2.2.3.

A map $f\colon M\to M$ is called _local homeomorphism_ at a point $x\in M$ if
there exists a neighborhood $U$ of $x$ such that restriction $f|_{U}$ is a
homeomorphism.

######  Definition 2.2.4.

A map $f\colon M\to M$ is called _covering_ at a point $x\in M$ if there
exists an open neighborhood $U$ of the point $f(x)$ such that
$f^{-1}(U)=\cup_{x_{i}\in f^{-1}(x)}V_{x_{i}}$ where

  * •

$V_{x_{i}}\cap V_{x_{j}}=\emptyset$ if $x_{i}\not=x_{j}$;

  * •

for each $x_{i}\in f^{-1}(x)$ $f(V_{x_{i}})=U$;

  * •

the restriction $f|_{V_{x_{i}}}$ is a homeomorphism.

For the finite-to-one maps the notions of local homeomorphism and a covering
at a point are equivalent.

######  Definition 2.2.5.

A map $f\colon M\to M$ is called

  * •

_local homeomorphism_ if $f$ is local homeomorphism at every point $x\in M$;

  * •

_covering_ if $f$ is covering at every point $x\in M$;

  * •

_branched covering_ if $f$ is covering at every point $x\in M$ except from a
nowhere dense set known as branch set.

######  Definition 2.2.6.

A point $x\in M$ is called _singular_ point of $f$ if $f$ is not a local
homeomorphism at the point $x$.

Denote by $\operatorname{Sing}(f)$ the set of singular points of $f$.

###  2.3. Some properties of inner mappings.

Let $f\colon M\to M$ be an inner mapping of a compact connected two-
dimensional manifold $M$ without boundary.

######  Lemma 2.3.1 (a collection of facts about inner mappings, see [12, 13,
15]).

Let $f\colon M\to M$ be an inner mapping of a compact connected two-
dimensional manifold $M$ with topological defree $\deg f$. Then

  1. (1)

$f$ is finite-to-one map;

  2. (2)

$f$ is closed map;

  3. (3)

$\operatorname{Sing}(f)$ is finite;

  4. (4)

for any point from $M\setminus\operatorname{Sing}(f)$ the number of preimages
is constant and is equal to $|\deg f|$;

  5. (5)

for any point $p\in\operatorname{Sing}(f)$ the number of preimages
$|f^{-1}(p)|$ is less then $|\deg f|$;

  6. (6)

$f$ is branched covering with the branch set $\operatorname{Sing}(f)$.

######  Definition 2.3.2.

For a $p\in M$ define its _defect number_ $e(p)$: $e(p)=|\deg f|-|f^{-1}(p)|$.

######  Lemma 2.3.3 (Stoilov Lemma, S. Stoilov [12]).

Let $p\in\operatorname{Sing}(f)$. Then there exists an open neighborhood
$U(p)$ and a homeomorphism $h\colon U\to U$ such that $h(p)=f(p)$ and
$f|_{U}=(z-f(p))^{e(p)+1}\circ h$ in some local coordinates on $U$.

It follows that points with defect number $e(p)=0$ are the points of local
homeomorphism, whereas points with defect number $e(p)>0$ are the singular
points of $f$. The local behavior of an inner mapping around a singular point
$p$ is determined by complex function $z^{e(p)+1}$. In particular, singular
points are isolated.

##  3\. Topological dynamics of inner mappings and its difference in case of
homeomorphisms

Let $M$ be a compact connected two dimensional manifold without boundary. Let
$f\colon M\to M$ be an inner mapping, i.e. a continuous non-invertible open
(an image of an open set is open) finite-to-one surjective map of $M$. The
iterations of $f$ create a dynamical system on $M$.

Topological theory of dynamical systems study maps up to topological
conjugacy. For a map $f$ its conjugacy class is the set $\\{h^{-1}\circ f\circ
h|h\in\textit{Homeo}(M)\\}$. However, since topological dynamics methods were
developed for homeomorphisms, they cannot be directly applied to inner
mappings. We will address this challenge by adapting classical topological
dynamics definitions to work with non-invertible inner mappings.

For more detail, see the author’s book [15].

###  3.1. Trajectories.

######  Definition 3.1.1.

Consider a point $x\in M$.

  * •

$O^{+}_{f}(x)=\\{f^{n}(x)|\ n\geq 0\\}$ is a _positive semitrajectory_ of $x$;

  * •

$O^{-}_{f}(x)=\\{f^{-n}(x)|\ n\geq 0\\}$ is a _negative semitrajectory_ of
$x$;

  * •

A sequence $(x_{n})_{n\in\mathbb{Z}}$, where $x_{0}=x$, $f(x_{n})=x_{n+1}$ is
called a _partial trajectory_ of $x$;

  * •

$O_{f}(x)=\cup_{y\in O^{+}_{f}(x)}O^{-}_{f}(y)$ is a _full trajectory_ of $x$;

  * •

$O^{\perp}_{f}(x)=\\{f^{-n}\left(f^{n}(x)\right)|\ n\geq 0\\}$ is a _neutral
section_ of (the full trajectory of) $x$;

Figure 2.  Periodic and generic $O_{f}(x)$ for homeomorphisms.

Figure 3.  Periodic and generic $O_{f}(x)$ for inner mappings.

By definition, $O^{+}_{f}(x)\subset O_{f}(x)$, $O^{-}_{f}(x)\subset O_{f}(x)$,
$O^{\perp}_{f}(x)\subset O_{f}(x)$. Note that partial trajectories of $x$ are
not uniquely defined: in the case of non-invertible maps the point $x$ can
have infinitely many different partial trajectories. Any partial trajectory of
$x$ is an invariant set and is contained in $O_{f}(x)$. Also notice that
$O_{f}(x)$ is a totally invariant set.

If $f$ is invertible, then $O^{\perp}_{f}(x)$ coincide with $\\{x\\}$, while
$O_{f}(x)$ and partial trajectories turn into the set $\\{f^{n}(x)|\
n\in\mathbb{Z}\\}$.

Evidently, the full trajectory, $O_{f}(x)$ is the smallest topological totally
invariant set preserved under topological conjugacy. However, for general non-
invertible maps, their full trajectories are unique sets that can differ
significantly from each other topologically. This makes describing the
topological conjugacy classes of general non-invertible maps significantly
more challenging. In contrast, for homeomorphisms, there are only a few
possible trajectory types: a generic trajectory without periodic points and
periodic trajectories of period $n$, $n\geq 1$. This fact explains why
classical topological dynamics is well-developed for homeomorphisms but not
for general non-invertible maps.

Among non-invertible maps, the class of inner mappings exhibits dynamical
systems most similar to homeomorphisms. For inner mappings, by definition, the
full trajectory, $O_{f}(x)$, is a closed and discrete set. If there are no
singular points, then for each degree $d$, $|d|>1$, for an inner mapping $f$
of degree $d$, each point has exactly $|d|$ preimages. Furthermore, the set of
all possible trajectories consists only of the generic trajectory without
periodic points and periodic trajectories of period $n$, $n\geq 1$, which is
similar to the case of homeomorphisms. The presence of a finite number of
singular points introduces additional complexities, but due to their finite
number, accounting for all possible classes of trajectories of inner mappings
remains manageable.

###  3.2. Neutrally saturated sets.

######  Definition 3.2.1.

A set $X\subset M$ is called _neutrally saturated_ , if $\forall x\in X$
$O^{\perp}(x)\subset X$.

In particular, a totally invariant set is neutrally saturated.

For any set $X$ the set $X^{\perp}=\cup_{n\geq 0}f^{-n}\circ f^{n}(X)$ is a
smallest neutrally saturated set such that $X\subset X^{\perp}$.

######  Definition 3.2.2.

The set $X^{\perp}$ for a set $X\subset M$ is called the _neutral saturation_
of the set $X$.

For example, consider an attractor $A$ such that $A$ is not totally invariant,
i.e. $f^{-1}(A)\not=A$. Then the set $A^{\perp}\setminus A$ consists of
additinal preimages of $A$: $A^{\perp}=\cup_{n\geq 0}f^{-n}\circ f^{n}(A)$,
$f^{n}(A)=A$ $\Rightarrow$ $A^{\perp}=\cup_{n\geq 0}f^{-n}(A)$.

###  3.3. Wandering and nonwandering points.

General non-invertible maps do not preserve neighborhoods of points. For
example, a simple map $x^{2}$ maps the open neighborhood $(-1,1)$ of $0$ to
the semi-interval $[0,1)$ which is neither open set, nor closed set, and not a
neighborhood of $0$ (it does not contain a metric $\varepsilon$-ball for any
$\varepsilon>0$). It only gets worse in dimensions 2 and above. In contrast,
in the case of homeomorphisms a neighborhood $U(x)$ of a point $x\in M$ can be
mapped to any point of $O_{f}(x)$ by $f^{n}$ or $f^{-n}$, and the image of
$U(x)$ is, by definition, homeomorphic.

In this regard inner mappings resemble homeomorphisms. As inner mappings are
open and close maps, the image and preimage of open or closed neighborhood is
also open or closed neighborhood. Even more, inner mappings have the structure
of branched coverings. This means that if $U(x)$ is small enough and
$O_{f}(x)$ does not have any singular points of $f$, then we can build a chain
of local homeomorphisms, moving $U(x)$ to any point of $O_{f}(x)$
homeomorphically. If there is a singular point, we can still build an image of
$U(x)$ locally which is not a homeomorphic image, but still remains open or
closed connected neighborhood.

a) b)

Figure 4.  Dynamics of neighborhoods along $O_{f}(x)$. a) homeomorphisms b)
inner mappings.

######  Definition 3.3.1.

Denote by $U(x)$ an open neighborhood of a point $x$. A point $x$ is called

  * •

_nonwandering_ point of $f$ if for any neighborhood $U(x)$ and for all
$m\in{\mathbb{Z}}$, $m\not=0$: $f^{m}(U)\cap U\not=\emptyset$.

  * •

_still wandering_ point of $f$ if $x$ isn’t nonwandering.

  * •

_forever wandering_ or, for short, _wandering_ point of $f$ if there exists a
neighborhood $U(x)$ such that $\forall n\geq 0$, $\forall m\in{\mathbb{Z}}$,
$m\not=n$: $f^{m}\left(f^{n}\left(U\right)\right)\cap U=\emptyset$.

  * •

_pre-nonwandering_ point of $f$ if $x$ is not forever wandering.

  * •

_pre-nonwandering wandering_ if $x$ is pre-nonwandering but not nonwandering.

