# Renormalization of Unicritical Diffeomorphisms
of the Disk

Sylvain Crovisier, Mikhail Lyubich, Enrique Pujals, Jonguk Yang

###### Abstract.

We introduce a class of infinitely renormalizable unicritical diffeomorphisms
of the disk (with a non-degenerate “critical point”). In this class of
dynamical systems, we show that under renormalization, maps eventually become
Hénon-like, and then converge super-exponentially fast to the space of one-
dimensional unimodal maps. We also completely characterize the local geometry
of every stable and center manifolds that exist in these systems. The theory
is based upon a quantitative reformulation of the Oseledets-Pesin theory
yielding a unicritical structure of the maps in question comprising regular
Pesin boxes co-existing with “critical tunnels” and “valuable crescents”. In
forthcoming notes we will show that infinitely renormalizable perturbative
Hénon-like maps of bounded type belong to our class.

The first author was partially supported by the ERC project 692925 – NUHGD.
The second author was partly supported by the NSF, the Hagler and Clay
Fellowships, the Institute for Theoretical Studies at ETH (Zurich), and SLMSI
(formerly MSRI Berkeley). The fourth author was partially supported by SLMSI,
Institut Mittag-Leffler, and the thematic programs ‘Topological, smooth and
holomorphic dynamics, ergodic theory, fractals’ of the Simons Foundation Award
No. 663281 for IMPAN

###### Contents

  1. 1 Introduction
  2. 2 Presentation of the Main Concepts and Arguments
  3. 3 Quantitative Pesin Theory
  4. 4 Homogeneity
  5. 5 Critical Orbits
  6. 6 Regularity Near a Critical Orbit
  7. 7 Mild Dissipativity
  8. 8 Infinite Renormalizability
  9. 9 Regular Unicriticality
  10. 10 Total Ordering Inside a Unicritical Cover
  11. 11 Connected Components of the Limit Set
  12. 12 Recurrence of the Critical Orbit
  13. 13 Total Order on Renormalization Pieces
  14. 14 Regular Hénon-like Returns
  15. A Regular Unicriticality in 1D
  16. B Pliss Lemma

##  1\. Introduction

Critical points play a fundamental role in one-dimensional dynamics as a
natural source of non-linearity. They allow even very simple maps to have
extremely rich and fascinatingly complicated dynamical behaviors.

The simplest setting for studying the dynamical properties of critical points
is the class of unimodal maps. A unimodal map is a $C^{2}$-map $f:I\to I$ on
an interval $I\subset\mathbb{R}$ with a unique critical point $c_{0}\in I$. In
this paper, we always assume that $c_{0}$ is of quadratic type:
$f^{\prime}(c_{0})=0$ and $f^{\prime\prime}(c_{0})\neq 0$. We say that $f$ is
normalized if $c_{0}=0$ and $f(c_{0})=1$. Let $r\geq 2$ be an integer. The
space of normalized $C^{r}$-unimodal maps is denoted $\mathfrak{U}^{r}$.

The model examples of unimodal maps are given by the quadratic family:

$\mathfrak{Q}:=\\{f_{a}(x):=x^{2}+a\;|\;a\in\mathbb{R}\\}.$

The study of this family has led to some incredibly deep mathematics that
spans over two decades of research by numerous authors. At the heart of this
topic lies the celebrated renormalization theory of unimodal maps completed by
Sullivan, McMullen, Lyubich and Avila.

There have been several works that extend this theory to a higher dimensional
setting (see e.g. [CoEK], [GvST] and [dCLM]). However, the limitation of these
approaches is that they are all based on perturbative methods (relying on the
robustness of the hyperbolic one-dimensional renormalization operator). Hence,
they only apply to systems that are sufficiently close to the space of one-
dimensional unimodal maps.

The main goal of this paper is to develop a non-perturbative renormalization
theory of “unicritical” diffeomorphisms in dimension two. The most immediate
challenge is to identify what being “unicritical” should mean in this context.
Drawing motivations from the one-dimensional and the perturbative two-
dimensional cases, we define a critical point as a point whose local stable
and center manifolds are well-defined and form a quadratic tangency. Clearly,
any invariant set that feature such a point cannot be uniformly partially
hyperbolic. However, if the failure of uniform regularity is restricted to a
slow exponentially shrinking neighborhood of the backward orbit of the
critical point, then we say that the map is unicritical. One may think of this
condition as a relaxation of uniform partial hyperbolicity that allows for the
existence of a tangency between stable and center manifolds. The main result
of this paper is that renormalizations of two-dimensional unicritical
diffeomorphisms converge super-exponentially fast to the space of one-
dimensional unimodal maps.

###  1.1. Hénon-like maps

The two-dimensional analog of the quadratic family is given by the Hénon
family:

$\mathfrak{H}:=\\{F_{a,b}(x,y):=(x^{2}+a-by,x)\;|\;a,b\in\mathbb{R}\\}.$

We identify the degenerate Hénon map $F_{a,0}$ with $f_{a}$. This way,
${\mathfrak{H}}$ can be viewed as a two-dimensional extension of the quadratic
family.

As in the one-dimensional case, we would like to consider Hénon maps as the
model example of a more general class of maps. The simplest and the most
obvious generalization is given by Hénon-like maps. Let $V=I\times
I\subset\mathbb{R}^{2}$. A $C^{2}$-diffeomorphism $F:V\to F(V)\Subset V$ is
Hénon-like if it is of the form

$F(x,y)=(f(x,y),x)\hskip 14.22636pt\text{for}\hskip 14.22636pt(x,y)\in V,$

where for all $y\in I$, the map $x\mapsto f(x,y)$ is unimodal. We say that $F$
is a normalized Hénon-like map if $0\in V$, and $f(\cdot,0)$ is a normalized
unimodal map. The space of normalized $C^{r}$-Hénon-like maps is denoted
${\mathfrak{H}\mathfrak{L}}^{r}$.

For $\beta\in(0,1]$, we say that $F$ is $\beta$-thin (in $C^{r}$) if

$\|\operatorname{Jac}F\|=\|\partial_{y}f\|_{C^{r-1}}<\beta.$

The space of $\beta$-thin maps in ${\mathfrak{H}\mathfrak{L}}^{r}$ is denoted
${\mathfrak{H}\mathfrak{L}}^{r}_{\beta}$. We say that
$F\in{\mathfrak{H}\mathfrak{L}}^{r}_{\beta}$ is a perturbative Hénon-like map
if $\beta\ll 1$.

For any 1D map $f:I\to I$, define a degenerate 2D map $\iota(f):V\to V$ by

$\iota(f)(x,y):=(f(x),x).$ (1.1)

We denote ${\mathfrak{H}\mathfrak{L}}^{r}_{0}:=\iota({\mathfrak{U}}^{r})$.

###  1.2. Renormalizability

Let $F:\Omega\to F(\Omega)\Subset\Omega$ be a dissipative
$C^{r}$-diffeomorphism defined on a Jordan domain
$\Omega\Subset\mathbb{R}^{2}$ (i.e. $\Omega$ is a topological disk whose
boundary is a topological circle). We say that $F$ is renormalizable if there
exists an $R$-periodic Jordan subdomain $\mathcal{D}\Subset\Omega$ for some
integer $R>1$:

$F^{i}(\mathcal{D})\cap\mathcal{D}=\varnothing\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq i<R,\hskip
14.22636pt\text{and}\hskip 14.22636ptF^{R}(\mathcal{D})\Subset\mathcal{D}.$

The map $F$ is infinitely renormalizable if there exist a nested sequence of
of Jordan domains and an increasing sequence of natural numbers

$\Omega=:\mathcal{D}^{0}\Supset\mathcal{D}^{1}\Supset\ldots\hskip
14.22636pt\text{and}\hskip 14.22636pt1=:R_{0}<R_{1}<\ldots$

such that for $n\in\mathbb{N}$, the domain $\mathcal{D}^{n}$ is
$R_{n}$-periodic, and

$r_{n-1}:=R_{n}/R_{n-1}\geq 2.$

We refer to $\mathcal{D}^{n}$ as the $n$th renormalization domain. The primary
set of interest is the renormalization limit set, defined as

$\Lambda_{F}:=\bigcap_{n=0}^{\infty}\bigcup_{i=0}^{R_{n}-1}F^{i}(\mathcal{D}^{n}).$

The $n$th pre-renormalization of $F$ is the first return map to the domain
$\mathcal{D}^{n}$:

$p\mathcal{R}^{n}(F):=F^{R_{n}}|_{\mathcal{D}^{n}}.$

Let $\Psi^{n}:\mathcal{D}^{n}\to\Omega^{n}$ be an appropriately defined
“rescaling map” (non-linear in general) that turns $\mathcal{D}^{n}$ into a
domain $\Omega^{n}\subset\mathbb{R}^{2}$ of some “standard size.” The $n$th
renormalization of $F$ is then:

$\mathcal{R}^{n}(F):=\Psi^{n}\circ p\mathcal{R}^{n}(F)\circ(\Psi^{n})^{-1}.$

Since $F$ is assumed to be dissipative:
$\|\operatorname{Jac}F\|\leq\lambda<1,$ a heuristic computation shows that

$\|\operatorname{Jac}\mathcal{R}^{n}(F)\|\approx\|\operatorname{Jac}F^{R_{n}}\|\leq\lambda^{R_{n}}\rightarrow
0\hskip 14.22636pt\text{as}\hskip 14.22636ptn\to\infty.$

Thus, one expects that under renormalization, the dissipative two-dimensional
map $F$ becomes more and more one-dimensional. Our main goal is to rigorously
establish a sufficiently stronger version of this statement for the class of
maps that we will introduce below.

###  1.3. Mild dissipativity

The map $F$ is mildly dissipative if for any ergodic measure $\mu$ (not
supported on a hyperbolic sink), the strong-stable manifold $W^{ss}(p)$ of
$\mu$-almost every point $p\in\Omega$ is proper in the following sense: if
$W_{\Omega}^{ss}(p)$ is the connected component of $W^{ss}(p)\cap\Omega$
containing $p$, then $\overline{W^{ss}_{\Omega}(p)}$ is a Jordan arc whose
endpoints are contained in $\partial\Omega$. See Figure 1.

Figure 1. Proper strong-stable manifold $W^{ss}(p)$ in $\Omega$.

All Hénon maps $F_{a,b}$ with $|b|<1/4$ are known to be mildly dissipative
(see [DL] and [CP]). In [CPT], Crovisier-Pujals-Tresser give an explicit
description of the topological structure of mildly dissipative diffeomorphisms
of the disk with zero topological entropy. Specifically, they show that any
such map must either be infinitely renormalizable (of eventually period-
doubling type), or is a generalized Morse-Smale system.

###  1.4. Unique ergodicity

Let $F$ be infinitely renormalizable. Suppose that its renormalization limit
set $\Lambda_{F}$ has a unique invariant measure $\mu$ (this is always true if
$F$ is mildly dissipative, see Proposition 8.2). The Lyapunov exponents of $F$
with respect to $\mu$ are $0$ and $\log\lambda_{F}<0$ for some
$\lambda_{F}\in(0,1)$ (see Proposition 8.3). By the Oseledets theorem, the
tangent space at $\mu$-a.e. point $p\in\Lambda_{F}$ splits as
$E^{ss}_{p}\oplus E^{c}_{p}$, so that

$\lim_{n\to+\infty}\frac{1}{n}\log\|DF^{n}|_{E_{p}^{ss}}\|=\log\lambda_{F}\hskip
14.22636pt\text{and}\hskip
14.22636pt\lim_{n\to+\infty}\frac{1}{n}\log\|DF^{-n}|_{E_{p}^{c}}\|=0.$ (1.2)

Since $F$ is $C^{r}$ with $r>1$, it follows from (1.2) and the unique
ergodicity that $p$ has $C^{r}$-smooth strong-stable and center manifolds
$W^{ss}(p)$ and $W^{c}(p)$ respectively (see Subsection 3.6 and Proposition
4.6). The former is tangent to $E^{ss}_{p}$ and the latter to $E^{c}_{p}$. The
center manifold $W^{c}(p)$ is not uniquely defined. However, its $C^{r}$-jet
at $p$ is unique (see Proposition 3.20).

###  1.5. Regular unicriticality

Understanding the dynamics of smooth maps of the interval is centered around
the analysis of their critical points. Therefore, when studying smooth maps in
dimension two, it is natural to try and identify the “critical sets” of these
maps that play a similar role.

This strategy is implemented in the landmark paper of Benedicks-Carleson [BCa]
to prove the existence of strange attractors for a positive measure set of
Hénon maps. In their setting, the critical set is defined as a certain set of
tangencies between stable and unstable manifolds obtained as the limit of an
inductive procedure. This topic has since been further developed by numerous
authors, including Wang-Young [WYo], Palis-Yoccoz [PaYo] and Berger [Be].

A much more general (albeit less explicit) way to identify the critical set is
to characterize it as the obstruction locus for dominated splitting (see
Definition 5.1). This idea is explored by Pujals-Rodriguez Hertz in [PR], and
Crovisier-Pujals in [CP2].

In our paper, we are mainly interested in two-dimensional maps that are
infinitely renormalizable. It turns out that such maps can be unicritical—the
critical set can consist of a single point. The notion of the critical set we
use incorporates aspects from the two aforementioned bodies of work. The
novelty of our approach is the emphasis on the regularity of points that are
not “too close” to the critical orbit. Since the critical orbit is expected to
be dense in the renormalization limit set, this property is rather delicate to
formulate. The rigorous definitions are given below.

Let $F$ be an infinitely renormalizable map as considered in Subsection 1.4.

######  Definition 1.1.

The orbit of a point $p\in\Lambda_{F}$ is regular critical if $p$ admits a
direction

$E^{*}_{p}:=E^{ss}_{p}=E^{c}_{p}$

that satisfies both conditions in (1.2). The criticality is quadratic if,
additionally, $W^{ss}(p)$ and $W^{c}(p)$ have a quadratic tangency at $p$.

######  Definition 1.2.

For $N\in\mathbb{N}$; $L\geq 1$ and $\varepsilon\in(0,1)$, we say that the
point $p$ is $N$-times forward $(L,\varepsilon)$-regular if it admits a
direction $E_{p}$ such that

$\|DF^{n}|_{E_{p}}|\|\leq L\lambda_{F}^{\varepsilon n}\hskip
14.22636pt\text{for each}\hskip 14.22636pt1\leq n\leq N.$ (1.3)

For $t>0$ and $p\in\mathbb{R}^{2}$, we denote the ball

$\mathbb{D}_{p}(t):=\\{q\in\mathbb{R}^{2}\;|\;\operatorname{dist}(q,p)<t\\}.$

######  Definition 1.3.

For $0<\varepsilon<\delta<1$, we say that $F$ is
$(\delta,\varepsilon)$-regularly unicritical on the limit set $\Lambda_{F}$ if
the following conditions hold.

  1. i)

There is a regular quadratic critical orbit point $c_{1}\in\Lambda_{F}$
(referred to as the critical value).

  2. ii)

For all $t>0$, there exists $L(t)\geq 1$ such that for any $N\in\mathbb{N}$,
if

$p\in\Lambda\setminus\bigcup_{n=0}^{N-1}\mathbb{D}_{F^{-n}(c_{1})}(t\lambda_{F}^{\varepsilon
n}),$ (1.4)

then $p$ is $N$-times forward $(L(t),\delta)$-regular.

When $\delta$ and $\varepsilon$ are implicit, we simply say that $F$ is
regularly unicritical on $\Lambda_{F}$.

######  Remark 1.4.

By Proposition 6.5, the measure of the set of points satisfying (1.4) for all
$N\in\mathbb{N}$ goes to $1$ as $t$ goes to $0$. This shows that the condition
imposed by Definition 1.3 ii) is not vacuous. Moreover, by Proposition 9.22, a
regularly unicritical system has a unique critical orbit as the name implies.

######  Remark 1.5.

The analog of Definition 1.3 holds in the 1D case. See Appendix A.

######  Remark 1.6.

In the 1D case, an important motivation for the introduction of unimodal maps
is that a pre-renormalization of a unimodal map is again unimodal. Observe
that in the 2D case, regular unicriticality is likewise preserved under pre-
renormalization.

######  Remark 1.7.

In [CLPY2], we will show that any infinitely renormalizable
$C^{2}$-diffeomorphisms of bounded type whose renormalizations are eventually
contained in a $C^{1}$-compact subset of ${\mathfrak{H}\mathfrak{L}}^{2}$ is
regularly unicritical on its renormalization limit set.

###  1.6. Statements of the main results

Consider a Hénon-like map $G:V\to V$ defined on a square $V=I\times
I\subset\mathbb{R}^{2}$ containing $0$. Denote the critical point of
$g(\cdot):=G(\cdot,0)$ by $c\in I$. Let
$\mathbb{S}_{G}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the affine scaling map
given by

$\mathbb{S}_{G}(x,y):=|g(c)-c|^{-1}(x-c,y).$

Observe that $\mathbb{S}_{G}\circ G\circ\mathbb{S}_{G}^{-1}$ is normalized.

######  Main Theorem.

There exist universal constants $\varepsilon_{0},\alpha\in(0,1)$ such that the
following holds. For an integer $r\geq 2$, let $F:\Omega\to
F(\Omega)\Subset\Omega\Subset\mathbb{R}^{2}$ be a mildly dissipative,
infinitely renormalizable $C^{r}$-diffeomorphism with renormalization limit
set

$\Lambda_{F}:=\bigcap_{n=1}^{\infty}\bigcup_{i=0}^{R_{n}-1}F^{i}(\mathcal{D}^{n})$

Suppose $F$ is $(\varepsilon,\varepsilon^{1/\alpha})$-regularly unicritical on
$\Lambda_{F}$ for some $\varepsilon\in(0,\varepsilon_{0})$ with the critical
value $c_{1}$. Then for any $n\in\mathbb{N}$ sufficiently large, there exists
a $R_{n}$-periodic quadrilateral $\mathcal{V}^{n}$ containing $c_{1}$, and a
$C^{r}$-diffeomorphism

$\Psi^{n}:(\mathcal{V}^{n},c_{1})\to(V^{n},0)$

from $\mathcal{V}^{n}$ to a square $V^{n}=I^{n}\times
I^{n}\subset\mathbb{R}^{2}$ such that the following properties hold.

  1. i)

We have
$\Lambda_{F}\cap\mathcal{D}^{n}\Subset\mathcal{V}^{n}\Subset\mathcal{V}^{n-1}$.

  2. ii)

We have

$\|(\Psi^{n})^{\pm 1}\|_{C^{r}}=O(1)\hskip 14.22636pt\text{and}\hskip
14.22636pt\|\Psi^{n}-\Psi^{n-1}|_{\mathcal{V}^{n}}\|_{C^{r}}<\lambda^{(1-\varepsilon^{\alpha})R_{n}}.$

  3. iii)

Define the $n$th pre-renormalization and the renormalization of $F$ as

$F_{n}:=\Psi^{n}\circ F^{R_{n}}|_{\mathcal{V}^{n}}\circ(\Psi^{n})^{-1}\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{R}^{n}(F):=\mathbb{S}_{F_{n}}\circ
F_{n}\circ\mathbb{S}_{F_{n}}^{-1}$

respectively. Then $\mathcal{R}^{n}(F)$ is a $\delta_{n}$-thin $C^{r}$-Hénon-
like map with $\delta_{n}<\lambda^{(1-\varepsilon^{\alpha})R_{n}}$.

As a prerequisite for the proof of the Main Theorem, we obtain an extremely
detailed understanding of the regularity of $F$ at every point in
$\Lambda_{F}$ (not just on unspecified sets of large or full measure). As a
result, we are also able to explicitly describe the local geometry of every
strong stable and center manifolds that exist in the system. See Theorem 2.1
and the discussion in Subsection 2.2.

In a sequel to this paper ([CLPY2]), we prove a priori bounds for the class of
bounded type infinitely renormalizable $C^{r}$-diffeomorphisms of the disk
that satisfy the conclusion of the Main Theorem. As a consequence, we have the
following corollaries for the map $F$.

######  Corollary 1.8.

Suppose $F$ is of bounded type. Then the Hausdorff dimension of $\Lambda_{F}$
is strictly less than $1$. In particular, $\Lambda_{F}$ is totally
disconnected and minimal.

######  Corollary 1.9.

Suppose $F$ is of bounded type and at least $C^{4}$. Then the sequence of
renormalizations of $F$ given in the Main Theorem converge exponentially fast
to a compact subset of ${\mathfrak{H}\mathfrak{L}}^{r}_{0}$ that can be
identified with the renormalization horseshoe of 1D unimodal maps.

###  1.7. The boundary of chaos in the Hénon family

For $\beta\in(0,1]$, denote

${\mathfrak{H}}_{\beta}:=\\{F_{a,b}\in{\mathfrak{H}}\;|\;b\in[0,\beta)\\}\quad\text{and}\quad{\mathfrak{H}}_{0}:=\\{F_{a,0}\in{\mathfrak{H}}\\}.$

Denote by $\partial^{1}{\mathfrak{H}}_{\beta}$ the set of all maps in
${\mathfrak{H}}_{\beta}$ that have zero topological entropy and can be
approximated by Hénon maps with positive entropy (note that
$\partial^{1}{\mathfrak{H}}_{\beta}\subsetneq\partial{\mathfrak{H}}_{\beta}$).
We refer to $\partial^{1}{\mathfrak{H}}_{\beta}$ as the boundary of chaos in
${\mathfrak{H}}_{\beta}$. If $F_{a,b}\in\partial^{1}{\mathfrak{H}}_{1/4}$,
then $F_{a,b}$ is mildly dissipative and has zero entropy. Hence $F_{a,b}$ is
infinitely renormalizable by [CPT].

A priori, the boundary of chaos could be a highly pathological (disconnected)
continuum. However, in [dCLM], de Carvalho-Lyubich-Martens characterize this
set in the perturbative case ($\partial^{1}{\mathfrak{H}}_{\delta}$ with
$\delta\ll 1$) in the following way.

In the quadratic family, the transition from zero to positive entropy happens
precisely at the limit $f_{a_{*}}$ of period-doubling bifurcations. Moreover,
under renormalization, $\mathcal{R}^{n}(f_{a_{*}})$ converges to some
universal map $f_{*}\in{\mathfrak{U}}^{r}$ as $n\to\infty$. Since
${\mathfrak{H}}_{0}$ can be identified as the embedding of the quadratic
family under $\iota$ (see (1.1)), we have
$\partial^{1}{\mathfrak{H}}_{0}=\\{F_{a_{*},0}=\iota(f_{a_{*}})\\}$, and
$\mathcal{R}^{n}(F_{a_{*},0})$ converges to some universal degenerate
$0$-Hénon-like map $F_{*}=\iota(f_{*})$ as $n\to\infty$. By the hyperbolicity
of renormalization, it follows that there is a real analytic curve

$\\{F^{*}_{b}:=F_{a_{*}(b),b}\;|\;b\in[0,\delta)\\}$

of infinitely renormalizable Hénon maps of period-doubling type extending from
$F^{*}_{0}=F_{a_{*},0}$ such that $\mathcal{R}^{n}(F^{*}_{b})\to F_{*}$ as
$n\to\infty$. De Carvalho-Lyubich-Martens show that this curve coincides with
$\partial^{1}{\mathfrak{H}}_{\delta}$.

We conclude this subsection with the following motivating open problem.

######  Open Problem.

Characterize the boundary of chaos $\partial^{1}{\mathfrak{H}}_{\beta}$ for
some definite non-perturbative value of $\beta\in(0,1]$.

Conjecturally, $\partial^{1}{\mathfrak{H}}^{0}_{1}$ consists of infinitely
renormalizable Hénon maps that converge to the period-doubling renormalization
fixed point $F_{*}$ under renormalization. Note that by Remark 1.7, all such
maps must be regularly unicritical on its renormalization limit set. If the
conjeture holds, then one would be able to conclude that
$\partial^{1}{\mathfrak{H}}_{1}$ is a piecewise analytic curve. Towards this
open problem, we state the following corollary of the Main Theorem and
Corollary 1.9.

######  Corollary 1.10.

Let $r\geq 4$ be an integer. Consider a $C^{r}$-diffeomorphism $F:\Omega\to
F(\Omega)\Subset\Omega$ defined on a Jordan domain
$\Omega\Subset\mathbb{R}^{2}$. Suppose $F$ is mildly dissipative, infinitely
renormalizable and regularly unicritical on its renormalization limit set. If
$F$ has zero topological entropy, then $F$ is in the boundary of chaos. More
precisely, in any neighborhood of $F$ in the $C^{r}$-topology, there exists a
diffeomorphism with positive topological entropy.

###  1.8. Conventions

Unless otherwise specified, we adopt the following conventions.

Any diffeomorphism on a domain in $\mathbb{R}^{2}$ is assumed to be
orientation-preserving. The projective tangent space at a point
$p\in\mathbb{R}^{2}$ is denoted by $\mathbb{P}^{2}_{p}$.

We typically denote constants by $K\geq 1$, $k>0$ (and less frequently $C\geq
1$, $c>0$). Given a number $\kappa>0$, we use $\bar{\kappa}$ to denote any
number that satisfy

$\kappa<\bar{\kappa}<C\kappa^{D}$

for some uniform constants $C>1$ and $D>1$ (if $\kappa>1$) or $D\in(0,1)$ (if
$\kappa<1$). We allow $\bar{\kappa}$ to absorb any uniformly bounded
coefficient or power. So for example, if $\bar{\kappa}>1$, then we may write

$\text{``}\;\;10\bar{\kappa}^{5}=\bar{\kappa}\;\;\text{''}.$

Similarly, we use $\underline{\kappa}$ to denote any number that satisfy

$c\kappa^{d}<\underline{\kappa}<\kappa$

for some uniform constants $c\in(0,1)$ and $d\in(0,1)$ (if $\kappa>1$) or
$d>1$ (if $\kappa<1$). As before, we allow $\underline{\kappa}$ to absorb any
uniformly bounded coefficient or power. So for example, if
$\underline{\kappa}>1$, then we may write

$\text{``}\;\;\tfrac{1}{3}\underline{\kappa}^{1/4}=\underline{\kappa}\;\;\text{''}.$

These notations apply to any positive real number: e.g.
${\bar{\varepsilon}}>\varepsilon$, $\underline{\delta}<\delta$, ${\bar{L}}>L$,
etc.

Note that $\bar{\kappa}$ can be much larger than $\kappa$ (similarly,
$\underline{\kappa}$ can be much smaller than $\kappa$). Sometimes, we may
avoid the $\bar{\kappa}$ or $\underline{\kappa}$ notation when indicating
numbers that are somewhat or very close to the original value of $\kappa$. For
example, if $\kappa\in(0,1)$ is a small number, then we may denote
$\kappa^{\prime}:=(1-\bar{\kappa})\kappa$. Then
$\underline{\kappa}\ll\kappa^{\prime}<\kappa$.

For any set $X_{m}\subset\Omega$ with a numerical index $m\in\mathbb{Z}$, we
denote

$X_{m+l}:=F^{l}(X_{m})$

for all $l\in\mathbb{Z}$ for which the right-hand side is well-defined.
Similarly, for any direction $E_{p_{m}}\in\mathbb{P}^{2}_{p_{m}}$ at a point
$p_{m}\in\Omega$, we denote

$E_{p_{m+l}}:=DF^{l}(E_{p_{m}}).$

We use $n,m,i,j$ to denote integers (and less frequently $l,k$). Typically
(but not always), $n\in\mathbb{N}$ and $m\in\mathbb{Z}$. We sometimes use
$l>0$ for positive geometric quantities (such as length). The letter $i$ is
never the imaginary number.

We typically use $N,M$ to indicate fixed integers (often related to variables
$n,m$).

We use calligraphic font $\mathcal{U},\mathcal{T},\mathcal{I},$ etc, for
objects in the phase space and regular fonts $U,T,I,$ etc, for corresponding
objects in the linearized/uniformized coordinates. A notable exception is for
the invariant manifolds $W^{ss},W^{c}$.

We use $p,q$ to indicate points in the phase space, and $z,w$ for points in
linearized/uniformized coordinates.

##  2\. Presentation of the Main Concepts and Arguments

We now present dynamical and geometrical features of unicritical
diffeomorphisms that are developed in the following sections, and explain key
concepts for the proofs.

Let $r\geq 2$ be an integer. Consider a mildly dissipative, infinitely
renormalizable $C^{r}$-diffeomorphism $F:\Omega\to F(\Omega)\Subset\Omega$ of
a Jordan domain $\Omega\subset\mathbb{R}^{2}$, which is
$(\varepsilon,\underline{\varepsilon})$-regularly unicritical on its
renormalization limit set

$\Lambda:=\bigcap_{n=1}^{\infty}\bigcup_{i=0}^{R_{n}-1}F^{i}(\mathcal{D}^{n})$

Let $0$ and $\log\lambda<0$ be the Lyapunov exponents of $F$ with respect to
the unique invariant probability measure $\mu$ on $\Lambda$.

###  2.1. Regular blocks

Let $\varepsilon\in(0,1)$ be a constant such that ${\bar{\varepsilon}}<1$. We
may assume without loss of generality that $F$ is sufficiently homogeneous
(see Section 4).

For $L\geq 1$, we say that $p\in\Lambda$ is (infinitely) forward
$(L,\varepsilon)$-regular if it has a strong-stable direction
$E^{ss}_{p}\in\mathbb{P}^{2}_{p}$ such that

$\|DF^{n}|_{E_{p}^{ss}}\|<L\lambda^{(1-\varepsilon)n}\hskip
14.22636pt\text{for all}\hskip 14.22636ptn\in\mathbb{N}.$ (2.1)

Similarly, $p$ is (infinitely) backward $(L,\varepsilon)$-regular if it has a
center direction $E^{c}_{p}\in\mathbb{P}^{2}_{p}$ such that

$\|DF^{-n}|_{E_{p}^{c}}\|<L\lambda^{-\varepsilon n}\hskip 14.22636pt\text{for
all}\hskip 14.22636ptn\in\mathbb{N}.$ (2.2)

Lastly, we say that $p$ is Pesin $(L,\varepsilon)$-regular if (2.1) and (2.2)
hold, and $\measuredangle(E^{ss}_{p},E^{c}_{p})>1/L$. The constants $L$ and
$\varepsilon$ are referred to as a regularity factor and a marginal exponent
respectively.

If $p$ is forward/backward $(L,\varepsilon)$-regular, then the
forward/backward iterates of $F$ can be nearly linearized in a neighborhood
$\mathcal{U}_{p}$ of $p$, called a regular neighborhood at $p$, with
$\operatorname{diam}(\mathcal{U}_{p})=1/{\bar{L}}$ (see Theorem 3.6). If $p$
is Pesin $(L,\varepsilon)$-regular, then the local linearization can be done
for iterates of $F$ going in both directions (see Theorem 3.6).