Note that if $x$ is pre-nonwandering point of $f$ then from definition there
exists $n\geq 0$ such that $f^{n}(x)$ is a nonwandering point of $f$.

Denote by $\overline{\Omega}(f)$, $\widetilde{\Omega}(f)$, $\overline{W}(f)$
and $\widetilde{W}(f)$ the sets of nonwandering, pre-nonwandering, still
wandering and forever wandering points of $f$. By definition,
$\overline{\Omega}(f)\subset\widetilde{\Omega}(f)$,
$\overline{W}(f)\subset\widetilde{W}(f)$. The set $\overline{\Omega}(f)$ is
invariant but not totally invariant. The set $\overline{W}(f)$ is not
invariant when $f$ is not invertible due to existsence of pre-nonwandering
wandering points. The sets $\widetilde{\Omega}(f)$ and its counterpart
$\widetilde{W}(f)$ are totally invariant.

When $f$ is a homeomorphism, the sets $\widetilde{W}(f)$ and
$\widetilde{\Omega}(f)$ both become the classic wandering and nonwanderings
sets of homeomorphisms $W(f)$ and $\Omega(f)$. The same is true for
$\overline{W}(f)$ and $\overline{\Omega}(f)$, because the definitions of
$\overline{W}(f)$ and $\overline{\Omega}(f)$ are just classic definitions of
wandering and nonwanderings sets applied verbatim to non-invertible case. We
focus on the set $\widetilde{W}(f)$ as the non-invertible analog to the
classical set of wandering points, $W(f)$, because the alternative choice the
set $\overline{W}(f)$ is not invariant.

###  3.4. Decomposition of wandering set of homeomorphisms.

Essentially, when restricted to their wandering sets, dynamical systems of
homeomorphisms become discrete $\mathbb{Z}$-actions. In contrast to some
simple $\mathbb{Z}$-actions, such as $z\mapsto z+1$ on $\mathbb{C}$ or
$z\mapsto 2z$ on $\mathbb{C}\setminus\\{0\\}$, a typical wandering set of
homeomorphisms as a whole is still topologically intricate and can be further
decomposed into ‘‘elementary blocks’’, open pieces with simple topology and
with $\mathbb{Z}$-actions that have uniform $-\infty$ and $+\infty$
directions, and some ‘‘separatrix sets’’, which divide the wandering set into
those ‘‘elementary blocks’’.

An example of such decomposition is the classical theory of regular subsets of
the wandering set, introduced by Birkhoff [2]. Another example is filtration
of the wandering set by attractor-repeller pairs described in Section 2.1. We
won’t go into the details regarding those decompositions, as they are used
purely as motivation of the research. Let’s limit ourselves to the class of
‘‘elementary blocks’’ which consists of invariant open connected sets.

a) b)

Figure 5.  Homeomorphisms on $S^{2}$ and their wandering sets.

######  Example 3.4.1 (Figure 5a).

Consider a dynamical system on a sphere where the nonwandering set consists
only of two points: a sink (attracting fixed point) $\omega$ and a source
(repelling fixed point) $\alpha$ which create attractor-repeller pair
$(\omega,\alpha)$. The rest of the sphere is wandering set which is
homeomorphic to an open annulus. This wandering set itself is an ‘‘elementary
block’’. Any pair of such homeomorphisms is topologically conjugate.

######  Example 3.4.2 (Figure 5b).

Consider a Morse-Smale diffeomorphism of a sphere $S^{2}$. Its nonwandering
set consists of four points: a sink $\omega$, a saddle $\chi$ and two sources
$\alpha_{1}$ and $\alpha_{2}$. There are 3 attractor-repeller pairs. The 1st
pair is attractor $\\{\omega\\}$ and repeller $\\{\alpha_{1}$, $s_{3}$,
$\chi$, $s_{4}$, $\alpha_{2}\\}$. This pair decompose the wandering set into a
‘‘separatrix set’’ of $s_{3}$ and $s_{4}$. and a single ‘‘elementary block’’,
which lies in the basin of attraction of $\omega$ and called an
$\omega$-regular component by Birkhoff. The 2nd and 3rd pairs are one of the
sources $\alpha_{1}$, $\alpha_{2}$ as repeller and closure of the basin of
repulsion of another source as attractor. They decompose the wandering set
into a ‘‘separatrix set’’ of $s_{1}$ and $s_{2}$. and two ‘‘elementary
blocks’’, which lie in the basins of repulsion of $\alpha_{1}$ and
$\alpha_{2}$ and called an $\alpha$-regular components by Birkhoff.

In the examples above all the ‘‘elementary blocks’’ are homeomorphic to
annuli. In fact, Birkhoff and Smith [1, 2] proved that ‘‘elementary blocks’’
of homeomorphisms, which are invariant open connected sets, on compact
surfaces can only be homeomorphic either to a disc or an annulus. Those
results also apply to invariant open connected sets cut from the wandering set
by an attractor-repeller pair.

However, the results of Birkhoff and Smith regarding the homeomorphisms do not
apply to non-invertible inner mappings of surfaces. A simple example is the
holomorphic map $z^{2}+1$ of the sphere $\overline{\mathbb{C}}$. Its Julia set
is the repeller and a Cantor set. $\infty\in\overline{\mathbb{C}}$ is the
attractor. Their complement is the Fatou set of $z^{2}+1$, which for this map
coincides with its wandering set. Note that the wandering set of $z^{2}+1$ is
totally invariant connected open set and its complement is the attractor-
repeller pair. Then it consists of single ‘‘elementary block’’ analog for
inner mappings. However, its topological type is not a disc or an annulus, but
a disc with a Cantor set punctured.

Let $f\colon M\to M$ be an inner mapping of a compact surface $M$ such that
$f$ has an totally invariant connected open subset $S$ of its wandering set
cut off by a attractor-repeller pair $(A,R)$. The question arises: what are
possible topological types of $S$? Consider a special case first. Let’s there
be a neighborhood $U$ of the attractor $A$ such that $U\cap S$ is an annulus.
What are possible topological types of $S$? These questions are the motivation
for this article.

###  3.5. Topological conjugacy problem of wandering set.

The key difference between wandering sets of homeomorphisms and inner mappings
lies in how they relate to topological conjugacy. For homeomorphisms,
topological invariants, which distinguish conjugacy classes, are concentrated
within the nonwandering set. In contrast, inner mappings exhibit an abundance
of conjugacy classes even when restricted to their wandering set.

For instance, the non-invertible analog of the example 3.4.1 is an $n$-sheeted
branched covering on a sphere where the nonwandering set consists of two
branch points, an attracting singular sink and a repelling singular source,
and the rest of the sphere are forever wandering points. There is an example
of a countable sequence of such inner mappings which are not pairwise
topologically conjugate [16].

Presumably there is, in fact, uncountable number of inner mappings which are
not pairwise topologically conjugate. To explain this, consider a set
$O^{\perp}_{f}(x)=\\{f^{-n}\left(f^{n}(x)\right)|\ n\geq 0\\}$ for a point
$x\in M$ (Definition 3.1.1). When $f$ is an invertible map,
$O^{\perp}_{f}(x)=\\{x\\}$. However, in the non-invertible case,
$O^{\perp}_{f}(x)$ typically is a countable set. We can consider the main
dynamical system with the map $f$ as the action of $\mathbb{Z}$. Then we can
consider $O^{\perp}_{f}(x)$ as the action of another group, the iterated
monodromy group, that acts locally in a neighborhood of $O^{\perp}_{f}(x)$
using single-valued branches of the family of multi-value maps $f^{-n}\circ
f^{n}$. Topological conjugacy has to preserve not only $\mathbb{Z}$-action on
$x$, but $O^{\perp}_{f}(x)$ too. And $O^{\perp}_{f}(x)$ as a countable set
will have a closure, which is some fancy closed fractal set due to
aforementioned monodromy action. This closure should also be preserved by
continuity. A small perturbation of $f$ will cause a countable set
$O^{\perp}_{f}(x)$ to have another closure, making the perturbed map
topologically different.

Consequently, the wandering set of non-invertible inner mappings exhibits a
level of complexity comparable to the nonwandering set of invertible maps. The
set of equivalence classes of wandering sets up to conjugacy is most likely
uncountable and can not possibly be described. It is a so called ‘‘wild’’
topological problem. Therefore, we can only compromise and reduce complexity
by imposing restrictions on the dynamics in the wandering set.

###  3.6. Restrictions on the wandering set.

As mentioned in Section 3.5, we want to impose restrictions on the wandering
set to reduce complexity. Holomorphic maps are the most well-studied subclass
of non-invertible inner mappings. Let us create the restrictions using
holomorphic maps as an example.

Let an analytic function $f$ has an attracting critical point (in the
terminology of holomorphic dynamics — a superattracting point). One can always
choose a local uniformizing parameter $z$ such that $z=0$ will be a
superattracting fixed point of the function $f(z)$. Then $f(z)$ has the form

$f(z)=a_{n}z^{n}+a_{n+1}z^{n+1}+\dots,$ (3.1)

where $n\geq 2$, $a_{n}\not=0$.

######  Theorem 3.6.1 (Böttcher’s theorem, see [7]).

Let the function $f$ be of the form (3.1). Then there exists a local
holomorphic change of coordinate $w=\phi(z)$ which conjugates $f$ to the
$n$-th power map $w\mapsto w^{n}$ throughout some neighborhood of $\phi(0)=0$.

Böttcher’s theorem has an interesting consequence in topological dynamics of
inner mappings, as pointed in [17]. Consider a map $z^{n}$, $n\geq 2$. It is
easy to verify by direct calculation that for any $z_{0}\in\mathbb{C}$ its
neutral section $O^{\perp}(z_{0})$ is everywhere dense in the level curve
$|z|=|z_{0}|$ of the real function $|z|$. As $O^{\perp}(x)$ is topological
invariant of inner mappings, then its closure $\overline{O^{\perp}(x)}$ is
also the topological invariant. For the map $z^{n}$ it means that the
foliation on the level curves of the function $|z|$ is the topological
invariant of $z^{n}$. Not only that, the function $|z|$ can be considered as a
special Lyapunov function in the neighborhood of $0$ such that its foliation
on the level curves is invariant under the action of $z^{n}$: a circle
$|z|=const$ moves onto the circle $|z|=const^{n}$.

Together with Böttcher’s theorem 3.6.1 it means that any superattracting point
$p$ of a holomorphic map $f$ has a distinguished Lyapunov function $\phi$ in a
neighborhood of $p$ such that the foliation on the level curves of $\phi$ is
$f$-invariant, and this foliation itself is a topological invariant of the
superattracting point $p$, which is preserved under topological conjugacy.