The sets of all points $p\in\Lambda$ such that $p$ is infinitely forward,
backward and Pesin $(L,\varepsilon)$-regular are referred to as the forward,
backward and Pesin $(L,\varepsilon)$-regular blocks, and are denoted by
$\Lambda^{+}_{L,\varepsilon}$, $\Lambda^{-}_{L,\varepsilon}$ and
$\Lambda^{P}_{L,\varepsilon}$ respectively.

If $p\in\Lambda^{+/-}_{L,\varepsilon}$, then $p$ has a $C^{r}$ strong-
stable/center manifold $W^{ss/c}(p)$. The local strong-stable/center manifold
$W^{ss/c}(p)$ at $p$ is defined as the connected component of
$W^{ss/c}(p)\cap\mathcal{U}_{p}$ containing $p$. The $C^{r}$-geometry of
$W^{ss/c}_{\operatorname{loc}}(p)$ is bounded by ${\bar{L}}$ (see Proposition
3.21). Moreover, $W^{ss/c}_{\operatorname{loc}}(p)$ varies
$C^{1+\alpha}$-continuously on $p\in\Lambda^{+/-}_{L,\varepsilon}$ (see
Propositions 3.22).

By classical Oseledets-Pesin theory, we have
$\mu(\Lambda^{P}_{L,\varepsilon})\to 1$ as $L\to\infty$. However, for the
proof of the Main Theorem, we need a much more complete description of the
regular blocks of $F$ in $\Lambda$. This is provided by the following theorem,
which is proved in Subsection 9.3.

######  Theorem 2.1.

Suppose that $F$ is $(\varepsilon,\underline{\varepsilon})$-regularly
unicritical on its renormalization limit set $\Lambda$. Let
$\\{c_{m}:=F^{m}(c_{0})\\}_{m\in\mathbb{Z}}$ be the regular critical orbit of
$F$. Denote $\varepsilon^{\prime}:=(1-{\bar{\varepsilon}})\varepsilon$. Then
there exists a constant $\mathbf{t}\in(0,1)$ such that for any
$t\in(0,\mathbf{t}]$, we have

  1. i)

$\Lambda^{+}_{1/\underline{t},\varepsilon}\supset\displaystyle\Lambda\setminus\bigcup_{n=0}^{\infty}\mathbb{D}_{c_{-n}}(t\lambda^{\varepsilon^{\prime}n})$,

  2. ii)

$\Lambda^{-}_{1/\underline{t},\varepsilon}\supset\displaystyle\Lambda\setminus\bigcup_{n=1}^{\infty}\mathbb{D}_{c_{n}}(t\lambda^{\varepsilon^{\prime}n})$,
and

  3. iii)

$\Lambda^{P}_{1/\underline{t},\varepsilon}\supset\displaystyle\Lambda\setminus\bigcup_{m=-\infty}^{\infty}\mathbb{D}_{c_{m}}(t\lambda^{\varepsilon^{\prime}|m|})$.

###  2.2. Local geometry of the stable and center manifolds

Using Theorem 2.1, we are able to explicitly describe the local geometry of
every strong stable and center manifolds that exist in the system. This
description centers around the critical orbit $\\{c_{m}\\}_{m\in\mathbb{Z}}$,
and relies on two key results: Theorem 5.8 and Theorem 9.7. The first result
is about the local dynamics of $F$ near $\\{c_{m}\\}_{m\in\mathbb{Z}}$. The
second result is about the local geometry of $\Lambda$ near $c_{0}$.

In the following discussion, we designate the “vertical” direction as the
direction of strong contraction, and the “horizontal” direction as the neutral
direction (when such directions are well-defined). We also distinguish $c_{0}$
and $c_{1}$ as the critical point and the critical value (the choice of
$c_{0}$ within the critical orbit can be done arbitrarily).

Consider a cover of the critical orbit by slow exponentially shrinking disks

$c_{m}\in\mathbb{D}_{m}:=\mathbb{D}_{c_{m}}(\mathbf{t}\lambda^{\varepsilon^{\prime}|m|})\hskip
14.22636pt\text{for}\hskip 14.22636ptm\in\mathbb{Z},$ (2.3)

where $\mathbf{t}$ is given in Theorem 2.1. Theorem 5.8 states that under a
suitable change of coordinates, the dynamics of $F$ on these disks can be
uniformized in the following way. On $\mathbb{D}_{m}$ with $m\neq 0$, iterates
of $F$ become $C^{1}$-close to iterates under the linear map
$A(x,y)=(x,\lambda y)$. On $\mathbb{D}_{0}$, an application of $F$ uniformizes
to an application of the Hénon map $H(x,y)=(x^{2}-\lambda y,x)$. See Figure 2.
This is analogous to the one-dimensional case, where for a unimodal map $f$,
iterations away from the critical point is effectively linear, while at the
critical point, $f$ can be uniformized to the power map $P(x)=x^{2}$.

Figure 2. Uniformization of the local dynamics of $F$ near the critical orbit
$\\{c_{m}\\}_{m\in\mathbb{Z}}$.

In the uniformizing coordinates, the local strong stable manifold
$W^{ss}_{\operatorname{loc}}(c_{n})$ with $n\geq 1$ is a vertical line
segment, while the center manifold $W^{c}(c_{-m})$ with $m\leq 0$ is a
horizontal line segment. At the critical moment, $F$ maps straight horizontal
curves in $\mathbb{D}_{0}$ to quadratic vertical curves in $\mathbb{D}_{1}$,
and $F^{-1}$ maps straight vertical curves in $\mathbb{D}_{1}$ to quadratic
horizontal curves in $\mathbb{D}_{0}$. Observe that the curvature of
$W^{c}(c_{n})$ for $n\geq 1$ increases exponentially fast near $c_{n}$ under
forward iterates of $F$. Similarly, the curvature of $W^{ss}(c_{-n})$ for
$n\leq 0$ increases exponentially fast near $c_{-n}$ under backward iterates
of $F$.

Theorem 9.7 states that $\Lambda$ “pinches” at $c_{0}$. That is, if
$\Phi_{c_{0}}:\mathbb{D}_{0}\to\Phi_{c_{0}}(\mathbb{D}_{0})$ is the change of
coordinate on $\mathbb{D}_{0}$ that uniformizes the dynamics of $F$, then

$\Phi_{c_{0}}(\Lambda\cap\mathbb{D}_{0})\subset
T_{0}:=\\{(x,y)\;|\;|y|\leq|x|^{1/\varepsilon}\\}.$

Let $\mathcal{T}_{c_{0}}:=\Phi_{c_{0}}^{-1}(T_{0})$. It follows that
$\Lambda\cap\mathbb{D}_{m}$ is contained in a pinched region
$\mathcal{T}_{c_{m}}$ for all $m\in\mathbb{Z}$, so that for $n\geq 0$, we have

$F^{n}(\mathcal{T}_{c_{-n}})\subset\mathcal{T}_{c_{0}}\hskip
14.22636pt\text{and}\hskip
14.22636ptF^{-(1+n)}(\mathcal{T}_{c_{1+n}})\subset\mathcal{T}_{c_{0}}.$

We call $\mathcal{T}_{c_{-n}}$ the critical tunnel at $c_{-n}$, and
$\mathcal{T}_{c_{1+n}}$ the valuable crescent at $c_{1+n}$. We also show that
for all $m,l\in\mathbb{Z}$ with $|m|<|l|$ we have

$\mathcal{T}_{c_{m}}\cap\mathcal{T}_{c_{l}}=\varnothing\hskip
14.22636pt\text{or}\hskip
14.22636pt\mathcal{T}_{c_{m}}\supset\mathcal{T}_{c_{l}}.$

The generation of $\mathcal{T}_{c_{-n}}$ is the number of critical tunnels
that contain $\mathcal{T}_{c_{-n}}$. Similarly, the generation of
$\mathcal{T}_{c_{1+n}}$ is the number of valuable crescents that contain
$\mathcal{T}_{c_{1+n}}$.

Lastly, we introduce the vertical manifold

$W^{v}(c_{0}):=\Phi_{c_{0}}^{-1}(\\{(0,y)\;|\;y\in\mathbb{R}\\})\cap\mathbb{D}_{0}.$

For $n\geq 0$, define the vertical manifold $W^{v}(c_{-n})$ and the horizontal
manifold $W^{h}(c_{1+n})$ as the longest arcs in $\mathbb{D}_{-n}$ and
$\mathbb{D}_{1+n}$ containing $c_{-n}$ and $c_{1+n}$ respectively such that

$F^{n}(W^{v}(c_{-n}))\subset W^{v}(c_{0})\hskip 14.22636pt\text{and}\hskip
14.22636ptF^{-(1+n)}(W^{h}(c_{1+n}))\cap\mathbb{D}_{0}\subset W^{v}(c_{0}).$

One should consider $W^{v}(c_{-n})$ and $W^{h}(c_{1+n})$ as “stand ins” for
the strong stable and center manifolds for $c_{-n}$ and $c_{1+n}$
respectively. See Figure 3

Figure 3. Critical tunnels, valuable crescents, and vertical and horizontal
manifolds. The local geometry of the strong-stable manifold of
$p_{-i}\in\mathcal{T}_{c_{-i}}$ for $i\geq 0$ is initially distorted near the
critical orbit, but becomes vertical again under the backward dynamics of $F$.

We are now ready to describe the local geometry of all invariant manifolds
that exist in $\Lambda$. We begin with strong stable manifolds. Consider a
sufficiently small neighborhood $\mathcal{U}_{p}$ of a uniformly forward
regular point $p$. For $k\geq 0$, let $\mathbf{T}_{p}^{k}$ be the set of all
critical tunnels of generation $k$ contained in $\mathcal{U}_{p}$.

If $q\in\mathcal{U}_{p}$ is uniformly forward regular, then
$W^{ss}_{\operatorname{loc}}(q)$ is a proper straight vertical curve in
$\mathcal{U}_{p}$. Moreover, if $\mathcal{T}_{c_{-m}}\in\mathbf{T}_{p}^{0}$,
then $W^{v}(c_{-m})$ is also a proper straight vertical curve in
$\mathcal{U}_{p}$. If, on the other hand, $q$ is not uniformly forward
regular, then $q\in\mathcal{T}_{c_{-m}}$ for some
$\mathcal{T}_{c_{-m}}\in\mathbf{T}_{p}^{0}$. See Figure 4.

Figure 4. Stable lamination in a neighborhood $\mathcal{U}_{p}$ of a uniformly
forward regular point $p$. If $q\in\mathcal{U}_{p}$ is likewise uniformly
forward regular, then $W^{ss}_{\operatorname{loc}}(q)$ is proper and vertical
in $\mathcal{U}_{p}$. If $q$ is not uniformly forward regular, then
$q\in\mathcal{T}_{c_{-m}}$ for some $m\geq 0$. If $\mathcal{T}_{c_{-m}}$ is of
generation $0$, then $W^{v}(c_{-m})$ is also proper and vertical.

To complete our description, we need to understand what local strong stable
manifolds look like inside the critical tunnels of generation $0$. Towards
this end, let $p=c_{1}$ in the above description. For $k\geq 0$, applying
$F^{-1}$ increases the the generation of critical tunnels in
$\mathbf{T}_{c_{1}}^{k}$ by one:

$\mathbf{T}_{c_{0}}^{k+1}=\\{\mathcal{T}_{c_{-m-1}}\;|\;\mathcal{T}_{c_{-m}}\in\mathbf{T}_{c_{1}}^{k}\\}.$

Hence, we see that if $q_{0}\in\mathcal{U}_{c_{0}}$ is not contained in any
$\mathcal{T}_{c_{-m}}\in\mathbf{T}_{c_{0}}^{1}$, then
$q_{1}\in\mathcal{U}_{c_{1}}$ is uniformly forward regular, and
$W^{ss}_{\operatorname{loc}}(q_{1})$ is a proper straight vertical curve.
Thus, we conclude that

$W^{ss}_{\operatorname{loc}}(q_{0})\subset
F^{-1}(W^{ss}_{\operatorname{loc}}(q_{1}))\cap\mathcal{T}_{c_{0}}$

is a subarc of a quadratic horizontal curve (proper in $\mathcal{T}_{c_{0}}$).
See Figure 3. Pulling back under the dynamics of $F$, we obtain the local
stable manifolds of every point contained in a critical tunnel of generation
$0$, but not contained in a critical tunnel of generation $1$.

This observation also explains the mechanism by which poor forward regularity
can improve under the dynamics. By uniformization, we see that iterating by
$F^{-1}$ on $\mathcal{T}_{c_{0}}$ is effectively iterating by the linear map
$A^{-1}(x,y)=(x,\lambda^{-1}y)$. Under this strong vertical expansion, subarcs
of quadratic horizontal curves in $\mathcal{T}_{c_{0}}$ become straighter and
more vertical. So for $q_{0}\in\mathcal{T}_{c_{0}}$ not contained in any
critical tunnels of higher generation, at the moment $-n$ when
$q_{-n}\notin\mathcal{T}_{c_{-n}}$, the forward regularity at $q_{-n}$ is
fully recovered to be once again uniform. See Figure 3.

Figure 5. Local strong-stable manifold contained in a critical tunnel
$\mathcal{T}_{c_{-m}}\subset\mathcal{T}_{c_{0}}$ of generation $1$ (which is
the preimage of the critical tunnel
$\mathcal{T}_{c_{-m+1}}\subset\mathcal{T}_{c_{1}}$ of generation $0$).

Proceeding by induction, we obtain a description of the local geometry of the
strong stable manifold of every point in $\Lambda$ that is contained in at
most finitely many critical tunnels. See Figure 5. We conclude by observing
that for $q\in\Lambda$, the deeper the nest of critical tunnels which contains
$q$, the worse the forward regularity of $F$ at $q$ becomes. Consequently, the
forward regularity of $F$ deteriorates completely at points contained in
infinite nests of critical tunnels, and the portions of their stable manifolds
that have controlled geometry get reduced to nothing more than the points
themselves. The set $\Delta_{c_{0}}^{c}$ of all such points has zero measure
(see Proposition 6.11), and is referred to as the critical dust of $c_{0}$.

Replacing forward regularity with backward regularity, vertical with
horizontal, and critical tunnels with valuable crescents in the above
discussion, we obtain a description of the local geometry of the center
manifold of every point in $\Lambda$ except those contained in infinitely many
valuable crescents. These exceptional points have completely deteriorated
backward regularity. They form a zero-measure set $\Delta_{c_{1}}^{v}$
referred to as the valuable dust of $c_{1}$.

###  2.3. Summary of the proof of the main theorem

For $i\in\\{0,1\\}$, let
$\Phi_{c_{i}}:(\mathbb{D}_{i},c_{i})\to(\Phi_{c_{i}}(\mathbb{D}_{i}),0)$ be
the change of coordinates on $\mathbb{D}_{i}$ given in (2.3) that uniformizes
the local dynamics of $F$ along the critical orbit
$\\{c_{m}\\}_{m\in\mathbb{Z}}$. In particular,

$\Phi_{c_{1}}\circ F\circ\Phi_{c_{0}}^{-1}(x,y)=(x^{2}-y,x)\hskip
14.22636pt\text{for}\hskip 14.22636pt(x,y)\in\Phi_{c_{0}}(\mathbb{D}_{0}).$
(2.4)

In these coordinates, $W^{ss}_{\operatorname{loc}}(c_{1})$ becomes a genuine
vertical segment, and $W^{c}(c_{0})$ becomes a genuine horizontal segment.

For $n\geq 1$, let $\mathcal{D}^{n}$ be the renormalization domain of depth
$n$ that contains $c_{1}$. A renormalization piece of depth $n$ is the set

$\Lambda^{n}_{i}:=\Lambda\cap F^{i-1}(\mathcal{D}^{n})\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq i<R_{n}.$

Note that $\Lambda^{n}_{R_{n}+1}=\Lambda^{n}_{1}$. Hence,
$\Lambda^{n}_{R_{n}}\ni c_{0}$. We show that
$\displaystyle\bigcap_{n=1}^{\infty}\Lambda^{n}_{1}=\\{c_{1}\\}$ (see Theorem
12.2). To simplify our discussion, we will assume that $\Lambda$ is totally
disconnected (to make the argument work in the general case, we prove that any
non-trivial connected component of $\Lambda$ is a $C^{r}$-curve that coincides
with the center manifold, see Theorem 11.2).

The main tool used in the proof of the Main Theorem is a structure called a
unicritical cover $\mathcal{C}$ of $\Lambda$. It is defined as follows. For
any uniformly Pesin regular point $p\in\Lambda$, one can define a regular
chart $\Phi_{p}:(\mathcal{B}_{p},p)\to(B_{p},0)$, which is a
$C^{r}$-diffeomorphism from a quadrilateral $\mathcal{B}_{p}\ni p$ (called a
regular box at $p$) to a rectangle $B_{p}$ with $\|\Phi_{p}^{\pm
1}\|_{C^{r}}=O(1)$, that satisfies the following properties:

  * •

The local strong stable manifold $W^{ss}_{\operatorname{loc}}(p)$ and the
center manifold $W^{c}(p)$ map to genuine vertical and horizontal line
segments respectively.

  * •

If there is another uniformly Pesin regular point $p^{\prime}\in\Lambda$ such
that $F(\mathcal{B}_{p})\cap\mathcal{B}_{p^{\prime}}\ni q$, then a horizontal
direction at $F^{-1}(q)$ (with respect to $\Phi_{p}$) is sent to a
sufficiently horizontal direction at $q$ (with respect to $\Phi_{p^{\prime}}$)
under $DF$. Similarly, a vertical direction at $q$ is sent to a sufficiently
vertical direction at $F^{-1}(q)$ under $DF^{-1}$.

The study of uniformly (partially) hyperbolic systems is greatly facilitated
by the fact that the invariant set can be entirely covered by a finite set of
regular boxes, which endows the system with what is referred to as a local
product structure.

In our setting, any union of regular boxes leaves out points in $\Lambda$ that
are too close to the critical orbit. To completely cover $\Lambda$, we include
critical tunnels and valuable crescents defined in Subsection 2.2. This leads
to the following definition of a unicritical cover of $\Lambda$:

$\mathcal{C}:=\\{\mathcal{B}_{p}\\}_{p\in\Lambda^{P}_{L,\varepsilon}}\cup\\{\mathcal{T}_{c_{m}}\\}_{m\in\mathbb{Z}}$

(where $L\geq 1$ is some sufficiently large regularity constant). By the
compactness of $\Lambda$, we may extract a finite subcover
$\mathcal{C}_{0}\subset\mathcal{C}$. If $\Lambda$ is totally disconnected,
then for any $n\geq N$ and $i\in\mathbb{N}$, there exists
$Q\in\mathcal{C}_{0}$ such that $\Lambda^{n}_{i}\Subset Q$.

For the Main Theorem, we need to ensure that a unicritical cover
$\mathcal{C}_{0}$ of $\Lambda$ for $F$ can fulfill a similar function as local
product structures for uniformly (partially) hyperbolic systems. This requires
the full use of our understanding of the forward and backward regularity of
$F$ on $\Lambda$ summarized in Subsection 2.2.

For each $Q\in\mathcal{C}_{0}$, we endow $\Lambda\cap Q$ with a total order
induced by the strong-stable lamination as follows. First consider the case
when $Q$ is either a regular box $\mathcal{B}_{p}$ for some uniformly Pesin
regular point $p\in\Lambda^{P}_{L,\varepsilon}$, or a valuable crescent
$\mathcal{T}_{c_{n}}$ for some $n\in\mathbb{N}$. If
$q\in\Lambda^{+}_{L,\varepsilon}\cap Q$ is a uniformly forward regular point,
then $W^{ss}_{\operatorname{loc}}(q)$ is a proper straight vertical curve in
$Q$. Hence, the strong-stable lamination clearly induces a total order on
$\Lambda^{+}_{L,\varepsilon}\cap Q$ (say, their order of appearance from left
to right).

Next, consider the case where $Q$ is a critical tunnel $\mathcal{T}_{c_{-m}}$
for some $m\geq 0$. Let $\Lambda^{+}(\mathcal{T}_{c_{-m}})$ be the set of all
points $q\in\mathcal{T}_{c_{-m}}$ such that $\mathcal{T}_{c_{-m}}$ is the
tunnel of the largest generation containing $q$ (i.e. there is no deeper
tunnel $\mathcal{T}_{c_{-k}}\subset\mathcal{T}_{c_{-m}}$ with $k>m$ containing
$q$).

If $q_{0}\in\Lambda^{+}(\mathcal{T}_{c_{0}})$, then its image
$q_{1}\in\mathcal{T}_{c_{1}}$ is not contained in any critical tunnel. Hence,
$q_{1}\in\Lambda^{+}_{L,\varepsilon}$, and
$W^{ss}_{\operatorname{loc}}(q_{1})$ is a proper straight vertical curve in a
neighborhood $\mathcal{U}_{c_{1}}$ of $c_{1}$. This means that, in particular,
the component $W^{ss}_{\mathcal{T}_{c_{1}}}(q_{1})$ of
$W^{ss}_{\operatorname{loc}}(q_{1})\cap\mathcal{T}_{c_{1}}$ containing $q_{1}$
is proper in $\mathcal{T}_{c_{1}}$. Define

$W^{ss}_{\operatorname{loc}}(q_{0}):=F^{-1}(W^{ss}_{\mathcal{T}_{c_{1}}}(q_{1})).$

It is easy to see that the strong-stable lamination

$\\{W^{ss}_{\operatorname{loc}}(q)\;|\;F(q)\in\Lambda^{+}_{L,\varepsilon}\cap\mathcal{T}_{c_{1}}\\}$

induces a total order on $\Lambda^{+}(\mathcal{T}_{c_{0}})$. Pulling this back
by the dynamics, we also obtain a total order on the set
$\Lambda^{+}(\mathcal{T}_{c_{-m}})$ for any $m\geq 0$.

We now explain how these total orders on separate subsets of $\Lambda$ fit
together to form a single total order on $\Lambda\cap Q$ for any
$Q\in\mathcal{C}_{0}$. Suppose $Q$ is either a regular box or a valuable
crescent. Let $\mathbf{T}_{Q}^{0}$ be the set of critical tunnels in $Q$ of
generation $0$. For $\mathcal{T}_{c_{-m}}\in\mathbf{T}_{Q}^{0}$, we consider a
suitable thin proper nearly-vertical strip $S_{c_{-m}}$ that contains
$\mathcal{T}_{c_{-m}}$. Then the collection

$\\{W^{ss}_{\operatorname{loc}}(q)\;|\;q\in\Lambda^{+}_{L,\varepsilon}\cap
Q\\}\cup\\{S_{c_{-m}}\;|\;\mathcal{T}_{c_{-m}}\in\mathbf{T}_{Q}^{0}\\}$

is pairwise disjoint. Hence, the total order on
$\Lambda^{+}_{L,\varepsilon}\cap Q$ can be extended to include
$\Lambda^{+}(\mathcal{T}_{c_{-m}})$ with
$\mathcal{T}_{c_{-m}}\in\mathbf{T}_{Q}^{0}$.

So far, for any $Q\in\mathcal{C}_{0}$, we have a total order on the set of all
points in $\Lambda\cap Q$ that are contained in at most one deeper critical
tunnel $\mathcal{T}_{c_{-m}}\subset Q$. Proceeding by induction, we obtain a
total order on the set of all points in $\Lambda\cap Q$ contained in at most
finitely many critical tunnels. In other words, we have a total order on the
set $(\Lambda\setminus\Delta_{c_{0}}^{c})\cap Q$, where $\Delta_{c_{0}}^{c}$
denotes the critical dust of $c_{0}$ defined in Subsection 2.2. Since the
intersection of any infinite nest of critical tunnels is a singleton, the
total order extends to $\Delta_{c_{0}}^{c}\cap Q$ as well. For more details,
see Section 10.

Now that we have the unicritical cover structure $\mathcal{C}_{0}$ with well-
defined total orders on its elements, we are able to give a general outline of
the proof of the Main Theorem. For $n\geq N$ with $N\in\mathbb{N}$
sufficiently large, let $\mathcal{V}_{1}^{n}$ be a suitable thin tubular
neighborhood of $W^{ss}_{\operatorname{loc}}(c_{1})$ in $\mathbb{D}_{1}\ni
c_{1}$ such that $\mathcal{V}_{1}^{n}\supset\Lambda_{1}^{n}$. Denote
$\mathcal{V}_{i}^{n}:=F^{i-1}(\mathcal{V}_{1}^{n})$ for $1\leq i<R_{n}$. We
have $\mathcal{V}^{n}_{R_{n}}=F^{R_{n}-1}(\mathcal{V}^{n}_{1})\ni c_{0}$. We
need to show that under $F^{R_{n}-1}$:

  1. 1)

There is a compression of $\mathcal{V}^{n}_{1}$ in the vertical direction by a
factor of $\lambda^{(1-{\bar{\varepsilon}})R_{n}}$.

  2. 2)

A horizontal foliation over $\mathcal{V}^{n}_{1}$ is mapped to a horizontal
foliation close to $W^{c}(c_{0})$.

Then by applying $F$ one more time in a neighborhood $\mathcal{U}_{c_{0}}$ of
the critical point $c_{0}$, the vertically-compressed, horizontally-aligned
rectangular set $\mathcal{V}^{n}_{R_{n}}\subset\mathcal{U}_{c_{0}}$ gets bent
into the image of a Hénon-like map. See (2.4) and Figure 6.

Figure 6. An illustration of the statement of the Main Theorem.

There are two crucial differences between the unicritical cover structure
$\mathcal{C}_{0}$ and the local product structure. The first is that because
the covering element $\mathcal{T}_{c_{0}}\in\mathcal{C}_{0}$ at $c_{0}$ is
pinched, it is a much more delicate problem to ensure that a domain of
interest stays contained inside elements in $\mathcal{C}_{0}$ under the
dynamics. The second is that unlike in the local product structure,
horizontal-like and vertical-like directions are not everywhere preserved.
Indeed, horizontal directions near $c_{0}$ switch to vertical-like directions
near $c_{1}$. These difficulties are navigated using the total order on
$\mathcal{C}_{0}$, which enables us to prove the following crucial facts.

  1. i)

For $1\leq i<R_{n}$, if $\Lambda^{n}_{i}\subset Q\in\mathcal{C}_{0}$, then the
extremal points of $\Lambda^{n}_{i}$ with respect to the total order on
$\Lambda\cap Q$ are $c_{1+i}$ and $c_{1+R_{n}+i}$. Moreover,
$\mathcal{V}^{n}_{i}$ stay horizontally confined between the strong-stable
manifolds of these extremal points.

  2. ii)

We have

$\Lambda^{n+1}_{1}\cap\mathcal{T}_{c_{-R_{n}+1}}=\varnothing\hskip
14.22636pt\text{for}\hskip 14.22636ptn\geq N.$

  3. iii)

We have

$\Lambda^{n+1}_{R_{n}}\cap\mathcal{T}_{c_{R_{n}}}=\varnothing\hskip
14.22636pt\text{for}\hskip 14.22636ptn\geq N.$

Facts i) and ii) are used to show that if $\Lambda^{n}_{i}\subset
Q\in\mathcal{C}_{0}$ for some $1\leq i<R_{n}$, then $\mathcal{V}^{n}_{i}$ is
contained in a slight thickening $\hat{Q}$ of $Q$. Fact iii) is used to show
that after passing through the critical value $c_{1}$, horizontal directions
in $\mathcal{V}^{n}_{i}$ recover their alignment to the local neutral
direction before the next closest approach to the critical point $c_{0}$. This
means that the vertical directions in $\mathcal{V}^{n}_{i}$ typically stay
transverse to the neutral direction. Hence, they must contract under the
dynamics. The Main Theorem follows. For more details, see Sections 13 and 14.

###  2.4. Outline of the paper

To help the reader, we outline now the main content of each section and we
refer to the main result proved in each one. Some of the sections revisit
results proved somewhere else.

In Section 3 we present the classical Pesin theory insisting in certain
quantitative properties that are needed in our context, in particular, to deal
with (forward/backward) “regular finite pieces of orbits” that are not
necessary part of regular orbits in the classical sense of Pesin theory. The
main result is Theorem 3.6 that provides linearizing coordinates for (finite)
regular points.

In Section 4 we revisit a quantitative notion of dissipation that was
introduced in [CP] and proved to be satisfied for odometers in [CPT]. It
allows us to show that the regularity conditions presented in Section 3 are
satisfied.

In Section 5 we obtain linearization coordinates at the critical point and
value (see Proposition 5.8) and we estimate the distance of the closest return
of the critical point to itself (see Proposition 5.9).

In the first part of Section 6 we show how the finite (forward/backward)
regularity of points in scaled neighborhood of the critical point is
controlled (Propositions 6.1 and 6.3 for obtaining regularity and Propositions
6.2 and 6.4 for obstructing regularity). In the second part (Section 6.1), we
introduce the notions of crescents and tunnels (and the scaled ones related to
the pre-iterates of the tunnels).

In Section 7 we revisit the notion of mild dissipativity. Recall that this
notion states that the stable manifold of any regular point separates the
attracting disk where the dynamics is defined. In this section we estimate
quantitatively how uniform pieces of local stable manifold separate forward
iterates of the attracting disk.

In Section 8 we recall a classical result about the unique ergodicity of
odometers and we characterize the trivial connected components of the limit
set.

In Section 9 we show that the odometer nearby the critical point is contained
inside the pinched tunnel (Theorem 9.7).

In Section 10 we show how scaled tunnels fits in the regular boxes and in the
initial tunnel, providing first a local total order and later a global total
order. More precisely, using regular boxes and scaled tunnels, in Section 10.3
we introduce the notion of unicritical cover and we show that such a cover is
“totally ordered” (Theorem 10.6).

##  3\. Quantitative Pesin Theory

Let $r\geq 2$ be an integer. Consider a dissipative $C^{r}$-diffeomorphism
$F:\Omega\to F(\Omega)\Subset\Omega$ defined on a Jordan domain
$\Omega\subset\mathbb{R}^{2}$. Let $\Lambda\Subset\Omega$ be a totally
invariant compact set, and let $\mu$ be an ergodic probability measure
supported on $\Lambda$. Suppose that the Lyapunov exponents of $F$ with
respect to $\mu$ are $0$ and $\log\lambda<0$. Let $\varepsilon>0$ be a small
constant such that ${\bar{\varepsilon}}<1$.