Let’s write a purely topological definition of this constraint on the
wandering set which is derived from the behavior of the holomorphic functions
in a neighborhood of an attracting singular point.

######  Definition 3.6.2.

An attracting singular point $p\in M$ is called _Böttcher attracting point_ of
an inner mapping $f\colon M\to M$, if there exists an open connected
neighborhood $V$ of $p$ and a Lyapunov function $\phi\colon
V\setminus\\{p\\}\to\mathbb{R}$ such that

  1. (1)

$|\deg f_{V\setminus\\{p\\}}|>1$;

  2. (2)

$\phi$ is monotonically decreases along the partial trajectories of $f$;

  3. (3)

Each level curve of $\phi$ is homeomorphic to a circle;

  4. (4)

foliation on the level curves of $\phi$ is $f$-invariant;

  5. (5)

For any point $x\in V\setminus\\{p\\}$, $O^{\perp}(x)$ is dense in the level
curve $\phi^{-1}(\phi(x))$.

This definition captures the key properties observed in holomorphic dynamics
near superattracting critical points, but formulated in a purely topological
context applicable to general inner mappings.

As we are only interested in the wandering set, an attracting singular point
can be replaced by an abstract attractor. In that case we get a definition of
_Böttcher component_ from [14]. The paper [14] studied conjugacy of Böttcher
components, but not their topological type.

However, these constraints imposed on the wandering set for Böttcher
components are too restrictive. In the absence of singular points, such
subsets of the wandering set belong to single conjugacy class [14]. Imposing
such strict limitations on the wandering set might limit the generality of the
analysis. Ideally, we would like to consider the questions raised in Section
3.4 in the most general case possible. Any restriction on the wandering set is
a technical step back. In this paper we take a step forward and lax the
condition 5 in Definition 3.6.2, so that $O^{\perp}(x)$ do not need to be
dense in the level curve. In that case we also do not need the condition 1 and
can allow homeomorphisms as well.

The obtained wandering set restriction is natural to call _Pseudo-Böttcher
components_ by analogy with Böttcher components. Its formal definition will be
given in Section 3.7 as Definition 3.7.1. Inner mappings with such restriction
already show an abundance of conjugacy classes in their wandering set [16].

###  3.7. Pseudo-Böttcher components.

From now on let $f\colon M\to M$ be an inner mapping of compact two-
dimensional manifold $M$ without boundary.

Let $A$ be an attractor such that its region of strict wandering $D$
(Definition 2.1.2) is connected totally invariant set. Then $D$ consists of
the only component of strict wandering $S$ (Definition 2.1.3) and
$S=D=B\setminus\cup_{n\geq 0}f^{-n}(A)$.

######  Definition 3.7.1.

$S$ is called _Pseudo-Böttcher_ component if there exists a strictly
attracting (Definition 2.1.1) neighborhood $U_{0}$ of the attractor $A$ such
that

  1. (1)

$A$ is generated by $U_{0}$, i.e. $A=\cap_{n\geq 0}f^{n}(U_{0})$;

  2. (2)

the open set $V_{0}=U_{0}\setminus A$ is connected;

  3. (3)

$V_{0}$ does not have singular points of $f$;

  4. (4)

There exists a continuous function $\tau_{0}\colon V_{0}\to(-\infty,0)$ such
that

    1. i)

$\tau_{0}$ is a trivial fiber bundle with fiber $S^{1}$; i.e. the foliation on
the level curves of $\tau_{0}$ is regular and any level curve
$\tau_{0}^{-1}(\lambda)$ is homeomorphic to a circle;

    2. ii)

$\forall x\in V_{0}$ $\tau(f^{n}(x))=\tau(x)-n$, $n\in\mathbb{N}$.

    3. iii)

$\tau_{0}$ is constant on neutral sections:

$\forall x\in V_{0}$ $\tau_{0}(O^{\perp}_{f}(x)\cap V_{0})=const$.

######  Lemma 3.7.2.

In the Pseudo-Böttcher component $S$ the foliation on level curves of
$\tau_{0}$ is invariant under the actions of $f$ and $f^{-1}$ restricted to
having images and preimages in $V_{0}$. In particular, the following equation
holds true: $\forall t\in(-\infty,0)$ $\forall n\geq 0$
$f^{n}(\tau_{0}^{-1}(t))=\tau_{0}^{-1}(t-n)$.

###### Proof.

Consider the level curves $\gamma_{t}=\tau_{0}^{-1}(t)$ and
$\gamma_{t-n}=\tau_{0}^{-1}(t-n)$. From the condition 4-4ii) of Definition
3.7.1,

$f^{n}(\gamma_{t})\subseteq\gamma_{t-n}$. $\gamma_{t}$ is closed set. Then
$f^{n}(\gamma_{t})$ also is closed set because $f^{n}$ is open closed map.

If $f^{n}(\gamma_{t})\not=\gamma_{t-n}$, then the image $f^{n}(\gamma_{t})$
has a boundary in $\gamma_{t-n}$. Let a point $f^{n}(x)$ be on the boundary of
$f^{n}(\gamma_{t})$ in $\gamma_{t-n}$. Then in local coordinates $f^{n}$ bends
an open segment neighborhood of $x$ in $\gamma_{t}$ that looks like
$(-\delta,\delta)$ to a half-open segment in $\gamma_{t-n}$ with $f^{n}(x)$ as
its boundary that looks like $[f^{n}(x),\varepsilon)$. Then $f^{n}$ is not a
local homeomorphism in $x$. But from the condition 3 of Definition 3.7.1,
$V_{0}\cap\operatorname{Sing}(f)=\emptyset$,
$\operatorname{Sing}(f^{n})=f^{-n}(\operatorname{Sing}(f))$ $\Rightarrow$
$V_{0}\cap\operatorname{Sing}(f^{n})=\emptyset$. It means that $f^{n}$ is a
local homeomorphism in $V_{0}$. We got a contradiction.

Therefore, $f^{n}(\gamma_{t})=\gamma_{t-n}$. As $\gamma_{t}=\tau_{0}^{-1}(t)$
and $\gamma_{t-n}=\tau_{0}^{-1}(t-n)$, we got the equation
$f^{n}(\tau_{0}^{-1}(t))=\tau_{0}^{-1}(t-n)$. ∎

######  Remark 3.7.3.

Since $M$ is compact, the condition 3 of Definition 3.7.1 is redundant because
the number of singular points of $f$ is at most finite and we can always
shrink $V_{0}$ to avoid them.

######  Remark 3.7.4.

From the condition 4-4ii) of Definition 3.7.1, $\forall x\in V_{0}$
$\tau_{0}(f(x))<\tau_{0}(x)$, which means that $\tau_{0}$ is a Lyapunov
function for the restriction $f|_{V_{0}}$.

###  3.8. Expansion of $\tau_{0}$ onto the Pseudo-Böttcher component.

Let $M$ be a compact connected 2-dimensional manifold without boundary. Let
$f\colon M\to M$ be an inner mapping of $M$. Let $S$ be a totally invariant
Pseudo-Böttcher component of strict wandering of $f$ (Definition 3.7.1).

######  Lemma 3.8.1.

There exists a function $\tau\colon S\to(-\infty,+\infty)$ such that

  1. i)

$\forall x\in V_{0}$ $\tau(x)=\tau_{0}(x)$.

  2. ii)

$\tau(f^{n}(x))=\tau(x)-n$, $n\in\mathbb{Z}$, i.e. partitioning on level
curves of $\tau$ is invariant under the actions of $f$ and $f^{-1}$.

  3. iii)

For any $c\in(-\infty,+\infty)$ the level curve $\tau^{-1}(c)$ is neutrally
saturated. As a result, $\tau$ is constant on neutral sections.

###### Proof.

There is already a function $\tau_{0}(x)$ on $V_{0}\subset S$ from Definition
3.7.1. Let’s expand $\tau$ from $V_{0}$ onto $S$ in 4 steps.

1) To satisfy the condition i), put $\forall x\in V_{0}$
$\tau(x)=\tau_{0}(x)$.

2) Put $\forall x\in V_{0}^{\perp}$ $\tau(x)=\tau_{0}(O^{\perp}(x)\cap
V_{0})$. This formula is well defined because $\tau_{0}$ is constant on
neutral sections in $V_{0}$ (Definition 3.7.1 4-4iii)). Note that in this step
the condition 4-4ii) of Defintiton 3.7.1 turns into the condition ii) of this
Lemma.

3) By continuity, put $\tau(x)=0$ $\forall
x\in\overline{V_{0}^{\perp}}\setminus V_{0}^{\perp}$. On this step the
condition ii) $\tau(f^{n}(x))=\tau(x)-n$ still hold.

4) Define $\tau$ onto $S$: if
$f^{-m}\left(f^{n}\left(x\right)\right)\in\overline{V_{0}^{\perp}}$,
$m,n\in\mathbb{Z}^{+}$ then

$\tau\left(f^{-m}\left(f^{n}\left(x\right)\right)\right)=\tau_{0}(x)+m-n.$
(3.2)

$S$ does not have periodic points of $f$ (if the basin of attraction $B$
contains a periodic point $p$ of $f$ then $p$ belongs to the attractor $A$ by
construction). Therefore, $\forall x\in S$
$O(x)/O^{\perp}(x)\approx\mathbb{Z}$. From the formula (3.2), it follows that
$\forall x\in S$ the function $\tau$ has the same value on $O^{\perp}(x)$,
i.e. the function $\tau$ is constant in $S$ on neutral sections. We can
consider the function $\tau$ as a function defined on equivalence classes of
the partitioning of $S$ by $O^{\perp}$. By definition, the function $\tau$ is
coordinated with the action of $f$ on $S$. Using commutative diagrams, we can
write it as
$\textstyle{O(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[O/O^{\perp}]}$$\scriptstyle{f}$$\textstyle{O(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[O/O^{\perp}]}$$\textstyle{S/O^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{f}$$\textstyle{S/O^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{f}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-1}$$\textstyle{\mathbb{Z}}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-1}$$\textstyle{\mathbb{R}}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-1}$$\textstyle{\mathbb{R}}$

It follows that for any point $x$ for any
$m_{1},n_{1},m_{2},n_{2}\in\mathbb{Z}^{+}$ such that both
$f^{-m_{1}}\left(f^{n_{1}}\left(x\right)\right)\in\overline{V_{0}^{\perp}}$,
and
$f^{-m_{2}}\left(f^{n_{2}}\left(x\right)\right)\in\overline{V_{0}^{\perp}}$,
$\tau$ has the same value in $x$. Thus, $\tau$ is well defined in $x$. The
function $\tau$ is defined in (3.2) as the composition of continuous
functions, thus for any $x\in S$ $\tau$ is continuous in $x$. The condition
ii) follows from the formula (3.2). The function $\tau$ is well defined in $S$
with the properties i)\- iii). This completes the proof. ∎

######  Corollary 3.8.2.

$f^{n}(\tau^{-1}(t))=\tau^{-1}(t-n)$.