###  3.1. Definitions of regularity

Let $N,M\in\mathbb{N}\cup\\{0,\infty\\}$ and $L\geq 1$. A point $p\in\Lambda$
is $N$-times forward $(L,\varepsilon)_{v}$-regular along
$E^{+}_{p}\in\mathbb{P}^{2}_{p}$ if for $s\in\\{-r,r-1\\}$, we have

$L^{-1}\lambda^{(1+\varepsilon)n}\leq\frac{(\operatorname{Jac}_{p}F^{n})^{s}}{\|DF^{n}|_{E^{+}_{p}}\|^{s-1}}\leq
L\lambda^{(1-\varepsilon)n}\hskip 14.22636pt\text{for all}\hskip
14.22636pt1\leq n\leq N.$ (3.1)

Similarly, $p$ is $M$-times backward $(L,\varepsilon)_{v}$-regular along
$E^{-}_{p}\in\mathbb{P}^{2}_{p}$ if for $s\in\\{-r,r-1\\}$, we have

$L^{-1}\lambda^{-(1-\varepsilon)n}\leq\frac{(\operatorname{Jac}_{p}F^{-n})^{s}}{\|DF^{-n}|_{E^{-}_{p}}\|^{s-1}}\leq
L\lambda^{-(1+\varepsilon)n}\hskip 14.22636pt\text{for all}\hskip
14.22636pt1\leq n\leq M.$ (3.2)

If (3.1) and (3.2) are both satisfied with $E_{p}^{v}:=E_{p}^{+}=E_{p}^{-}$,
then we say that $p$ is $(M,N)$-times $(L,\varepsilon)_{v}$-regular along
$E_{p}^{v}$. If additionally, we have $M=N=\infty$, then we say that $p$ is
Pesin $(L,\varepsilon)_{v}$-regular along $E_{p}^{v}$.

We say that $p$ is $N$-times forward $(L,\varepsilon)_{h}$-regular along
$\tilde{E}_{p}^{+}\in\mathbb{P}_{p}^{2}$ if for $s\in\\{-r+1,r\\}$, we have

$L^{-1}\lambda^{(1+\varepsilon)n}\leq\frac{\operatorname{Jac}_{p}F^{n}}{\|D_{p}F^{n}|_{\tilde{E}_{p}^{+}}\|^{s+1}}\leq
L\lambda^{(1-\varepsilon)n}\hskip 14.22636pt\text{for}\hskip 14.22636pt1\leq
n\leq N.$ (3.3)

Similarly, we say that $p$ is $M$-times backward $(L,\varepsilon)_{h}$-regular
along $\tilde{E}_{p}^{-}\in\mathbb{P}_{p}^{2}$ if for $s\in\\{-r+1,r\\}$, we
have

$L^{-1}\lambda^{-(1-\varepsilon)n}\leq\frac{\operatorname{Jac}_{p}F^{-n}}{\|D_{p}F^{-n}|_{\tilde{E}_{p}^{-}}\|^{s+1}}\leq
L\lambda^{-(1+\varepsilon)n}\hskip 14.22636pt\text{for}\hskip 14.22636pt1\leq
n\leq M.$ (3.4)

If both (3.3) and (3.4) hold with
$E_{p}^{h}:=\tilde{E}_{p}^{+}=\tilde{E}_{p}^{-}$, then $p$ is $(M,N)$-times
$(L,\varepsilon)_{h}$-regular along $E_{p}^{h}$. If, additionally, we have
$M=N=\infty$, then $p$ is Pesin $(L,\varepsilon)_{h}$-regular along
$E_{p}^{h}$.

######  Proposition 3.1 (Vertical forward regularity $=$ horizontal forward
regularity).

Suppose $p$ is $N$-times forward $(L,\varepsilon)_{v}$-regular along
$E_{p}^{v}\in\mathbb{P}_{p}^{2}$. Let
$E_{p}^{h}\in\mathbb{P}_{p}^{2}\setminus\\{E_{p}^{v}\\}$. If
$\measuredangle(E_{p}^{v},E_{p}^{h})>\theta$, then the point $p$ is $N$-times
forward $({\bar{L}}/\theta^{2},\varepsilon)_{h}$-regular along $E_{p}^{h}$.

Conversely, if $p$ is $N$-times forward $(L,\varepsilon)_{h}$-regular along
$E_{p}^{h}\in\mathbb{P}_{p}^{2}$, then there exists
$E_{p}^{v}\in\mathbb{P}_{p}^{2}$ such that $p$ is $N$-times forward
$({\bar{L}},{\bar{\varepsilon}})_{v}$-regular along $E_{p}^{v}$.

######  Proposition 3.2 (Horizontal backward regularity $=$ vertical backward
regularity).

Suppose $p$ is $M$-times backward $(L,\varepsilon)_{h}$-regular along
$E_{p}^{h}\in\mathbb{P}_{p}^{2}$. Let
$E_{p}^{v}\in\mathbb{P}_{p}^{2}\setminus\\{E_{p}^{h}\\}$. If
$\measuredangle(E_{p}^{h},E_{p}^{v})>\theta$, then the point $p$ is $M$-times
backward $({\bar{L}}/\theta^{2},\varepsilon)_{v}$-regular along $E_{p}^{v}$.

Conversely, if $p$ is $M$-times backward $(L,\varepsilon)_{v}$-regular along
$E_{p}^{v}\in\mathbb{P}_{p}^{2}$, then there exists
$E_{p}^{h}\in\mathbb{P}_{p}^{2}$ such that $p$ is $M$-times backward
$({\bar{L}},{\bar{\varepsilon}})_{h}$-regular along $E_{p}^{h}$.

######  Proposition 3.3 (Pesin regularity $=$ vertical forward regularity $+$
horizontal backward regularity $+$ transversality).

Suppose $p$ is $N$-times forward $(L,\varepsilon)_{v}$-regular along
$E_{p}^{v}\in\mathbb{P}_{p}^{2}$ and $M$-times backward
$(L,\varepsilon)_{h}$-regular along $E_{p}^{h}\in\mathbb{P}_{p}^{2}$ with
$\theta:=\measuredangle(E_{p}^{v},E_{p}^{h})>0$. Let $\mathcal{L}\geq 1$ be
the minimum value such that $p$ is $(M,N)$-times
$(\mathcal{L},{\bar{\varepsilon}})_{v}$-regular along $E_{p}^{v}$ and
$(\mathcal{L},{\bar{\varepsilon}})_{h}$-regular along $E_{p}^{h}$. Then we
have $k/\theta^{2}<\mathcal{L}<{\bar{L}}/\theta^{2}$ for some uniform constant
$k>0$.

Suppose $p_{0}\in\Lambda$ is $(M,N)$-times $(L,\varepsilon)_{v/h}$-regular
along $E_{p_{0}}^{v/h}\in\mathbb{P}^{2}_{p_{0}}$. Define the regularity of
$p_{m}$ as the smallest value $\mathcal{L}_{p_{m}}\geq 1$ such that $p_{m}$ is
$(M+m,N-m)$-times $(\mathcal{L}_{p_{m}},\varepsilon)_{v/h}$-regular along
$E_{p_{m}}^{v/h}$.

######  Proposition 3.4 (Decay in regularity).

For $-M\leq m\leq N$, we have

$\mathcal{L}_{p_{m}}<L^{2}\lambda^{-2\varepsilon|m|}.$

######  Remark 3.5.

By Proposition 3.1, 3.2 and 3.3, we see that regularity along a vertical
direction and regularity along a horizontal direction are interchangeable
conditions (at the cost of increasing the constants of regularity by some
uniform amount). Henceforth, the vertical direction will be taken as the
default unless otherwise stated.

###  3.2. Linearization along regular orbits

For the remainder of this section, we assume that $p_{0}\in\Lambda$ is
$(M,N)$-times $(L,\varepsilon)_{v}$-regular along
$E_{p_{0}}^{v}\in\mathbb{P}^{2}_{p_{0}}$ for some
$M,N,\in\mathbb{N}\cup\\{\infty\\}$ and $L\geq 1$.

For $l,w>0$, denote

$\mathbb{I}(l):=(-l,l)\subset\mathbb{R},\hskip
14.22636pt\mathbb{B}(l,w):=\mathbb{I}(l)\times\mathbb{I}(w)\subset\mathbb{R}^{2}\hskip
14.22636pt\text{and}\hskip 14.22636pt\mathbb{B}(l):=\mathbb{B}(l,l).$

######  Theorem 3.6.

For $-M\leq m\leq N$, let

$l_{p_{m}}:={\bar{L}}^{-1}\lambda^{{\bar{\varepsilon}}|m|}\hskip
14.22636pt\text{and}\hskip 14.22636ptU_{p_{m}}:=\mathbb{B}(l_{p_{m}}).$

Then there exists a $C^{r}$-diffeomorphism
$\Phi_{p_{m}}:(\mathcal{U}_{p_{m}},p_{m})\to(U_{p_{m}},0)$ such that

$\|\Phi_{p_{m}}^{\pm
1}\|_{C^{r}}=O({\bar{L}}\lambda^{-{\bar{\varepsilon}}|m|}),\hskip
14.22636ptD\Phi_{p_{m}}(E^{v}_{p_{m}})=E^{gv}_{0},$

and $\Phi_{p_{n+1}}\circ F|_{\mathcal{U}_{p_{m}}}\circ\Phi_{p_{m}}^{-1}$
extends to a globally defined $C^{r}$-diffeomorphism
$F_{p_{m}}:(\mathbb{R}^{2},0)\to(\mathbb{R}^{2},0)$ satisfying the following
properties.

  1. i)

We have $\displaystyle\|F_{p_{m}}^{\pm 1}\|_{C^{r}}=O(1)$.

  2. ii)

The map $F_{p_{m}}$ is uniformly $C^{1}$-close to

$D_{0}F_{p_{m}}=A_{m}=\begin{bmatrix}a_{m}&0\\\ 0&b_{m}\end{bmatrix},$

with $\lambda^{1+{\bar{\varepsilon}}}<b_{m}<\lambda^{1-{\bar{\varepsilon}}}$
and $\lambda^{\bar{\varepsilon}}<a_{m}<\lambda^{-{\bar{\varepsilon}}}$.

  3. iii)

We have

$F_{p_{m}}(x,y)=(f_{p_{m}}(x),e_{p_{m}}(x,y))\hskip 14.22636pt\text{for}\hskip
14.22636pt(x,y)\in\mathbb{R}^{2},$

where $f_{p_{m}}:(\mathbb{R},0)\to(\mathbb{R},0)$ is a $C^{r}$-diffeomorphism,
and $e_{p_{m}}:\mathbb{R}^{2}\to\mathbb{R}$ is a $C^{r}$-map with
$e_{p_{m}}(\cdot,0)\equiv 0$.

The construction in Theorem 3.6 is referred to as a linearization of $F$ along
the $(M,N)$-orbit of $p_{0}$ with vertical direction $E^{v}_{p_{0}}$. For
$-M\leq m\leq N$, we refer to $l_{p_{m}}$, $\mathcal{U}_{p_{m}}$,
$\Phi_{p_{m}}$ and $F_{p_{m}}$ as a regular radius, a regular neighborhood, a
regular chart and a linearized map at $p_{m}$ respectively. For $1\leq n\leq
N-m$, we denote

$F_{p_{m}}^{n}:=F_{p_{m+n-1}}\circ\ldots\circ F_{p_{m+1}}\circ F_{p_{m}}.$

Let $X_{m}\subset U_{p_{m}}$ be a set. We denote

$X_{m+l}:=F_{p_{m}}^{l}(X_{m})$

for all $l\in\mathbb{Z}$ for which the right-hand side is well-defined.
Similarly, for any direction $E_{z_{m}}\in\mathbb{P}^{2}_{z_{m}}$ at a point
$z_{m}\in U_{p_{m}}$, we denote

$E_{z_{m+l}}:=DF_{p_{m}}^{l}(E_{z_{m}}).$

######  Remark 3.7.

Note that we have the freedom to make the regular radii used in Theorem 3.6
uniformly smaller whenever necessary (i.e. replace $l_{p_{m}}$ by
$\underline{l_{p_{m}}}$ for all $-M\leq m\leq N$).

######  Proposition 3.8.

For $-M\leq m\leq N$, we have

$\operatorname{diam}(\mathcal{U}_{p_{m}})\asymp{\bar{L}}^{-1}\lambda^{{\bar{\varepsilon}}|m|}.$

The following result states that restricted to the regular neighborhoods,
iterates of $F$ are nearly linear.

######  Proposition 3.9 (Linear dynamics inside regular neighborhoods).

For any constant $k>0$, the regular radii in Theorem 3.6 can be chosen
sufficiently small so that the following holds. Let $-M\leq m\leq N$ and
$-M-m\leq l\leq N-m$. Suppose that $q_{m+i}\in\mathcal{U}_{p_{m+i}}$ for
$i\in[m,m+l]\cap\mathbb{Z}$. Write $z_{m}:=\Phi_{p_{m}}(q_{m})\in U_{p_{m}}$.
Then for all $v\in\mathbb{R}^{2}$, we have

$\|D_{z_{m}}F_{p_{m}}^{l}(v)-D_{0}F_{p_{m}}^{l}(v)\|<k\|D_{0}F_{p_{m}}^{l}(v)\|$

and

$\|D_{q_{m}}F^{l}(v)-D_{p_{m}}F^{l}(v)\|<k\|D_{p_{m}}F^{l}(v)\|.$

Moreover,

$1-k<\frac{\operatorname{Jac}_{z_{m}}F_{p_{m}}^{l}}{\operatorname{Jac}_{0}F_{p_{m}}^{l}},\frac{\operatorname{Jac}_{q_{m}}F^{l}}{\operatorname{Jac}_{p_{m}}F^{l}}<1+k.$

###  3.3. Vertical and horizontal directions

For $z\in\mathbb{R}^{2}$, denote the genuine vertical/horizontal direction at
$z$ by $E^{gv/gh}_{z}\in\mathbb{P}^{2}_{z}$ respectively.

Let $-M\leq m\leq N$. For $q\in\mathcal{U}_{p_{m}}$, write
$z:=\Phi_{p_{m}}(q)$. The vertical/horizontal direction at $q$ in
$\mathcal{U}_{p_{m}}$ is defined as
$E^{v/h}_{q}:=D\Phi_{p_{m}}^{-1}(E^{gv/gh}_{z})$. For $t\geq 0$, we say that
$E_{q}\in\mathbb{P}^{2}_{q}$ is $t$-vertical/horizontal in
$\mathcal{U}_{p_{0}}$ if

$\frac{\measuredangle(E_{q},E_{q}^{v/h})}{\measuredangle(E_{q},E_{q}^{h/v})}<t.$

Similarly, we say that a $C^{1}$-curve $\gamma\subset\mathcal{U}_{p_{0}}$ is
$t$-vertical/horizontal in $\mathcal{U}_{p_{0}}$ if

$\frac{\measuredangle(\gamma^{\prime}(s),E_{\gamma(s)}^{v/h})}{\measuredangle(\gamma^{\prime}(s),E_{\gamma(s)}^{h/v})}<t\hskip
14.22636pt\text{for}\hskip 14.22636pts\in(0,|\gamma|).$

By the construction of regular charts in Theorem 3.6, vertical directions are
invariant under $F$:

$\text{i.e.}\hskip 14.22636ptDF(E_{q}^{v})=E_{F(q)}^{v}\hskip
14.22636pt\text{for}\hskip 14.22636ptq\in\mathcal{U}_{p_{m}}.$

Note that the same is not true for horizontal directions. However, the
following result states that they are still nearly invariant under $F$.

######  Proposition 3.10 (Near invariance of horizontal directions).

Let $-M\leq m\leq N$ and $-M-m\leq l\leq N-m$. Suppose that
$q_{m+i}\in\mathcal{U}_{p_{m+i}}\hskip 14.22636pt\text{for}\hskip
14.22636pti\in[m,m+l]\cap\mathbb{Z}$. Let
$\tilde{E}_{q_{m+l}}^{h}:=D_{q_{m}}F^{l}(E_{q_{m}}^{h})$. Write
$z_{m}=(x_{m},y_{m}):=\Phi_{p_{m}}(q_{m})$ and

$\tilde{E}_{z_{m+l}}^{h}:=D_{z_{m}}F_{p_{m}}^{l}(E_{z_{m}}^{gh})=D_{q_{m+l}}\Phi_{p_{m+l}}(\tilde{E}_{q_{m+l}}^{h}).$

Then we have

$\measuredangle(\tilde{E}_{z_{m+l}}^{h},E_{z_{m+l}}^{gh}),\measuredangle(\tilde{E}_{q_{m+l}}^{h},E_{q_{m+l}}^{h})<K|y_{m+l}|^{1-{\bar{\varepsilon}}}$

for some uniform constant $K>1$.

######  Proposition 3.11 (Vertical alignment of forward contracting
directions).

Let $q_{0}\in\mathcal{U}_{p_{0}}$ and
$\tilde{E}_{q_{0}}^{v}\in\mathbb{P}^{2}_{q_{0}}$. Suppose for some $0<n\leq
N$, we have $q_{i}\in\mathcal{U}_{p_{i}}$ for $0\leq i\leq n$. If
$\nu:=\|DF^{n}|_{\tilde{E}_{q_{0}}^{v}}\|<{\bar{L}}^{-1}\lambda^{{\bar{\varepsilon}}n}$,
then

$\measuredangle(\tilde{E}_{q_{0}}^{v},E_{q_{0}}^{v})<{\bar{L}}\lambda^{-{\bar{\varepsilon}}n}\nu+{\bar{L}}\lambda^{(1-{\bar{\varepsilon}})n}.$

######  Proposition 3.12 (Horizontal alignment of backward neutral
directions).

Let $q_{0}\in\mathcal{U}_{p_{0}}$ and
$\tilde{E}_{q_{0}}^{h}\in\mathbb{P}^{2}_{q_{0}}$. Suppose for some $0<m\leq
M$, we have $q_{-i}\in\mathcal{U}_{p_{-i}}$ for $0\leq i\leq m$. If
$\mu:=\|DF^{-m}|_{\tilde{E}_{q_{0}}^{h}}\|<{\bar{L}}^{-1}\lambda^{-(1-{\bar{\varepsilon}})m}$,
then

$\measuredangle(\tilde{E}_{q_{0}}^{h},E_{q_{0}}^{h})<{\bar{L}}\lambda^{(1-{\bar{\varepsilon}})m}(1+\mu).$

Let $\mathcal{E}:\mathcal{D}\to T^{1}\mathcal{D}$ be a unit vector field on a
domain $\mathcal{D}\subset\Omega$. Define

$DF_{*}(\mathcal{E})(p):=\frac{DF(\mathcal{E}(p))}{\|DF(\mathcal{E}(p))\|}\in
T^{1}_{F(p)}F(\mathcal{D})\hskip 14.22636pt\text{for}\hskip
14.22636ptp\in\mathcal{D}.$

If $\mathcal{E}(p)$ is $t$-vertical for all
$p\in\mathcal{D}\cap\mathcal{U}_{p_{0}}$, then $\mathcal{E}$ is said to be
$t$-vertical in $\mathcal{U}_{p_{0}}$.

For $-N\leq m\leq N$, define $\mathcal{E}^{v}_{p_{m}}:\mathcal{U}_{p_{m}}\to
T^{1}(\mathcal{U}_{p_{m}})$ to be the unit vector field given by

$\mathcal{E}^{v}_{p_{m}}(q)\in E^{v}_{q}\hskip 14.22636pt\text{for}\hskip
14.22636ptq\in\mathcal{U}_{p_{m}}.$

######  Proposition 3.13 (Vertical alignment of non-horizontal vector fields
under backward iterates).

Let $\mathcal{D}_{0}\subset\mathcal{U}_{p_{0}}$ be a domain, and $0\leq n\leq
N$. Suppose

$\mathcal{D}_{i}:=F^{i}(\mathcal{D}_{0})\subset\mathcal{U}_{p_{i}}\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq i\leq n.$

Let $\mathcal{E}:\mathcal{D}_{n}\to T^{1}(\mathcal{D}_{n})$ be a
$C^{r-1}$-unit vector field. If $\mathcal{E}$ is $t$-vertical in
$\mathcal{U}_{p_{n}}$ for some $t\geq 0$, then we have

$\|DF^{-n}_{*}(\mathcal{E})-\mathcal{E}^{v}_{p_{0}}|_{\mathcal{D}_{0}}\|_{C^{r-1}}\leq(1+t^{2})(1+\|\mathcal{E}\|_{C^{r-1}}){\bar{L}}\lambda^{(1-{\bar{\varepsilon}})n}.$

Let $\gamma\subset\mathbb{R}^{2}$ be a $C^{r}$-curve. We define

$\|\gamma\|_{C^{r}}:=\|\phi_{\gamma}\|_{C^{r}},$

where $\phi_{\gamma}:[0,|\gamma|]\to\gamma$ is the parameterization of
$\gamma$ by its arclength.

######  Proposition 3.14 (Horizontal alignment of non-vertical curves under
forward iterates).

Let $\gamma_{0}\subset\mathcal{U}_{p_{0}}$ be a $C^{r}$-curve, and $0\leq
n\leq N$. Suppose

$\gamma_{i}:=F^{i}(\gamma_{0})\subset\mathcal{U}_{p_{i}}\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq i\leq n.$

If $\gamma_{0}$ is $t$-horizontal in $\mathcal{U}_{p_{0}}$ for some $t\geq 0$,
then $\Phi_{p_{n}}(\gamma_{n})$ is the horizontal graph of some $C^{r}$-map
$g_{n}:J_{n}\subset\mathbb{I}(l_{p_{n}})\to\mathbb{I}(l_{p_{n}})$ with

$\|g_{n}\|_{C^{r}}\leq(1+t^{r+1})\|\gamma_{0}\|_{C^{r}}{\bar{L}}\lambda^{(1-{\bar{\varepsilon}})n}.$

###  3.4. Truncated and pinched regular neighborhoods

Let $\delta\in({\bar{\varepsilon}},1)$ be a small constant such that
${\bar{\delta}}<1$. For $n\geq 0$, the $n$-times $\delta$-truncated regular
neighborhood of $p_{0}$ is defined as

$\mathcal{U}_{p_{0}}^{\delta n}:=\Phi_{p_{0}}^{-1}\left(U_{p_{0}}^{\delta
n}\right)\subset\mathcal{U}_{p_{0}},\hskip 14.22636pt\text{where}\hskip
14.22636ptU_{p_{0}}^{\delta n}:=\mathbb{B}\left(\lambda^{\delta
n}l_{p_{0}},l_{p_{0}}\right).$ (3.5)

The purpose of truncating a regular neighborhood is to ensure that its
iterated images stay inside regular neighborhoods.

######  Lemma 3.15.

For $0\leq n\leq N$, we have
$F^{i}(\mathcal{U}_{p_{0}}^{{\bar{\varepsilon}}n})\subset\mathcal{U}_{p_{i}}$
for $0\leq i\leq n$.

For $\omega>1$ and ${t}\in(0,l_{p_{0}})$, define

$T_{p_{0}}^{\omega}({t}):=\\{(x,y)\in
U_{p_{0}}\;|\;|x|<{t}\;\text{and}\;|y|<|x|^{\omega}\\}.$

We refer to

$\mathcal{T}_{p_{0}}^{\omega}({t}):=\Phi_{p_{0}}^{-1}(T_{p_{0}}^{\omega}({t}))\subset\mathcal{U}_{p_{0}}$

as a $\omega$-pinched regular neighborhood of length ${t}$ at $p_{0}$. Note
that $p_{0}\not\in\mathcal{T}_{p_{0}}^{\omega}({t})$. For $n\geq 0$, we also
denote

$T_{p_{0}}^{\omega,\delta n}({t}):=T_{p_{0}}^{\omega}({t})\cap
U_{p_{0}}^{\delta n}\hskip 14.22636pt\text{and}\hskip
14.22636pt\mathcal{T}_{p_{0}}^{\omega,\delta
n}({t}):=\Phi_{p_{0}}^{-1}(T_{p_{0}}^{\omega,\delta
n}({t}))=\mathcal{T}_{p_{0}}^{\omega}({t})\cap\mathcal{U}_{p_{0}}^{\delta n}.$

If the length ${t}$ is not indicated, then it is assumed to be equal to
$l_{p_{0}}$ (i.e.
$\mathcal{T}_{p_{0}}^{\omega}:=\mathcal{T}_{p_{0}}^{\omega}(l_{p_{0}})$).

The purpose of pinching a regular neighborhood is to ensure that its iterated
preimages stay inside regular neighborhoods.

######  Lemma 3.16.

Let $0\leq n\leq M$. If $\omega>(1+{\bar{\varepsilon}})/\delta$, then
$F^{-i}(\mathcal{T}_{p_{0}}^{\omega,\delta n})\subset\mathcal{U}_{p_{-i}}$ for
$0\leq i\leq n$.

######  Lemma 3.17.

Let $\omega\in(1,{\bar{\varepsilon}}^{-1}]$. For
$q_{0}\in\mathcal{U}_{p_{0}}\setminus\\{p_{0}\\}$, write
$z_{0}=(x_{0},y_{0}):=\Phi_{p_{0}}(q_{0})\in U_{p_{0}}$. If
$q_{-n}\in\mathcal{U}_{p_{-n}}$ for some

$n>(1+{\bar{\varepsilon}})\omega\log_{\lambda}|x_{0}|,$

then $q_{0}\in\mathcal{T}_{p_{0}}^{\omega}$.

###  3.5. Regular boxes

Let $L^{\prime}$ be a constant sufficiently uniformly larger than $L$. In this
subsection, we assume that

$M,N>\log_{\lambda}\overline{L^{\prime}}.$ (3.6)

A regular box at $p_{0}$ is defined as

$\mathcal{B}_{p_{0}}:=\Phi_{p_{0}}^{-1}(B_{p_{0}})\Subset\mathcal{U}_{p_{0}},\hskip
14.22636pt\text{where}\hskip 14.22636ptB_{p_{0}}:=\mathbb{B}(1/L^{\prime}).$

Denote

$\mathbb{L}^{h}_{b}:=\\{(x,b)\;|\;x\in\mathbb{R}\\}.$

The top-bottom boundary of $\mathcal{B}_{p_{0}}$ is defined as

$\hat{\partial}\mathcal{B}_{p_{0}}:=\Phi_{p_{0}}^{-1}(\hat{\partial}B_{p_{0}}),\hskip
14.22636pt\text{where}\hskip 14.22636pt\hat{\partial}B_{p_{0}}:=\partial
B_{p_{0}}\cap(\mathbb{L}^{h}_{1/L^{\prime}}\cup\mathbb{L}^{h}_{-1/L^{\prime}}).$

A Jordan curve $\Gamma:(0,1)\to\mathcal{B}_{p_{0}}$ is said to be vertically
proper in $\mathcal{B}_{p_{0}}$ if $\Gamma$ extends continuously to $[0,1]$,
and $\Gamma(0)$ and $\Gamma(1)$ are in distinct components of
$\hat{\partial}\mathcal{B}_{p_{0}}$. More generally, a Jordan curve $\Gamma$
in $\Omega$ is said to be vertically proper in $\mathcal{B}_{p_{0}}$ if
$\Gamma\cap\mathcal{B}_{p_{0}}$ is.

Let $q\in\mathcal{U}_{p_{0}}$. We say that $E_{q}\in\mathbb{P}^{2}_{q}$ is
sufficiently vertical/horizontal in $\mathcal{U}_{p_{0}}$ if $E_{q}$ is
$1/\overline{L^{\prime}}$-vertical/horizontal in $\mathcal{U}_{p_{0}}$.
Similarly, we say that a $C^{1}$-curve $\Gamma\subset\mathcal{U}_{p_{0}}$ is
sufficiently vertical/horizontal in $\mathcal{U}_{p_{0}}$ if $\Gamma$ is
$1/\overline{L^{\prime}}$-vertical/horizontal in $\mathcal{U}_{p_{0}}$.

######  Proposition 3.18 (Consistent vertical and horizontal directions inside
regular boxes).

Let $\tilde{M},\tilde{N}\in\mathbb{N}\cup\\{0,\infty\\}$. Suppose
$\tilde{p}_{0}\in\mathcal{B}_{p_{0}}$ is $(\tilde{M},\tilde{N})$-times
$(L,\varepsilon)_{v}$-regular along
$\tilde{E}^{v}_{\tilde{p}_{0}}\in\mathbb{P}^{2}_{\tilde{p}_{0}}$. Denote the
vertical/horizontal direction and the local vertical/horizontal manifold at
$\tilde{p}_{0}$ associated with the linearization of $F$ along the
$(\tilde{M},\tilde{N})$-orbit of $\tilde{p}_{0}$ with vertical direction
$\tilde{E}^{v}_{\tilde{p}_{0}}$ by
$\tilde{E}_{\tilde{p}_{0}}^{v/h}\in\mathbb{P}^{2}_{\tilde{p}_{0}}$ and
$\tilde{W}^{v/h}_{\operatorname{loc}}(\tilde{p}_{0})\subset\mathcal{U}_{\tilde{p}_{0}}$
respectively.

  1. i)

If $\tilde{N}>\log_{\lambda}\overline{L^{\prime}}$, then
$\tilde{E}^{v}_{\tilde{p}_{0}}$ and $\tilde{W}^{v}(\tilde{p}_{0})$ are
sufficiently vertical in $\mathcal{U}_{p_{0}}$.

  2. ii)

If $\tilde{M}>\log_{\lambda}\overline{L^{\prime}}$, then
$\tilde{E}^{h}_{\tilde{p}_{0}}$ and $\tilde{W}^{h}(\tilde{p}_{0})$ are
sufficiently horizontal in $\mathcal{U}_{p_{0}}$.

######  Proposition 3.19 (Verticality and horizontality of curves preserved
inside regular boxes).

For $0\leq k\leq K<\infty$, let $p^{k}\in\Omega$ be $(M,N)$-times
$(L,\varepsilon)_{v}$-regular along $E^{v}_{p^{k}}$. If
$\Gamma_{0}\subset\Omega$ is a $C^{1}$-curve such that
$\Gamma_{k}\subset\mathcal{B}_{p^{k}}$ for all $0\leq k\leq K$, then the
following holds.

  1. i)

If $\Gamma_{0}$ is sufficiently horizontal in $\mathcal{U}_{p^{0}}$, then
$\Gamma_{k}$ is sufficiently horizontal in $\mathcal{U}_{p^{k}}$ for all
$0\leq k\leq K$. If, additionally, $\Gamma_{0}$ is $C^{r}$ and
$\|\Gamma_{0}\|_{C^{r}}<{\bar{L}}$, then $\|\Gamma_{K}\|_{C^{r}}<{\bar{L}}$.

  2. ii)

If $\Gamma_{K}\in\mathbb{P}^{2}_{q_{K}}$ is sufficiently vertical in
$\mathcal{U}_{p^{K}}$, then $\Gamma_{k}$ is sufficiently vertical in
$\mathcal{U}_{p^{k}}$ for all $0\leq k\leq K$. Moreover,

$|\Gamma_{0}|>{\bar{L}}^{-1}|\Gamma_{K}|.$ (3.7)

If, additionally, $\Gamma_{K}$ is $C^{r}$ and
$\|\Gamma_{K}\|_{C^{r}}<{\bar{L}}$, then $\|\Gamma_{0}\|_{C^{r}}<{\bar{L}}$.