##  4\. Base annulus and partitioning

Let $f\colon M\to M$ be an inner mapping of compact two-dimensional manifold
$M$ and $S\subset M$ be a totally invariant Pseudo-Böttcher component of
strict wandering.

This section lays the groundwork for understanding the topological structure
of $S$ by establishing a partitioning of $S$ into multiple connected compact
surfaces with boundary, called atoms.

This partitioning is analogous to the Birkhoff’s concept of ‘‘fundamental
neighborhoods’’ [1], used in studying homeomorphisms. However, ‘‘fundamental
neighborhoods’’ do not generalize easily to arbitrary non-invertible maps. In
our construction, we rely heavily on the properties of level curves of the
function $\tau_{0}$ from Definition 3.7.1.

We start from studying topology of single atoms. Then we focus on atom groups,
general ‘‘molecules’’ and special molecules called ‘‘chains’’, which are used
to study the action of $f$ on $S$. Later, we consider how this partitioning is
connected with the topology of $S$ as a whole.

These results will be instrumental in our analysis of $S$’s topology in
Section 5.

###  4.1. Partitioning of $V_{0}$.

Figure 6.  $V_{0}$, $V_{c}$ and $Q_{c}$.

Consider the neighborhood $V_{0}$ and the function $\tau_{0}\colon
V_{0}\to\mathbb{R}$ from Definition 3.7.1.

Define $V_{c}:=\tau_{0}^{-1}\left(\left(-\infty,c\right)\right)$ and
$Q_{c}:=\tau_{0}^{-1}\left(\left[c-1,c\right]\right)$ for any
$c\in(-\infty,0)$. $Q_{c}\subset\overline{V_{c}}$, $V_{c}\subset V_{0}$. By
Definition 3.7.1, $V_{c}$ and $Q_{c}$ are homeomorphic to the open and closed
annulus respectively.

Choose the orientation of $V_{c}$ towards the attractor $A$. As a closed
topological annulus, $Q_{c}$ is closed connected set. Its boundary $\partial
Q_{c}$ has 2 connected components: $\partial^{+}Q_{c}=\tau_{0}^{-1}(c)$ and
$\partial^{-}Q_{c}=\tau_{0}^{-1}(c-1)$. Each $\partial^{+}Q_{c}$ and
$\partial^{-}Q_{c}$ is homeomorphic to a circle. Let’s call
$\partial^{+}Q_{c}$ and $\partial^{-}Q_{c}$ _external_ and _internal_ part of
$\partial Q_{c}$.

$V_{c}$ is naturally partitioned with the images of $Q_{c}$ (Lemma 3.7.2):

$V_{c}=\cup_{n\in\mathbb{Z}^{+}}Q_{c-n}=\cup_{n\in\mathbb{Z}^{+}}f^{n}(Q_{c}).$

Note that not only $V_{0}$ with function $\tau_{0}$, but for any
$c\in(-\infty,0)$ $V_{c}$ with function $\tau_{c}=\tau_{0}+c$ also satisfy the
conditions of Definition 3.7.1. In particular, $U_{c}=V_{c}\cup A$ is a
strictly attracting neighborhood of the attractor $A$. Then $\forall x\in S$
$\exists m,n\in Z^{+}$: $f^{n}(x)\in Q_{c-m}\subset V_{c}$,
$Q_{c-m}=f^{m}(Q_{c})$ $\Rightarrow$ $\forall x\in S$ $O(x)\cap
Q_{c}\not=\emptyset$. Therefore,

$S=\cup_{l,m\geq 0}f^{-l}\left(Q_{c-m}\right)=\cup_{l,m\geq
0}f^{-l}\left(f^{m}\left(Q_{c}\right)\right).$ (4.1)

Conceptually, partitioning of $V_{0}$ is very easy to understand. We partition
$\mathbb{R}$ into segments $[c-1-n,c-n]_{n\in\mathbb{Z}}$ of length 1. Their
preimages with $\tau_{0}^{-1}$ become partitioning of $V_{0}$ into annuli. The
only complexity there is using a constant $c$. $c$ is required to offset the
boundaries of the annuli from the trajectories of singular points.

######  Lemma 4.1.1.

For each point $p\in\operatorname{Sing}(f)$, such that $O(p)\cap
V_{o}\not=\emptyset$, the function $\tau_{0}\colon V_{0}\to\mathbb{R}$ maps
the set $O(p)\cap V_{o}$ to a discrete set in $\mathbb{R}$.

###### Proof.

If $p\in\operatorname{Sing}(f)$, then $p\not\in V_{0}$. (Defintiton 3.7.1-3).
If $O(p)\cap V_{o}\not=\emptyset$, then there exists the smallest $m>0$ such
that $f^{m}(p)\in V_{o}$. From the conditions 4-4ii) and 4-4iii) of Defintiton
3.7.1, $\tau_{0}(O(p)\cap V_{o})=\\{\tau_{0}(f^{m}(p))-n|n\in\mathbb{N}\\}$. ∎

Let
$O\left(\operatorname{Sing}\left(f\right)\right):=\cup_{p\in\operatorname{Sing}(f)}O(p)$
be the full trajectory of the set of singular points. The set
$\operatorname{Sing}(f)$ is finite. Then

######  Corollary 4.1.2.

The function $\tau_{0}\colon V_{0}\to\mathbb{R}$ maps the set
$O\left(\operatorname{Sing}\left(f\right)\right)\cap V_{o}$ to a discrete set
in $\mathbb{R}$.

###  4.2. Introducing atoms.

Let’s select a constant $c$ for the set $Q_{c}$ such that
$c\not\in\tau_{0}\left(O\left(\operatorname{Sing}\left(f\right)\right)\right)$.
In that case, the boundary $\partial Q_{c}$ has no intersection with
trajectories of singular points.

Let’s call the selected $Q_{c}$ a _base annulus_ and build a partitioning of
$S$ based on $Q_{c}$.

######  Definition 4.2.1.

Connected components of the sets $f^{-l}(f^{m}(Q_{c}))$, where $l,m\geq 0$,
are called _atoms_.

Note that $f^{-l}(f^{m}(Q_{c}))\subset f^{-(l+1)}(f^{m+1}(Q_{c}))$. Thus, for
any given atom $A$ the numbers $l,m\geq 0$ are defined up to the difference
$m-l$.

######  Definition 4.2.2.

For an atom $A\subset f^{-l}(f^{m}(Q_{c}))$ the number $m-l$ is called $A$’s
_generation_.

Atoms defined by numbers $l,m$ do not change when defined by numbers
$l+k,m+k$. One way to prove it is to consider that the multivalue maps
$f^{-n}\circ f^{n}$ are the identity maps locally, as they might introduce new
preimages under $f^{-n}$, but they do not modify the existing preimages when
$n$ increases. Another way is to define atoms is to use the function $\tau$
from Lemma 3.8.1:

######  Definition 4.2.3.

Connected components of the sets $\tau^{-1}([c+n-1,c+n])$, $n\in\mathbb{Z}$
are called _atoms_ (of _generation_ $n$).

From the condition ii) of Lemma 3.8.1, Definitions 4.2.1 and 4.2.3 are
equivalent: any atom $A$ which is a connected component of
$f^{-l}(f^{m}(Q_{c}))$ is a connected component of $\tau^{-1}([c-n-1,c-n])$,
where $n=m-l$.

There exist a lot of different partitionings of $S$ that depend on

  1. (1)

the choice of the neighborhood $V$ and the function $\tau_{0}$ in the
definition 3.7.1 of Pseudo-Böttcher component.

  2. (2)

the choice of $c$ in $Q_{c}$, that defines the base annulus.

Moreover, those partitionings might be different topologically: for example,
if $S$ has 2 singular points sufficiently close, then there could exist one
partitioning where those 2 singular points belong to a single atom, and
another partitioning, where those 2 singular points belong to two different
atoms. However, partitionings are just a tool to study $S$. In the end, the
results will not depend on the choice of partitioning.

From now on, we fix the previously selected constant $c$, the base annulus
$Q_{c}$ and the corresponding partitioning of $S$ so that any atom we speak of
can only belong to the partitioning based on $Q_{c}$.

######  Definition 4.2.4.

For an atom $A$ define the _internal_ and _external_ parts of the boundary
$\partial A$ of atom $A$ respectively as $\partial^{-}A=\tau^{-1}(c-n-1)\cap
A$ and $\partial^{+}A=\tau^{-1}(c-n)\cap A$.

######  Definition 4.2.5.

Two atoms $A_{1}$ and $A_{2}$ are called _adjacent_ , if they have a non-empty
intersection of their boundaries.

By construction, a common part of the boundary of adjacent atoms is a part of
external boundary for one atom and a part of internal boundary for another.
This intersection of boundaries induces a natural equivalence relation: A
point $x$ on the external part of the boundary of one atom is equivalent to a
point $y$ on the internal part of the boundary of the other atom if $x$ and
$y$ are mapped into the same point by natural embedding of their atoms into
$S$.

######  Definition 4.2.6.

Let’s call this natural equivalence a _gluing_ of atoms.

###  4.3. Atom’s topology and the Riemann-Hurvitz formula.

Recall the Riemann-Hurvitz formula (see. [6]). Let $M_{1}$, $M_{2}$ be
surfaces and $h\colon M_{1}\to M_{2}$ be a branched covering. Then

$\chi(M_{1})=|\deg(h)|\cdot\chi(M_{2})-\sum_{{p\in\operatorname{Sing}(h)}}(e_{p}-1)$
(4.2)

where $\chi(M)$ is the Euler characteristic of a surface $M$.

Consider an atom $A$. By atom’s definition, $A=f^{-l}(f^{m}(Q_{c}))$ for a
$l,m\in Z^{+}$. Then $f^{l}(A)=f^{m}(Q_{c})=Q_{c-m}$. Therefore, $A$ is a
branched covering of the closed annulus $Q_{c-m}$ by $f^{l}$.

$\operatorname{Sing}(f^{l})=f^{-l}\left(\operatorname{Sing}\left(f\right)\right)\subset
O\left(\operatorname{Sing}\left(f\right)\right)$. By the choice of $c$,
$c\not\in\tau_{0}\left(O\left(\operatorname{Sing}\left(f\right)\right)\right)$.
Then $\partial A\cap\operatorname{Sing}(f^{l})=\emptyset$ and both
$\partial^{+}A$ and $\partial^{-}A$ have no intersection with the trajectories
of singular points of $f$ and $f^{l}$. Therefore, they are regular coverings
(by $f^{l}$) of $\partial^{+}Q_{c-m}$ and $\partial^{-}Q_{c-m}$ respectively.

Consider the map $f^{l}|_{A}\colon A\to Q_{c-m}$. The Euler characteristic
$\chi(Q_{c-m})$ for the annulus $Q_{c-m}$ is 0, so using the Riemann-Hurvitz
formula 4.2 we got the following formula and Lemma 4.3.1 as the consequence:

$\chi(A)=-\sum_{{p\in A\cap\operatorname{Sing}(f^{l})}}(e_{p}-1)$ (4.3)

######  Lemma 4.3.1.

If $A\cap\operatorname{Sing}(f^{l})=\emptyset$ then $A$ is regular covering of
$Q_{c-m}$, $\chi(A)=0$, and $A$ itself is also homeomorphic to a closed
annulus.

Otherwise, $\chi(A)<0$, $A$ is a planar set (a disc with holes), a two-
dimensional manifold with boundary. Each connected component of $\partial A$
is homeomorphic to a circle, and $\partial A$ has at least 3 or more connected
components (the exact number depends on the set
$\operatorname{Sing}(f^{l})\cap A$).

Consider the function $\tau$ from Lemma 3.8.1, restricted on $A$.

######  Lemma 4.3.2.

If $A\cap\operatorname{Sing}(f^{l})=\emptyset$, then
$\tau|_{A}:A\to([c+l-m-1,c+l-m])$ is a trivial fiber bundle with fiber
$S^{1}$.

###### Proof.

We will now prove that if $A\cap\operatorname{Sing}(f^{l})=\emptyset$, then
the restriction of $\tau$ to $A$ defines a trivial fiber bundle. According to
Lemma 4.3.1, $A$ is a regular covering of $Q_{c-m}$. Similar to the proof of
Lemma 3.7.2, we get $\forall t\in[c+l-m-1,c+l-m]$ $f^{l}(\tau^{-1}(t)\cap
A)=\tau_{0}^{-1}(t-l+m)$. Then $\tau^{-1}(t)\cap A\subset
f^{-l}(\tau_{0}^{-1}(t-l+m))$. $\tau_{0}^{-1}(t-l+m)\approx S^{1}$
$\Rightarrow$ $f^{-l}(\tau_{0}^{-1}(t-l+m))$ is a finite number of $S^{1}$.

Lamination $\mathcal{L}$ on level curves of $\tau$ on $A$ is a regular
covering of the regular foliation on level curves of $\tau_{o}$ on $Q_{c-m}$.
Therefore, $\mathcal{L}$ itself is regular foliation where each level curve is
homeomorphic to a finite number of $S^{1}$. Its boundaries $\partial^{+}A$ and
$\partial^{-}A$ consist of one $S^{1}$ each. By continuity, each level curve
in $\mathcal{L}$ is also homeomorphic to one $S^{1}$ which gives us the
statement of the Lemma. ∎

By the choice of $c$, the boundary of any atom belongs to a regular level
curve of the function $\tau$. Therefore, when restricted on the boundary, $f$
is a regular covering and connected components of the boundary are circles.
The number of connected components is finite because $f$ is a finite-to-one
map.

######  Definition 4.3.3.

An atom $A$ is called to be of _(boundary) type $(a,b)$_ if $\partial^{-}A$
consists of $a$ circles and $\partial^{+}A$ consists of $b$ circles.

This lemma establishes how the boundary type of an atom changes under $f^{n}$
depending on the presence or absence of singular points within the atom.

######  Lemma 4.3.4.

Let $f^{n}$ takes an atom $A_{1}$ of type $(a_{1},b_{1})$ to an atom $A_{2}$
of type $(a_{2},b_{2})$.

If the atom $A_{1}$ does not have singular points of $f^{n}$, then

1-1) if the atom $A_{2}$ is of type $(1,1)$, then $A_{1}$ is of type $(1,1)$
too;

1-2) if the atom $A_{2}$ is of type $(a_{2},b_{2})$, where $a_{2}\not=1$ or
$b_{2}\not=1$, then $A_{1}$ is of type $(a_{2},b_{2})$ too and $f^{n}$ is a
homeomorphism from $A_{1}$ to $A_{2}$.

If the atom $A_{1}$ does have singular points of $f^{n}$, then 2)
$a_{1}>a_{2}$, or $b_{1}>b_{2}$ or both.

The proof is by direct application of the Riemann-Hurvitz formula 4.2 and the
Lemma 4.3.1.

######  Corollary 4.3.5.

Let $f^{n}$ takes an atom $A_{1}$ to an atom $A_{2}$. If the atom $A_{1}$ does
not have singular points of $f^{n}$, then $A_{1}$ and $A_{2}$ have
homeomorphic partitionings on level curves of $\tau$.

###  4.4. Molecules.

######  Definition 4.4.1.

A connected subset of $S$ which is obtained by gluing different atoms together
is called _molecule_.

A molecule that consists of finite number of atoms is called a finite
molecule.

######  Lemma 4.4.2.

A finite molecule is a planar set.

###### Proof.

Assume the converse. Let a molecule $D$ be a surface with non-zero genus. It
does not matter whether the genus is oriented or not. The molecule $D$ is
finite. Then $\exists m,n\in Z$: $D\subset\tau^{-1}([m,n])$. According to the
Riemann-Hurvitz formula 4.2, a surface with genus can only cover a surface
with genus. Therefore, $\forall k\geq 0$ the molecule $f^{k(n-m)}(D)$ also is
a surface with genus. Then $S$ is a surface of infinite genus because $\forall
k\geq 0$ $f^{k(n-m)}(D)\subset S$. But $S$ is a part of compact oriented
surface $M$. This contradiction proves the Lemma. ∎

If an infinite molecule $E$ has a genus, then $E$ contains a finite
submolecule $D$ that has a genus, because both a handle and a Möbius strip are
compact sets. As a consequence, we got

######  Corollary 4.4.3.

An infinite molecule is a planar set.

######  Corollary 4.4.4.

Any atom disconnects $S$ into multiple connected components.

######  Lemma 4.4.5.

An atom $A$ and a molecule $B$ can only be adjacent with a single common
connected component of their boundary.

###### Proof.

Assume the converse. Suppose we consider the gluing of $A$ and $B$ using 2 or
more components of their boundary. But this will create a handle and a
resulting molecule would get a genus, contradicting Lemma 4.4.2. ∎

######  Lemma 4.4.6.

Let $B_{1}$, $B_{2}$ be molecules. Then the set $B_{1}\cap B_{2}$, if not
empty, is either molecule or a single common connected component of their
boundary.

The proof’s idea is the same as in Lemma 4.4.5.

###  4.5. A chain of atoms.

######  Definition 4.5.1.

Let $I$ be an interval in $\mathbb{Z}$. A sequence of atoms $(A_{i})_{i\in I}$
is called a _forward chain_ of atoms if $\forall$ $i\in I$
$\partial^{-}A_{i}\cap\partial^{+}A_{i+1}\not=\emptyset$.

By definition, adjacent atoms in the forward chain differ in generation by 1:
if an atom $A_{i}$ belongs to generation $k_{i}$, then the adjacent atom
$A_{i+1}$ belongs to generation $k_{i}+1$.

######  Definition 4.5.2.

A sequence of atoms $(A_{i})_{i\in I}$ (where $I$ is an interval in
$\mathbb{Z}$), such that for all $i\in I$
$\partial^{-}A_{i}\cap\partial^{+}A_{i-1}\not=\emptyset$, is called a
_backward chain_ of atoms.

By definition, a backward chain is a forward chain where atoms are enumerated
in reverse order. A chain where $I$ is finite interval in $\mathbb{Z}$ such as
$0\dots n$ we call a finite chain. A chain where $I$ is semi-infinite interval
such as $-\infty\dots 0$ we call a semi-infinite chain.

Atoms of the chain glued together create a molecule. We will refer to such a
molecule as a "chain" in contexts where the specific order of the atoms is not
important. To emphasize that a chain $(A_{i})_{i\in I}$ defines a molecule
$C$, we will write this as $C=\cup_{i\in I}A_{i}$.

######  Lemma 4.5.3.

Let ($A_{0}$, $A_{1}$) be a forward chain of atoms. If $f(A_{0})\cap
A_{1}\not=\emptyset$, then $f(A_{0})=A_{1}$.

###### Proof.

Let $A_{0}$ belongs to generation $k_{0}$ (Definition 4.2.2). By Definition
4.2.3, $A_{0}$ is a connected component of $\tau^{-1}([c-k_{0}-1,c-k_{0}])$.
Then $A_{1}$ belongs to generation $k_{0}+1$ and is a connected component of
$\tau^{-1}([c-k_{0}-2,c-k_{0}-1])$. From Corollary 3.8.2,
$f(\tau^{-1}([c-k_{0}-2,c-k_{0}-1]))=\tau^{-1}([c-k_{0}-1,c-k_{0}])$. By
continuity, $f$ maps $A_{0}$ onto a single connected component of
$\tau^{-1}([c-k_{0}-2,c-k_{0}-1])$. but $f(A_{0})\cap A_{1}\not=\emptyset$,
therefore, $f(A_{0})=A_{1}$. ∎

######  Lemma 4.5.4.

Let $(A_{i})$, $i=0\dots n$, be a forward chain of atoms. If
$f(A_{n-1})=A_{n}$, then $\forall i\in 1\dots n$ $f(A_{i-1})=A_{i}$.

###### Proof.

Let $A_{0}$ belongs to generation $k_{0}$. Then $\forall$ $i=0\dots n$ $A_{i}$
belong to generation $k_{0}+i$. By Definition 4.2.3, atoms of generation $k$
are connected components of $\tau^{-1}([c-k-1,c-k])$. By Lemma 3.7.2, $f$ maps
them onto atoms of generation $k+1$ (that are connected components of
$\tau^{-1}([c-k-2,c-k-1])$).

Prove the lemma by induction. The condition that $f(A_{n-1})=A_{n}$ is the
base of induction. Suppose $f(A_{m-1})=A_{m}$.