###  3.6. Stable and center manifolds

If $N=\infty$, then Proposition 3.11 implies that $E_{p_{0}}^{v}$ is the
unique direction along which $p_{0}$ is infinitely forward
$(L,\varepsilon,\lambda)$-regular. In this case, we denote

$E_{p_{0}}^{ss}:=E_{p_{0}}^{v},$

and refer to this direction as the strong stable direction at $p_{0}$.
Moreover, we define the local strong stable manifold at $p_{0}$ as

$W^{ss}_{\operatorname{loc}}(p_{0}):=\Phi_{p_{0}}^{-1}(\\{(0,y)\in
U_{p_{0}}\\}),$

and the strong stable manifold at $p_{0}$ as

$W^{ss}(p_{0}):=\\{q\in\Omega\;|\;F^{n}(q)\in
W^{ss}_{\operatorname{loc}}(p_{m})\;\;\text{for some}\;\;n\geq 0\\}.$

If $M=\infty$, we denote

$E_{p_{0}}^{c}:=E_{p_{0}}^{h},$

and refer to this direction as the center direction at $p_{0}$. Moreover, we
define the (local) center manifold at $p_{0}$ as

$W^{c}(p_{0}):=\Phi_{p_{0}}^{-1}(\\{(x,0)\in U_{p_{0}}\\}).$

Unlike strong-stable manifolds, center manifolds are not uniquely and
canonically defined. However, the following result states that it still has a
canonical jet.

######  Proposition 3.20 (Canonical $C^{r}$-jets of center manifolds).

Suppose $M=\infty$. Let $\Gamma_{0}:(-t,t)\to\mathcal{U}_{p_{0}}$ be a
$C^{r}$-curve parameterized by its arclength such that $\Gamma_{0}(0)=p_{0}$,
and for all $n\in\mathbb{N}$, we have

$\|DF^{-n}|_{\Gamma_{0}^{\prime}(t)}\|<\lambda^{-(1-{\bar{\varepsilon}})n}\hskip
14.22636pt\text{for}\hskip 14.22636pt|t|<\lambda^{\varepsilon n}.$

Then $\Gamma_{0}$ has a degree $r$ tangency with $W^{c}(p_{0})$ at $p_{0}$.

######  Proposition 3.21.

If $p$ is infinite-time forward/backward $(L,\varepsilon)$-regular. Then

$\left\|W^{ss/c}_{\operatorname{loc}}(p)\right\|_{C^{r}}<{\bar{L}}.$

######  Proposition 3.22.

The strong stable manifold $W^{ss}(p)$ depends $C^{r}$-continuously on
$p\in\Lambda^{+}_{L,\varepsilon}$. The center manifold $W^{c}(p)$ depends
$C^{r-{\bar{\varepsilon}}}$-continuously on $p\in\Lambda^{-}_{L,\varepsilon}$.

######  Remark 3.23.

Henceforth, we say that $p$ is infinite-time forward/backward
$(L,\varepsilon)$-regular (without specifying $v$/$h$ in the subscript) if $p$
is infinite-time forward/backward $(L,\varepsilon)_{v/h}$-regular along
$E^{ss/c}_{p}$. Furthermore, we say that $p$ is Pesin
$(L,\varepsilon)$-regular if $p$ is infinite-time forward and backward
$(L,\varepsilon)$-regular, and we have
$\measuredangle(E^{ss}_{p},E^{c}_{p})>1/L$ (see Proposition 3.3). The sets of
all infinite-time forward/backward/Pesin $(L,\varepsilon)$-regular points in
$\Lambda$ are referred to as the forward, backward and Pesin
$(L,\varepsilon)$-regular blocks, and are denoted by
$\Lambda^{+}_{L,\varepsilon}$, $\Lambda^{-}_{L,\varepsilon}$ and
$\Lambda^{P}_{L,\varepsilon}$ respectively.

######  Proposition 3.24.

We have $\mu(\Lambda^{P}_{L,\varepsilon})\to 1$ as $L\to\infty$.

##  4\. Homogeneity

Let $F$ be the diffeomorphism considered in Section 3. For $\eta\in(0,1)$ with
$\bar{\eta}<1$, we say that $F$ is $\eta$-homogeneous on $\Lambda$ if for all
$p\in\Lambda$ and $E_{p}\in\mathbb{P}_{p}^{2}$, we have:

  1. i)

$\lambda^{1+\eta}<\|D_{p}F|_{E_{p}}\|<\lambda^{-\eta}$ and

  2. ii)

$\lambda^{1+\eta}<\operatorname{Jac}_{p}F<\lambda^{1-\eta}$.

Homogeneity considerably simplifies the regularity conditions given in (3.1),
(3.2), (3.3) and (3.4). Let $N,M\in\mathbb{N}\cup\\{0,\infty\\}$; $L\geq 1$
and $\varepsilon\in(\bar{\eta},1)$ with ${\bar{\varepsilon}}<1$. Then a point
$p\in\Lambda$ is:

  * •

$N$-times forward $(L,\varepsilon)_{v}$-regular along
$E^{v}_{p}\in\mathbb{P}^{2}_{p}$ if

$\|DF^{n}|_{E^{v}_{p}}\|\leq L\lambda^{(1-\varepsilon)n}\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq n\leq N;$

  * •

$M$-times backward $(L,\varepsilon)_{v}$-regular along $E^{v}_{p}$ if

$\|DF^{-n}|_{E^{v}_{p}}\|\geq L^{-1}\lambda^{-(1-\varepsilon)n}\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq n\leq M;$

  * •

$N$-times forward $(L,\varepsilon)_{h}$-regular along
$E^{h}_{p}\in\mathbb{P}^{2}_{p}$ if

$\|DF^{n}|_{E^{h}_{p}}\|\geq L^{-1}\lambda^{\varepsilon n}\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq n\leq N;$

  * •

$M$-times backward $(L,\varepsilon)_{h}$-regular along
$E^{h}_{p}\in\mathbb{P}^{2}_{p}$ if

$\|DF^{-n}|_{E^{h}_{p}}\|\leq L\lambda^{-\varepsilon n}\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq n\leq M.$

######  Proposition 4.1.

Suppose $F|_{\Lambda}$ is uniquely ergodic. Then for any $\eta>0$, there
exists $N=N(\eta)\in\mathbb{N}$ such that if $n\geq N$, then the map $F^{n}$
is $\eta$-homogeneous on $\Lambda$.

###  4.1. Pliss moments

Let $\eta,\varepsilon,\delta\in(0,1)$ with $\bar{\eta}<\varepsilon<\delta$ and
${\bar{\delta}}<1$. Suppose that $F$ is $\eta$-homogeneous on $\Lambda$. We
show that for such maps, regularity is a statistically common occurrence.

Let $p_{0}\in\Lambda$. Suppose that along its forward/backward/full orbit,
$p_{0}$ exhibits asymptotically regular behavior with marginal exponent
$\varepsilon$ (in the sense made rigorous in the propositions stated below).
Loosely speaking, an integer $m$ is a Pliss moment for the orbit of $p_{0}$ if
$p_{m}$ is $(1,\delta)$-regular.

######  Proposition 4.2 (Frequency of Pliss moments along forward orbit).

Let $p_{1}\in\Lambda$. Suppose there exist $E_{p_{1}}^{v}$ or
$E_{p_{1}}^{h}\in\mathbb{P}^{2}_{p_{1}}$ such that

$\liminf_{n\to\infty}\frac{1}{n}\log_{\lambda}\|DF^{n}|_{E_{p_{1}}^{v}}\|>1-\varepsilon$

or

$\limsup_{n\to\infty}\frac{1}{n}\log_{\lambda}\|DF^{n}|_{E_{p_{1}}^{h}}\|<\varepsilon.$

Let $\\{n_{i}\\}_{i=1}^{\infty}\subset\mathbb{N}$ be the increasing set of all
moments such that $p_{n_{i}}$ is $(n_{i},\infty)$-times
$(1,\delta)_{v/h}$-regular along $E_{p_{n_{i}}}^{v/h}$. Then

$\liminf_{i\to\infty}\frac{i}{n_{i}}>1-{\bar{\varepsilon}}/\delta$

If $p_{1}$ is infinitely forward $(L,\varepsilon)_{v/h}$-regular along
$E_{p_{1}}^{v/h}$ for some $L\geq 1$, then

$i>\left(1-{\bar{\varepsilon}}/\delta\right)n_{i}+\delta^{-1}\log_{\lambda}L\hskip
14.22636pt\text{and}\hskip
14.22636ptn_{i}>\left(1-{\bar{\varepsilon}}/\delta\right)n_{i+1}+\delta^{-1}\log_{\lambda}L.$

######  Proposition 4.3 (Frequency of Pliss moments along backward orbit).

Let $p_{-1}\in\Lambda$. Suppose there exists $E_{p_{-1}}^{v}$ or
$E_{p_{-1}}^{h}\in\mathbb{P}^{2}_{p_{-1}}$ such that

$\liminf_{m\to-\infty}\frac{1}{m}\log_{\lambda}\|DF^{m}|_{E_{p_{-1}}^{v}}\|>1-\varepsilon$

or

$\limsup_{m\to-\infty}\frac{1}{m}\log_{\lambda}\|DF^{m}|_{E_{p_{-1}}^{h}}\|<\varepsilon.$

Let $\\{-m_{i}\\}_{i=1}^{\infty}\subset-\mathbb{N}$ be the decreasing set of
all moments such that $p_{-m_{i}}$ is $(\infty,m_{i})$-times
$(1,\delta)_{v/h}$-regular along $E_{p_{-m_{i}}}^{v/h}$. Then

$\liminf_{i\to\infty}\frac{i}{m_{i}}>1-{\bar{\varepsilon}}/\delta.$

If $p_{-1}$ is infinitely backward $(L,\varepsilon)_{v/h}$-regular along
$E_{p_{-1}}^{v/h}$ for some $L\geq 1$, then

$i>(1-{\bar{\varepsilon}}/\delta)m_{i}+\delta^{-1}\log_{\lambda}L\hskip
14.22636pt\text{and}\hskip
14.22636ptm_{i}>(1-{\bar{\varepsilon}}/\delta)m_{i+1}+\delta^{-1}\log_{\lambda}L.$

######  Lemma 4.4.

Let $p_{0}\in\Lambda$ be Pesin $(L,\varepsilon)_{v/h}$-regular along
$E^{v/h}_{p_{0}}\in\mathbb{P}^{2}_{p_{0}}$, and let

$n>-\delta^{-1}\log_{\lambda}{\bar{L}}.$

If $p_{n}$ is $(n,\infty)$-times $(1,\underline{\delta})_{v/h}$-regular along
$E^{v/h}_{p_{n}}$, then $p_{n}$ is Pesin $(1,\delta)_{v/h}$-regular along
$E_{p_{n}}^{v/h}$. Similarly, if $p_{-n}$ is $(\infty,n)$-times
$(1,\underline{\delta})_{v/h}$-regular along $E^{v/h}_{p_{-n}}$, then $p_{-n}$
is Pesin $(1,\delta)_{v/h}$-regular along $E_{p_{-n}}^{v/h}$.

######  Proposition 4.5 (Frequency of Pliss moments along full orbit).

Let $p_{0}\in\Lambda$. Suppose there exists $E_{p_{0}}^{v}$ or
$E_{p_{0}}^{h}\in\mathbb{P}^{2}_{p_{0}}$ such that

$\liminf_{|m|\to\infty}\frac{1}{m}\log_{\lambda}\|DF^{m}|_{E_{p_{0}}^{v}}\|>1-\varepsilon$

or

$\limsup_{|m|\to\infty}\frac{1}{m}\log_{\lambda}\|DF^{m}|_{E_{p_{0}}^{h}}\|<\varepsilon.$

Let $\\{m_{i}\\}_{i=-\infty}^{\infty}\subset\mathbb{Z}$ be the increasing set
of all moments with

$m_{n}\geq 0\hskip 14.22636pt\text{and}\hskip 14.22636ptm_{-n}\leq 0\hskip
14.22636pt\text{for}\hskip 14.22636ptn\geq 0,$

such that $p_{m_{i}}$ is Pesin $(1,\delta)_{v/h}$-regular along
$E_{p_{m_{i}}}^{v/h}$. Then

$\liminf_{|i|\to\infty}\frac{i}{m_{i}}>1-{\bar{\varepsilon}}/\delta.$

The numbers $n_{i}$ in Proposition 4.2, $-m_{i}$ in Proposition 4.3 and
$m_{i}$ in Proposition 4.5 are referred to as Pliss moments.

Define

$\Lambda^{\pm}_{\varepsilon}:=\bigcup_{L\geq
1}\Lambda^{\pm}_{L,\varepsilon}\hskip 14.22636pt\text{and}\hskip
14.22636pt\Lambda^{P}_{\varepsilon}:=\bigcup_{L\geq
1}\Lambda^{P}_{L,\varepsilon}.$

Note that these sets are totally invariant. Also, recall that
$\Lambda^{\pm}_{\operatorname{Ly}}$ and $\Lambda^{P}_{\operatorname{Ly}}$
denote the set of forward/backward and Pesin Lyapunov regular points in
$\Lambda$ respectively.

######  Proposition 4.6.

We have $\Lambda^{\pm}_{\operatorname{Ly}}\subset\Lambda^{\pm}_{\varepsilon}$
and $\Lambda^{P}_{\operatorname{Ly}}\subset\Lambda^{P}_{\varepsilon}.$

######  Proposition 4.7.

Let $p_{0}\in\Lambda$ be a Lyapunov forward/backward/Pesin regular point. Then
there exists $N=N(p_{0})\geq 0$ such that for $n\geq N$, the point $p_{0}$ is
infinitely forward/backward/Pesin $(1,\varepsilon)$-regular for the map
$F^{n}$.

######  Proposition 4.8.

Suppose that $\mu$ is the unique invariant probability measure for
$F|_{\Lambda}$. Then

$\mu(\Lambda^{\pm}_{1,\varepsilon})>\frac{\varepsilon}{\varepsilon+\eta}\hskip
14.22636pt\text{and}\hskip
14.22636pt\mu(\Lambda^{P}_{1,\varepsilon})>\frac{\varepsilon-\eta}{\varepsilon+\eta}.$

##  5\. Critical Orbits

Let $F:\Omega\to F(\Omega)\Subset\Omega$ be the diffeomorphism considered in
Section 3 (not necessarily homogeneous). For $p\in\Lambda$ and
$n\in\mathbb{N}$, let $r_{p}:\mathbb{R}/\mathbb{Z}\to\mathbb{R}^{+}$ and
$\theta_{p}:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ be given by

$r_{p}(t)(\cos(2\pi\theta_{p}(t)),\sin(2\pi\theta_{p}(t))):=D_{p}F(\cos(2\pi
t),\sin(2\pi t)).$

For

$E_{p}^{t}=\\{l(\cos(2\pi t),\sin(2\pi
t))\;|\;l\in\mathbb{R}\\}\in\mathbb{P}^{2}_{p},$

the projective derivative of $F$ at $E_{p}^{t}$ is defined as

$\partial_{\mathbb{P}}F(E_{p}^{t}):=\theta_{p}^{\prime}(t)=\frac{\operatorname{Jac}_{p}F}{\|D_{p}F|_{E_{p}^{t}}\|^{2}}.$

######  Definition 5.1.

A point $c\in\Lambda$ is a (general) critical point if there exists a tangent
direction $E_{c}^{*}\in\mathbb{P}_{c}^{2}$ such that

$\partial_{\mathbb{P}}F^{\pm n}(E_{c}^{*})\geq 1\hskip
14.22636pt\text{for}\hskip 14.22636ptn\in\mathbb{N}.$

In this case, $E_{c}^{*}$ is referred to as a (general) critical direction at
$c$.