Let’s prove that $f(A_{m-2})=A_{m-1}$. Consider the atom $A_{m-2}$. Let
$\Delta=\partial^{+}A_{m-1}\cap\partial^{-}A_{m-2}$. $\Delta\subset A_{m-1}$,
$f(A_{m-1})=A_{m}$. Then $f(\Delta)\subset A_{m}$. But $\Delta\subset A_{m-2}$
$\Rightarrow$ $f(A_{m-2})\cap A_{m-1}\not=\emptyset$. By Lemma 4.5.3,
$f(A_{m-2})=A_{m-1}$.

By induction, if $f(A_{n-1})=A_{n}$ then for all $i=1\dots n$
$f(A_{i-1})=A_{i}$. This completes the proof. ∎

######  Definition 4.5.5.

An atom $A$ is called _main_ if the pair $A$ and $f(A)$ is a forward chain of
atoms. Otherwise it is called _auxiliary_.

Atoms contained in $\overline{V_{c}}$ are an example of main atoms.

######  Definition 4.5.6.

A chain that consists of main atoms is called _main_. A chain that consists of
auxiliary atoms is called _auxiliary_. Otherwise the chain is called _mixed_.

######  Lemma 4.5.7.

Let $A$ be a main atom. Then $f(A)$ is also a main atom.

###### Proof.

By definition of main atom, $(A,f(A))$ is a forward chain. Then
$\partial^{-}A\cap\partial^{+}f(A)\not=\emptyset$. Then
$f(\partial^{-}A\cap\partial^{+}f(A))=f(\partial^{-}A)\cap
f(\partial^{+}f(A))=\partial^{-}f(A)\cap\partial^{+}f(f(A))\not=\emptyset$.
Then $f(A)$ and $f(f(A))$ also create a forward chain. By definition, $f(A)$
is main. ∎

######  Corollary 4.5.8.

Let $C=\cup_{i=0\dots n}A_{i}$ be a main forward chain of atoms. Then $f(C)$
is also a main forward chain of atoms.

######  Lemma 4.5.9.

Let $A$ be a main atom. Then

  1. (1)

any atom $A_{i}$ adjacent to $A$ by $\partial^{+}A$ is main;

  2. (2)

for atoms adjacent to $A$ by $\partial^{-}A$, $f(A)$ is main and the rest is
auxiliary.

###### Proof.

(1) follows from Lemma 4.5.4. $\\{A,f(A),f(f(A))\\}$ is main forward chain by
Lemma 4.5.7. Consider an atom $B$ such that $B\not=f(A)$
$\partial^{-}A\cap\partial^{+}B\not=\emptyset$. From 4.4.4, $A$ split $S$ into
connected components. Denote the connected component of $S\setminus A$ that
contains $f(f(A))$ and $B$ as $K_{f(f(A))}$ and $K_{B}$ correspondingly. As
$A$ maps onto $f(A)$, then $B$ maps onto $f(f(A))$. Then $K_{B}$ maps onto
$K_{f(f(A))}$. By definition, $B$ is auxiliary. ∎

######  Corollary 4.5.10.

If the Pseudo-Böttcher component $S$ don’t have atoms of type $(k,*)$, where
$k>1$, then any atom of $S$ is main.

######  Lemma 4.5.11.

Let $\\{A_{n}\\}$, $n\geq 0$ be a main backward infinite chain such that the
first atom $A_{0}$ is of type $(1,*)$, but $\forall n>1$ $A_{n}$ are of type
$(a_{n},1)$, where $a_{n}>1$. Then $f$ has infinitely many singular points in
$S$.

###### Proof.

Consider $A_{1}$. $a_{1}>1$ $\Rightarrow$ $a_{1}$ at least 2. $A_{0}$ is of
type $(1,*)$. Then $f\colon A_{1}\to A_{0}$ is at least 2-list covering. By
Riemann-Hurvitz formula (4.2), $A_{1}$ must have a singular point.

Continue by induction. Suppose $f\colon A_{n-1}\to A_{n-2}$ is at least
$n$-list covering $\Rightarrow$ $f|_{\partial^{+}A_{n-1}}$ is at least
$n$-list covering. $A_{n-1}$ is of type $(a_{n-1},1)$ $\Rightarrow$
$\partial^{+}A_{n-1}$ is connected.

Consider $A_{n}$. Atoms $A_{n-1}$ and $A_{n}$ are adjacent.
$\partial^{+}A_{n-1}\subset\partial^{-}A_{n}$. $a_{n}>1$ $\Rightarrow$ $a_{n}$
at least 2. One component of $\partial^{-}A_{n}$ is exactly
$\partial^{+}A_{n-1}$, and there exists at least one other component of
$\partial^{-}A_{n}$, that does not belong to $A_{n-1}$. Then $f\colon A_{n}\to
A_{n-1}$ is at least $n+1$-list covering. $A_{n}$ is of type $(1,*)$. By
Riemann-Hurvitz formula (4.2), $A_{n}$ must have a singular point. By
induction, we got a singular point in each $A_{n}$, $n\geq 1$. Lemma is
proved. ∎

Note that on $n$-th step we got that $f$ is at least $n$-list covering.
Therefore, not only $f$ has infinitely many singular points, but also $f$ is
not a finite-to-one map. As we consider $f$ on the compact manifold $M$, both
those cases are forbidden.

######  Corollary 4.5.12.

Let the Pseudo-Böttcher component $S$ have singular points. Let $\\{A_{m}\\}$,
$m\geq 0$ be a main backward infinite chain. Then there exists $N>0$ such that
$\forall n\geq N$

  1. (1)

atoms $A_{n}$ are of type $(*,a_{n})$, where $a_{n}>1$.

  2. (2)

$f$ restricted on $A_{n}$ is homeomorphism.

###  4.6. Partitioning of the Pseudo-Böttcher component $S$.

######  Lemma 4.6.1.

There exists a sequence of nested connected molecules $\\{P_{c+n}\\}$ such
that $\cup_{n\geq 0}P_{c+n}=S$.

###### Proof.

$Q_{c}\subset V_{o}$ is selected to be a base annulus. $S$ is represented as
gluing of atoms: $S=\cup_{l,m\geq 0}f^{-l}(f^{m}((Q_{c}))$, $l,m\geq 0$
(Formula 4.1).

Let $V_{c}=\tau_{0}^{-1}((-\infty,c))$,
$\overline{V_{c}}=\tau_{0}^{-1}((-\infty,c])$.
$Q_{c}\subset\overline{V_{c}}\subset V_{0}$. $\overline{V_{c}}$ is molecule
because $\overline{V_{c}}=\cup_{n\geq 0}Q_{c-n}=\cup_{n\geq 0}f^{n}(Q_{c})$.

Let’s build a sequence $\\{P_{c+n}\\}$ by induction. Molecules
$P_{c+0}:=\overline{V_{c}}$ and $P_{c+-1}:=\overline{V_{c-1}}$ are the base of
induction.

By induction hypothesis, on $n$-th step we have a molecule $P_{c+n}$ such that
$f(P_{c+n})=P_{c+n-1}$, $\partial P_{c+n}\not=\emptyset$, $\partial
P_{c+n}\subset\tau^{-1}(c+n)$.

Consider the step of induction. Let $P_{c+n+1}$ be a connected component of
the set $\tau^{-1}([-\infty,c+n+1])$ that contains $P_{c+n}$. By construction,
$\partial P_{c+n+1}\subset\tau^{-1}(c+n+1)$. From connectivity and the fact
that $f(P_{c+n})=P_{c+n-1}$ we have $f(P_{c+n+1})=P_{c+n}$. Therefore, the
induction hypothesis is true for the induction step.

By induction, we got a sequence of sets $P_{c+n}$, $n\geq 0$. Their union the
set $K=\cup_{n\geq 0}P_{c+n}$ does not have a boundary and is a connected
component of $S$. If an atom sill exists such that it does not belong to $K$
then $S$ is not connected set, which contradicts the definition of $S$. Hence,
$K=S$ and the Lemma is proved. ∎

######  Lemma 4.6.2.

if $S$ does not have singular points of $f$ then $S$ is homeomorphic to an
open annulus (an annulus without boundary).

###### Proof.

Let’s prove Lemma using induction. Consider a sequence $\\{P_{c+n}\\}$ from
Lemma 4.6.1. The molecule $P_{c+0}:=\overline{V_{c}}$ is the base of
induction.

By induction hypothesis, on $n$-th step we have an open set $V_{c+n}$ and a
function $\tau_{c+n}:V_{c+n}\to(-\infty,0)$ such that $V_{c+n}$ with
$\tau_{c+n}$ satisfies the conditions of Definition 3.7.1, its closure
$\overline{V_{c+n}}$ is the molecule $P_{c+n}$ from Lemma 4.6.1, and boundary
$\partial\overline{V_{c+n}}$ is a circle.

Consider the step of induction. As $\partial\overline{V_{c+n}}$ is a circle,
there $\exists!$ atom $A_{c+n+1}$ adjacent to $\overline{V_{c+n}}$. Denote by
$\overline{V_{c+n+1}}$ the gluing of $\overline{V_{c+n}}$ and $A_{c+n+1}$. By
construction, $\overline{V_{c+n+1}}$ is connected component of the set
$\tau^{-1}([-\infty,c+n+1])$. Therefore, $\overline{V_{c+n+1}}=P_{c+n+1}$. Let
$V_{c+n+1}=\operatorname{Int}(\overline{V_{c+n+1}})$. By this Lemma’s
condition, $S\cap\operatorname{Sing}(f)=\emptyset$. Then from Lemma 4.3.1,
$A_{c+n+1}$ is homeomorphic to a closed annulus. Therefore,
$\overline{V_{c+n+1}}$ is homeomorphic to $\overline{V_{c+n}}$, and
$V_{c+n+1}$ is homeomorphic to $V_{c+n}$. Take the function $\tau$ from Lemma
3.8.1 and define a function $\tau_{c+n+1}:V_{c+n+1}\to(-\infty,0)$ as the
restriction of $\tau(x)+c+n+1$ on $V_{c+n+1}$. From Lemma 4.3.2, any level
curve of $\tau$ (and, by construction, of $\tau_{c+n+1}$) in $A$ is a circle.
Therefore, $V_{c+n+1}$ with the function $\tau_{c+n+1}$ also satisfies the
conditions of Definition 3.7.1, and induction hypothesis is true for the
induction step.

By induction, we got a sequence of sets $V_{c+n}$, $n\geq 0$. Their union the
set $K=\cup_{n\geq 0}V_{c+n}$ is homeomorphic to an open annulus and is a
connected component of $S$. If an atom sill exists such that it does not
belong to $K$ then $S$ is not connected set, which contradicts the definition
of $S$. Hence, $K=S$ and the Lemma is proved. ∎

######  Corollary 4.6.3.

If $S$ has singular points of $f$, then in the sequence $\\{P_{c+n}\\}$ from
Lemma 4.6.1 there exists $n\geq 1$ such that $P_{c+n-1}=\overline{V_{c+n-1}}$,
but the adjacent atom $A^{0}_{c+n}$ contains some singular points.

###### Proof.

If for any $k$ the adjacent atom $A^{0}_{c+k}$ has no singular points, then we
continue the process of expanding the structure in Definition 3.7.1 infinitely
and in the end we get $S$ without singular points. ∎

######  Definition 4.6.4.

Let $S$ has singular points. Let’s call

  * •

the molecule $\overline{V_{c+n-1}}$ the _main trunk_ ;

  * •

the atom $A^{0}_{c+n}$ from Corollary 4.6.3 with singular points of $f$ the
_main stump_ ;

  * •

the molecule $RT$ that consists of the main stump and any atom that can be
connected to the main stump with a forward chain the _main root_.

Figure 7.  Pseudo-Böttcher components with partitioning.

By definition, any atom $A\subset RT$ is some preimage of the main stump
$A^{0}_{c+n}$: $\exists m\geq 0$ $A\subset f^{-m}(A^{0}_{c+n})$.

Recall that by $X^{\perp}$ we denote the neutral saturation of the set $X$
(Definition 3.2.2).

######  Definition 4.6.5.

Let’s call the connected components (if any) of the set
$\overline{V_{c+n-1}}^{\perp}\setminus\overline{V_{c+n-1}}$ _auxiliary trunks_
, $(A^{0}_{c+n})^{\perp}\setminus A^{0}_{c+n}$ _auxiliary stumps_ , and the
set $RT^{\perp}\setminus RT$ the _auxiliary root_.

By definition, the main trunk, stump and root are not empty, while auxiliary
trunks, stumps and root can be empty sets. Note that decomposition of $S$ into
main and auxiliary trunks, stumps and root is defined only when $S$ has
singular points.

Figure 7 shows examples of partitioning of Pseudo-Böttcher components: a) $S$
without singular points. All atoms are annuli. b) $S$ with a singular points.
All atoms are main. The first non-annulus atom is the stump that ends the
trunk and begins the root. c) $S$ with a lot of singular points. The main
stamp has two. An auxiliary trunk has one. Note that further down outside of
picture window $S$ also has auxiliary trunks and roots and can have other
singular points.

######  Lemma 4.6.6.

If the main stump $A^{0}_{c+n}$ has only one singular point of $f$ with defect
$n_{p}-1$, then $A^{0}_{c+n}$ is either of type $(n_{p},1)$ or $(1,n_{p})$,
but not both.

###### Proof.

According to Stoilov Lemma(see 2.3.3), $f$ in $p$ is equivalent to the
holomorphic function $z^{n_{p}}$ up to homeomorphisms in neighborhoods of $p$
and $f(p)$. In particular, the regular level curve that pass through $f(p)$ in
a neighborhood of $f(p)$ gets $n_{p}$ preimages that intersect in $p$ in a
neighborhood of $p$. Then $p$ is also a singular point of the function $\tau$
and a singular point of the level curve $A^{0}_{c+n}\cap\tau^{-1}\tau((p))$.
Topologically, there is only one singular level curve with one singular point.
It is a wedge of circles.

By the choice of $c$, $c+n-1<\tau(p)<c+n$. Consider the partitioning on the
level curves of the function $\tau$ in the atom $A^{0}_{c+n}$. $f$ takes
$A^{0}_{c+n}$ onto $Q_{c+n-1}$. Any level curve of $\tau$ in $Q_{c+n-1}$ is a
circle. $f$ is a local homeomorphism in $A^{0}_{c+n}$ except for $p$.
Therefore, in the atom $A^{0}_{c+n}$ any level set of $\tau$ is a finite
number of circles except for the singular level set $\tau^{-1}(\tau(p))$.
Denote by $\gamma_{1}$ the connected component of the singular level set
$\tau^{-1}(\tau(p))$ that belongs to $A^{0}_{c+n}$. $f$ restricted to
$\gamma_{1}$ is a branched covering of a circle with the branch point $p$.
Therefore, $\gamma_{1}$ is a bouquet of $n_{p}$ circles. As an atom,
$A^{0}_{c+n}$ is a planar surface with boundary. $\gamma_{1}$ is embedded in
$A^{0}_{c+n}$. As a bouquet of $n_{p}$ circles, $\gamma_{1}$ divide $A_{c+n}$
and, correspondingly, $\partial A^{0}_{c+n}$ onto $n_{p}+1$ connected
components such that $n_{p}$ connected components of the boundary belong to
one level set of the function $\tau$ and one connected component belongs to
another level set of $\tau$. In other words, $A^{0}_{c+n}$ is either of type
$(1,n_{p})$ or of type $(n_{p},1)$.

Figure 8. $A^{0}_{c+n}$ for $n_{p}$=3.

As both cases are mutually exclusive, that completes the proof. ∎

######  Lemma 4.6.7.

if $S$ has only one singular point of $f$ with defect $n_{p}-1$ then
$n_{p}=|\deg(f)|$, and $S$ consists of the main trunk and the main root, where
the atoms of main root all have type $(1,n_{p})$.

###### Proof.

From Corollary 4.6.3, the singular point of $S$ is contained in the main stamp
$A^{0}_{c+n}$. From Lemma 4.6.6, $A^{0}_{c+n}$ is either of type $(n_{p},1)$
or $(1,n_{p})$, but not both. According to Lemma 4.3.4, since $n_{p}>1$,
$\forall m\geq 1$ any atom from $f^{-m}(A^{0}_{c+n})$ in absence of other
singular points is homeomorphic to $A^{0}_{c+n}$. In particular, they have the
same type. If $A^{0}_{c+n}$ is of type $(n_{p},1)$, then we got situation of
Lemma 4.5.11, which is forbidden. Therefore, $A^{0}_{c+n}$ is of type
$(1,n_{p})$ and all the atoms of main trunk is also of type $(1,n_{p})$.

As $f$ is an inner mapping of manifolds, $\deg(f)$ is constant in $M$ and
$|\deg(f)|$ coincide with the number of preimages in each point of
$M\setminus\operatorname{Sing}(f)$ [13]. As $S$ is totally invariant set, for
any point the number of preimages locally coincides with the number of
preimages globally and $n_{p}=|\deg(f)|$. ∎

Note that components of strict wandering with a single singular point
permitted in Lemma 4.6.7 do exist. Basin of attraction toward infinity in the
holomorphic map $z^{n_{p}}+1$ is the example.

##  5\. Topology of pseudo-Böttcher components of strict wandering

Let $S$ be a pseudo-Böttcher component of strict wandering. Since $S$ is an
open subset of the compact oriented surface $M$ then $S$ is non-compact
surface without boundary. Let’s find the topological type of surface $S$.

Non-compact surfaces are classified in papers [11, 8, 10, 9].

Let’s recall the corresponding definitions following the paper [10].

Let $K_{1}\supset K_{2}\supset K_{3}\supset\dots$ be a nested sequence of
connected unbounded regions in a surface $S$ such that the following holds:

  1. a)

the boundary of $K_{n}$ in $S$ is compact for all n;

  2. b)

for any bounded subset $A$ of $S$, $K_{n}\cap A=\emptyset$ for $n$
sufficiently large.

We say that two sequences $K_{1}\supset K_{2}\supset K_{3}\supset\dots$ and
$Q_{1}\supset Q_{2}\supset Q_{3}\supset\dots$ are equivalent if for any $n$
there is a corresponding integer $N$ such that $K_{N}\subset Q_{n}$ and vice
versa.

######  Definition 5.0.1.

The equivalence class of sequences containing $k=K_{1}\supset K_{2}\supset
K_{3}\supset\dots$ is called an end and denoted by $k^{*}$.

######  Definition 5.0.2.

Let $k^{*}$, represented by $k=K_{1}\supset K_{2}\supset K_{3}\supset\dots$ be
an end of $S$. We say that $k^{*}$ is planar and$\backslash$or orientable if
the sets $K_{n}$ are planar and$\backslash$or orientable for all sufficiently
large $n$.

######  Definition 5.0.3.

The ideal boundary (the set of ends) $B(S)$ of a surface $S$ is a topological
space having the ends of $S$ as elements and endowed with the following
topology: for any set $U$ in $S$ whose boundary in $S$ is compact, we define
$U^{*}$ to be the set of all ends $k^{*}$, represented by some $k=K_{1}\supset
K_{2}\supset K_{3}\supset\dots$ such that $K_{n}\subset U$ for a sufficiently
large $n$; we take the set of all such $U^{*}$ to be a basis for the topology
of $B(S)$.

A triplet $B(S)\supset B^{\prime}(S)\supset B^{\prime\prime}(S)$, where $B(S)$
is the whole ideal boundary, $B^{\prime}(S)$ is the part which is not planar,
and $B^{\prime\prime}(S)$ is the part which is not orientable, is a
topological invariant of $S$ up to a compact subsurface.

Let $S$ be a totally invariant pseudo-Böttcher component of strict wandering.
Since $S$ is embedded in the compact oriented surface $M$ then its genus (the
number of handles) is finite and
$B^{\prime}(S)=B^{\prime\prime}(S)=\emptyset$. Then. according to the paper
[11], the topological type of $S$ is defined by its genus and by the topology
of the set $B(S)$.

Let $\alpha_{n}=\tau^{-1}((-\infty,-n))$, $\omega_{n}=\tau^{-1}((n,\infty))$,
where the function $\tau$ is from Lemma 3.8.1. $\omega_{n}$ in a sense is the
neighborhood of repeller $R$, whereas $\alpha_{n}$ – neighborhood of the set
$A^{\perp}$. $\\{\tau^{-1}([-n,n])|n\in\mathbb{Z}\\}$ is a sequence of bounded
sets of $S$. Therefore, for any representation $(K_{n})$ of an end $k^{*}$ of
$S$ $(K_{n})$ is majorized either by $(\alpha_{n})$ or by $(\omega_{n})$.

######  Definition 5.0.4.

If the end $K$ is majorized by $(\alpha_{n})$ we say that $K$ belongs to the
set $AIB$ (Attractor-side Ideal Boundary).

######  Definition 5.0.5.

If the end $K$ is majorized by $(\omega_{n})$, we say that $K$ belongs to the
set $RIB$ (Repeller-side Ideal Boundary).

By definition, $B(S)=AIB\cup RIB$. Both sets $AIB$ and $RIB$ are not empty, as
we can build an end from each sequence $(\alpha_{n})$ and $(\omega_{n})$ by
choosing a connected component from $\alpha_{n}$ and $\omega_{n}$ for each n.
In particular, $AIB$ always contains an end $V^{*}$ generated by the sequence
$V_{0}\supset V_{-1}\supset...\supset V_{-n}\supset...$, where
$V_{i}=\tau_{0}^{-1}(-\infty,i)$ are neighborhoods of the attractor $A$.
$V_{0}$ by definition is homeomorphic to an open annulus. Therefore, the end
$V^{*}$ is an isolated point in the ideal boundary $B(S)$.

From the Lemma 4.4.5, if $S$ has no singular points of $f$ then $B(S)$
consists of two ends, one from $AIB$ and the other from $RIB$.

######  Lemma 5.0.6.

If $S$ has an atom of type $(k,*)$, $k>1$, then $AIB$ is a zero-dimensional
infinite set. If $S$ has an atom of type $(*,k)$, $k>1$, then $RIB$ is a zero-
dimensional infinite set.

###### Proof.

By definition, any open unbounded connected set in $S$ is a neighborhood in
$B(S)$ and contains at least one point in $B(S)$.

Consider the oriented Conrod-Reeb graph $\Gamma$ of the function $\tau$.
According to Lemma 4.4.2, $\Gamma$ has no cycles, therefore, $\Gamma$ is a
tree. An atom of type $(p,q)$ corresponds to a bounded area of $\Gamma$ with
$q$ edges in and $p$ edges out. Let us ignore atoms of type $(1,1)$ as they
correspond to simple segments of $\Gamma$. An atom of type $(p,q)$ splits
$\Gamma$ into $p+q$ subgraphs that correspond to open unbounded connected sets
of $S$. $q$ subgraphs, which $\tau$ maps to intervals of the form
$[const,\infty)$, correspond to non-empty neighborhoods in $RIB$, therefore,
to at least $q$ ends in $RIB$. Also, $p$ subgraphs, which $\tau$ maps to
intervals of the form $(-\infty,const]$, correspond to non-empty neighborhoods
in $AIB$, therefore, to at least $p$ ends in $AIB$. If we got $k$ atoms of
type $(p,*)$, we got at least $k(p-1)$ different ends in $AIB$. If we got $k$
atoms of type $(*,q)$, we got at least $k(q-1)$ different ends in $RIB$.
According to Lemma 4.3.4, if we got one atom $A$ of type $(p,*)$, $p>1$, then
each preimage of $A$ will be an atom of type $(p^{\prime},*)$, where
$p^{\prime}\geq p$. Similarly, for an atom $A$ of type $(*,q)$, $q>1$, each
preimage of $A$ will be an atom of type $(*,q^{\prime})$, where
$q^{\prime}\geq q$.

As any atom has infinitely many preimages in $S$, we got infinitely many ends
in $AIB$ if $p>1$ and in $RIB$ if $q>1$.

A topological space is zero-dimensional if every open cover of the space has a
refinement which is a cover by disjoint open sets. For $AIB$ and $RIB$ this
refinement is provided by connected components of $(\alpha_{n})$ and
$(\omega_{n})$ correspondingly. Lemma is proved. ∎

As a consequence of Lemmas 4.6.7, 5.0.6, we got

######  Lemma 5.0.7.

if $S$ has only one singular point of $f$ then $|AIB|=1$ and $|RIB|=\infty$.

######  Lemma 5.0.8.

Any end that belong to $AIB$ is an isolated point in $B(S)$.

###### Proof.

Select $N\in\mathbb{Z}$ such that
$N+c<\mathrm{min}_{p\in\operatorname{Sing}(f)}(\tau(p))$. Consider a connected
component $M$ of the set $\tau^{-1}((-\infty,N+c))$. $M$ is either $V_{N+c}$
or $\exists m\in\mathbb{N}$ $f^{m}(M)=V_{N-m+c}$, because $V_{N+c}$ is a
strictly attracting neighborhood of attractor. In the former case $V_{N+c}$ is
homeomorphic to an open annulus by definition. In the latter case $f^{m}$ is a
regular covering of $V_{N-m+c}$ by the choice of $N$. Therefore, by Riemann-
Hurvitz formula, $M$ is also an open annulus. $M$ is an unbound connected set
by definition. Therefore, as an open annulus, each $M$ represents both a
single end in $AIB$ and its open neighborhood in $B(S)$ which does not have
other ends of $S$. Lemma is proved. ∎

After we chose $Q_{c}$ and fixed atom decomposition of $S$ it is convenient to
use end’s representations that are consistent with atom decomposition.

######  Definition 5.0.9.

A representation $(W_{n})$ of an end $k^{*}\in RIB$, $k^{*}=W_{0}\supset
W_{1}\supset W_{2}\supset\dots$, is called _canonical_ , if $\forall i\geq 0$
$W_{i}$ is a connected component of the closed set
$\tau^{-1}\left(\left[c+i,\infty\right)\right)$ (a molecule).

By definition, for any end $k^{*}\in RIB$ its canonical representation is
fixed and only depend on the choice of $Q_{c}$.

######  Lemma 5.0.10.

If $S$ has singular points, then no end in $RIB$ is isolated.

###### Proof.

Let $k^{*}$ be an end in $RIB$ with the canonical representation $(W_{i})$.
Consider a forward chain $C_{0}=\cup_{i\leq 0}A_{i}$ which belongs to
$(W_{i})$.

Suppose $C_{0}$ is a main forward chain. From Corollary 4.5.12 $\exists N$
such that $\forall n\geq N$ atoms $A_{-n}$ are of type $(*,a_{-n})$, where
$a_{-n}>1$. Then $\forall n\leq N$ $W_{n}$ contains another chain different
from the subchain $C_{n}=\cup_{i\geq n}A_{-i}$. As the sequence $(W_{n})$ is
the base of topology of point $k^{*}$ in $B(S)$, it means that $k^{*}$ is not
isolated in $B(S)$. Moreover, $k^{*}$ is not isolated in $RIB$ as $W_{n}$ are
neighborhoods of $k^{*}$ in $RIB$.

Let $C_{0}$ be an auxiliary forward chain. Then by definition $\exists m>0$
such that $f^{m}(C_{0})$ is a main forward chain. $f^{m}$ induces an inner
mapping on $RIB$. It is proved above that the end $f^{m}(k^{*})$ that
corresponds to the main forward chain $f^{m}(C_{0})$ is not isolated in $RIB$.
As one of its preimages, $k^{*}$ is also is not isolated in $RIB$. Lemma is
proved. ∎

######  Lemma 5.0.11.

If $S$ has singular points, then $RIB$ is closed set in $B(S)$.

###### Proof.

Let $(k^{*}_{j})$ be a convergent sequence of ends in $RIB$ with the canonical
representations $(W^{j}_{i})$. Connected components of $(\omega_{n})$ provide
the base of topology in $RIB$. Therefore, for the sequence $(k^{*}_{j})$ to be
convergent their canonical representations $(W^{j}_{i})$ should stabilize,
i.e. $\forall i\geq 0$ $\exists J\geq 0$ $\forall j>J$ $W^{j}_{i}$ are the
same ($W^{j}_{i}=W^{stabilized}_{i}$). The stabilized sequence
$W^{stabilized}_{i}$ by construction is also a canonical representation of
some end $k^{*}\in RIB$. Then $k^{*}$ is the limit point of $(k^{*}_{j})$. ∎

######  Lemma 5.0.12.

If $S$ has singular points, then $RIB$ is the Cantor set in $B(S)$.

###### Proof.

According to Lemmas 5.0.10, 5.0.11 $RIB$ is closed set and has no isolated
points. Then $RIB$ is perfect space. According to Lemma 5.0.6, $RIB$ is zero-
dimensional. Therefore, $RIB$ is the Cantor set. ∎

As a consequence of Lemmas 4.6.2, 5.0.7, 5.0.8, 5.0.12, we got

######  Theorem 5.0.13.

$S$ is a planar set such that the ideal boundary of $S$ consists of two
subsets $AIB$ and $RIB$. Both $AIB$ and $RIB$ are not empty.

  1. (1)

If $S$ has no singular points of $f$ then both $AIB$ and $RIB$ consist of
single end;

  2. (2)

If $S$ has a single singular point of $f$ then $AIB$ consists of single end
and $RIB$ is a Cantor set;

  3. (3)

otherwise $RIB$ is a Cantor set and $AIB$ either consists of a single end
(punctured point) or consists of a countable set of isolated points.

##  6\. Note on general Pseudo-Böttcher components

######  Definition 6.0.1.

A connected component $K$ of a totally invariant set $R$ is called

  * •

_periodic_ , if $\exists n\geq 1$ $f^{n}(K)=K$;

  * •

_pre-periodic_ , if $\exists m\geq 1$ $f^{m}(K)$ is periodic;

  * •

_forever wandering_ otherwise.

######  Definition 6.0.2.

A region of strict wandering $D$ is called _Pseudo-Böttcher_ if for any
connected component $S_{i}$ of $D$ there exist $m\geq 0$ and $n\geq 1$ such
that the set $f^{m}(S_{i})$ is totally invariant Pseudo-Böttcher component
with respect to the map $f^{n}$. Connected components $S_{i}$ of $B$ are
called _Pseudo-Böttcher_.

By construction, $D$ is totally invariant set. However, by Definition 6.0.2,
Pseudo-Böttcher region of strict wandering $D$ can’t have wandering
components. Then any connected component of $D$ is either periodic or pre-
periodic. $f$ eventually maps pre-periodic ones onto periodic ones. Any
periodic component $S$ of period $k$ is invariant for $f^{k}$. Moreover, $S$
is totally invariant for the restriction $f^{k}|_{S}$. The fact that the ends
of $S$’s ideal boundary belong to the attractor and the repeller also holds
true for any power of $f$. Therefore the statements 4.6.2 and 5.0.13 also hold
true for any $S$ of $D$.

Consider a pre-periodic connected component $S$ of $D$ such that $\exists
l\geq 1$ $f^{l}(S)$ is periodic. $f^{l}$ is a branched covering of $M$.
According to Theorem 5.0.13, $f^{l}(S)$ does not have handles. Using the
Riemann-Hurvitz formula, we obtain that $S$ does not have handles too.
However, the Riemann-Hurvitz formula does not prohibit the case when $S$ has
more ends then $f^{l}(S)$.

For example, if the topological type of $f^{l}(S)$ is a sphere with 2
punctured points, then the topological type of $S$ can be a sphere with 3
punctured points, because that kind of branched covering is allowed.

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