Related to critical point, the following theorem is proved in [CP2].

######  Theorem 5.2.

The set $\Lambda$ admits a dominated splitting if and only if it does not
contain any critical point of $F$.

######  Remark 5.3.

In particular, if $F|_{\Lambda}$ is dissipative, it follows that it contains a
critical point of $F$ if and only if $F|_{\Lambda}$ is not uniformly partially
hyperbolic. Moreover, as explained in [CPT], since odometers can not be
partially hyperbolic, then they has to contain (general) critical points.

###  5.1. Critical points obstruct linearization

Let $\varepsilon\in(0,1)$ be a small constant such that
${\bar{\varepsilon}}<1$.

######  Proposition 5.4 (No critical point in regular neighborhoods).

Let $p_{0}\in\Lambda$ be $(M,N)$-times $(L,\varepsilon)_{v}$-regular along
$E_{p_{0}}^{v}\in\mathbb{P}^{2}_{p_{0}}$ for some $M,N\in\mathbb{N}$ and
$L\geq 1$. Let

$\\{\Phi_{p_{m}}:\mathcal{U}_{p_{m}}\to U_{p_{m}}\\}_{m=-M}^{N}$

be a linearization of the $(M,N)$-orbit of $p_{0}$ with vertical direction
$E_{p_{0}}^{v}$. Suppose there exists a point $q_{0}\in\mathcal{U}_{p_{0}}$
such that $q_{m}\in\mathcal{U}_{p_{m}}$ for all $-M\leq m\leq N.$ If

$M,N>-K\log_{\lambda}L+K$

for some uniform constant $K\geq 1$, then $q_{0}$ cannot be a critical point.

###### Proof.

Suppose towards a contradiction that $q_{0}$ is a critical point with a
critical direction $E_{q_{0}}^{*}$. Then
$\partial_{\mathbb{P}}F^{N}(E_{q_{0}}^{*})\geq 1$ and this implies

$\|D_{q_{0}}F^{N}|_{E_{q_{0}}^{*}}\|\leq(\operatorname{Jac}_{q_{0}}F^{N})^{1/2}<\lambda^{(1-\eta)N/2}.$

By Proposition 3.9 and 3.11, we see that

$\measuredangle(E_{q_{0}}^{*},E_{q_{0}}^{v})<{\bar{L}}\lambda^{-{\bar{\varepsilon}}N}\lambda^{(1-\eta)N/2}<k$

for some sufficiently small uniform constant $k>0$. It follows that

$\|D_{q_{0}}F^{-M}|_{E_{q_{0}}^{*}}\|>{\bar{L}}^{-1}\lambda^{-(1-{\bar{\varepsilon}})M}.$

Thus,

$\partial_{\mathbb{P}}F^{-M}(E_{q_{0}}^{*})=\frac{\operatorname{Jac}_{q_{0}}F^{-M}}{\|D_{q_{0}}F^{-M}|_{E_{q_{0}}^{*}}\|^{2}}<\frac{{\bar{L}}\lambda^{-(1+\varepsilon)M}}{{\bar{L}}^{-2}\lambda^{-2(1-{\bar{\varepsilon}})M}}<{\bar{L}}\lambda^{(1-{\bar{\varepsilon}})M}<1.$

This is a contradiction. ∎

######  Proposition 5.5.

Let $p_{0}\in\Lambda$ be Pesin $(L,\varepsilon)$-regular for some $L\geq 1$,
and let $\\{\Phi_{p_{m}}:\mathcal{U}_{p_{m}}\to U_{p_{m}}\\}_{m\in\mathbb{Z}}$
be a linearization of the full orbit of $p_{0}$. Then the regular radii
(specified in Theorem 3.6) can be chosen uniformly small so that
$\mathcal{U}_{p_{m}}$ does not contain a critical point for all
$m\in\mathbb{Z}$.

###### Proof.

Let
$\\{\Phi_{p_{m}}:\hat{\mathcal{U}}_{p_{m}}\to\hat{U}_{p_{m}}\\}_{m\in\mathbb{Z}}$
be a starting uniformization of the orbit of $p_{0}$. Recall that

$\hat{U}_{p_{m}}:=\mathbb{B}(cL^{-C}\lambda^{{\bar{\varepsilon}}|m|})$

for some uniform constants $c\in(0,1)$ and $C\geq 1$. Let $d\in(0,c)$ and
$D>C$, and denote

$U_{p_{m}}:=\mathbb{B}(dL^{-D}\lambda^{{\bar{\varepsilon}}|m|})\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{U}_{p_{m}}:=\Phi_{p_{m}}^{-1}(U_{c_{m}}).$

Suppose towards a contradiction that $q_{0}$ is a critical point and
$q_{0}\in\mathcal{U}_{p_{m}}$ for some $m\in\mathbb{Z}$. Then for
$i\in\mathbb{N}$ such that $q_{-i}\in\hat{\mathcal{U}}_{p_{m-i}}$, we have

$\operatorname{dist}(p_{m-i},q_{-i})<\lambda^{-(1+{\bar{\varepsilon}})i}dL^{-D}\lambda^{{\bar{\varepsilon}}|m|}.$

Thus, we have $q_{-i}\in\hat{\mathcal{U}}_{p_{m-i}}$ as long as

$cL^{-C}\lambda^{{\bar{\varepsilon}}|m-i|}>dL^{-D}\lambda^{-(1+\eta)i}\lambda^{{\bar{\varepsilon}}|m|},$

or more simply,

$(d/c)L^{C-D}<\lambda^{(1+{\bar{\varepsilon}})i}.$

Let $i=M$ be the first value for which the above inequality fails. Then

$M>(1+{\bar{\varepsilon}})^{-1}\left(\log_{\lambda}(d/c)-(D-C)\log_{\lambda}L\right).$

We also have

$\operatorname{dist}(p_{m+i},q_{i})<\lambda^{-{\bar{\varepsilon}}i}dL^{-D}\lambda^{{\bar{\varepsilon}}|m|}\hskip
14.22636pt\text{for}\hskip 14.22636pti\geq 0.$

Thus, we have $q_{i}\in\hat{\mathcal{U}}_{p_{m+i}}$ as long as

$cL^{-C}\lambda^{{\bar{\varepsilon}}|m+i|}>\lambda^{-{\bar{\varepsilon}}i}dL^{-D}\lambda^{{\bar{\varepsilon}}|m|},$

or more simply,

$(d/c)L^{C-D}<\lambda^{{\bar{\varepsilon}}i}.$

Let $i=N$ be the smallest value for which the above inequality fails. Then

$N>{\bar{\varepsilon}}^{-1}\left(\log_{\lambda}(d/c)-(D-C)\log_{\lambda}L\right).$

Hence, by choosing $d$ and $D$ sufficiently smaller and larger than $c$ and
$C$ respectively, the result follows from Proposition 5.4. ∎

###  5.2. Regular critical values

A point $c_{1}\in\Lambda$ is an $\varepsilon$-regular critical value if
$c_{1}$ is both infinite-time forward and backward $(1,\varepsilon)$-regular,
and

$E_{c_{1}}^{*}:=E_{c_{1}}^{ss}=E_{c_{1}}^{c}.$

We refer to $c_{0}:=F^{-1}(c_{1})$ as an $\varepsilon$-regular critical point.
Note that the strong stable manifold $W^{ss}(c_{1})$ and the center manifold
$W^{c}(c_{1})$ have a tangency at $c_{1}$. If this tangency is of order $2$,
then we say that $c_{1}$ is of quadratic type.

######  Proposition 5.6.

Let $c_{1}\in\Lambda$ be an $\varepsilon$-regular critical value. Then $c_{1}$
is the unique general critical point in its orbit.

######  Proposition 5.7.

Suppose $F$ is $\eta$-homogeneous for some
$\eta\in(0,\underline{\varepsilon})$. If $c_{1}\in\Lambda$ is a regular
critical orbit point, then there exists $N\in\mathbb{N}$ such that for $n\geq
N$, the point $c_{1}$ is an $\varepsilon$-regular critical value for the map
$F^{n}$.

###### Proof.

The result follows immediately from Proposition 4.7. ∎

For $a,b\in\mathbb{R}$, denote

$\mathbb{L}^{h}_{b}:=\\{(x,b)\;|\;x\in\mathbb{R}\\}\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathbb{L}^{v}_{a}:=\\{(a,y)\;|\;y\in\mathbb{R}\\}.$

Additionally, let $f_{a}(x):=x^{2}+a$, and denote

$\mathbb{P}^{h}_{a}:=\\{(x,f_{a}(x))\;|\;x\in\mathbb{R}\\}\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathbb{P}^{v}_{a}:=\\{(f_{a}(y),y)\;|\;y\in\mathbb{R}\\}.$

######  Theorem 5.8 (Uniformization along a regular critical orbit).

Let $c_{1}$ be an $\varepsilon$-regular critical value of quadratic type with
the critical direction $E_{c_{1}}^{*}\in\mathbb{P}^{2}_{c_{1}}$. Then there
exist linearizations

$\\{\Phi_{c_{1+n}}:\mathcal{U}_{c_{1+n}}\to
U_{c_{1+n}}\\}_{n=0}^{\infty}\hskip 14.22636pt\text{and}\hskip
14.22636pt\\{\Phi_{c_{-n}}:\mathcal{U}_{c_{-n}}\to
U_{c_{-n}}\\}_{n=0}^{\infty}$

of the forward orbit of $c_{1}$ and the backward orbit of $c_{0}$ respectively
that induce the following Hénon-like transition at the critical moment:

$F_{c_{0}}(x,y):=\Phi_{c_{1}}\circ
F|_{\mathcal{U}_{c_{0}}}\circ\Phi_{c_{0}}^{-1}(x,y)=(x^{2}-\lambda y,x)\hskip
14.22636pt\text{for}\hskip 14.22636pt(x,y)\in U_{c_{0}}.$

###### Proof.

Let $\\{\Psi_{c_{1+n}}\\}_{n=0}^{\infty}$ and
$\\{\Psi_{c_{-n}}\\}_{n=0}^{\infty}$ be some linearizations of the forward
orbit of $c_{1}$ and backward orbit of $c_{0}$. Note that

$\Psi_{c_{1}}(W^{ss}_{\operatorname{loc}}(c_{1}))\subset\mathbb{L}^{v}_{0}\hskip
14.22636pt\text{and}\hskip
14.22636pt\Psi_{c_{0}}(W^{c}(c_{0}))\subset\mathbb{L}^{h}_{0}.$

We may assume that the regular radius $\rho_{0}:=l_{c_{0}}$ at $c_{0}$ is
chosen uniformly smaller than the regular radius $\rho_{1}:=l_{c_{1}}$ at
$c_{1}$ such that $F(\mathcal{U}_{c_{0}})\subset\mathcal{U}_{c_{1}}$. Denote
$I^{i}:=(-\rho_{i},\rho_{i})$ for $i\in\\{0,1\\}$, and
$\hat{F}_{0}:=\Psi_{c_{1}}\circ F\circ\Psi_{c_{0}}^{-1}$.

The curve

$\hat{F}_{0}(I^{0}\times\\{0\\})=\Psi_{c_{1}}(W^{c}(c_{1}))$

is a quadratic vertical graph that has a unique point of genuine vertical
tangency given by

$v_{0}=(0,0)=\Psi_{c_{1}}(c_{1})\in W^{ss}_{\operatorname{loc}}(c_{1})\cap
W^{c}(c_{1}).$

Consequently, for $|y|<\rho_{0}$, the curve $\hat{F}_{0}(I^{0}\times\\{y\\})$
is a quadratic vertical graph that has a unique point $v_{y}$ whose tangent
vector is genuine vertical. Moreover, there exist an interval $J^{1}\subset
I^{1}$ containing $0$; and $C^{r}$-maps $g_{0}:I^{0}\to I^{0}$ and
$g_{1}:J^{1}\to I^{1}$ such that:

$\hat{F}_{0}(g_{0}(y),y)=v_{y}=(u,g_{1}(u)).$

The vertical graph $\Gamma^{v}(g_{0})$ can be straightened to $\\{0\\}\times
I^{0}$ by a map of the form

$\Phi_{0,0}(x,y)=(\phi_{0,0}(x,y),y).$

Likewise, the horizontal graph $\Gamma^{h}(g_{1})$ can be straightened to
$J^{1}\times\\{0\\}$ by a map of the form

$\Phi_{1,0}(u,w)=(u,\phi_{1,0}(u,w)).$

Since the derivatives $D\Phi_{0,0}$ and $D\Phi_{1,0}$ preserve the horizontal
and vertical directions respectively, we see that the former preserves
horizontal tangencies, while the latter preserves vertical tangencies.

Observe that $F_{0,0}:=\Phi_{1,0}\circ\hat{F}_{0}\circ\Phi_{0,0}^{-1}$ maps
$\\{0\\}\times I^{0}\subset\mathbb{L}^{v}_{0}$ to
$J^{1}\times\\{0\\}\subset\mathbb{L}^{h}_{0}$. Define

$\psi_{1,1}(y):=\pi_{h}\circ F_{0,0}(0,y)\hskip 14.22636pt\text{for}\hskip
14.22636pty\in I^{0};$

and let

$\phi_{1,1}(u):=\lambda\psi_{1,1}^{-1}(u)\hskip 14.22636pt\text{for}\hskip
14.22636ptu\in J^{1}.$

Extend $\phi_{1,1}$ to a $C^{r}$-diffeomorphism of $I^{1}\supset J^{1}$, and
let $\Phi_{1,1}(u,w)=(\phi_{1,1}(x),y)$. Then for $F_{0,1}:=\Phi_{1,1}\circ
F_{0,0}$, we have $F_{0,1}(0,y)=(-\lambda y,0)$.

The collection $\\{F_{0,1}(I^{0}\times\\{y\\})\\}_{y\in I^{0}}$ forms a
$C^{r}$-foliation of $F_{0,1}(U_{c_{0}})$ by quadratic vertical curves in
$U_{c_{1}}$. Extend this foliation to a $C^{r}$-foliation
$\\{\gamma^{v}_{z}\\}_{z\in\hat{J}^{1}}$ of a domain $\hat{U}_{c_{1}}\supset
U_{c_{1}}$ by quadratic vertical curves, so that
$\hat{U}_{c_{1}}\cap\mathbb{L}^{h}_{0}=\hat{J}^{1}$, and for $(u,w)\in
U_{c_{1}}$, there exists a unique value $\tau(u,w)\in\hat{J}^{1}$ such that
$(u,w)\in\gamma^{v}_{\tau(u,w)}$. Define

$\Phi_{1,2}(u,w):=(u,\phi_{1,2}(u,w))\hskip 14.22636pt\text{for}\hskip
14.22636pt(u,w)\in U_{c_{1}},$

where

$\phi_{1,2}(u,w):=\left\\{\begin{array}[]{cl}\sqrt{u-\tau(u,w)}&:w\geq 0\\\
-\sqrt{u-\tau(u,w)}&:w<0;\end{array}\right.$

and let $F_{0,2}:=\Phi_{1,2}\circ F_{0,1}$. Then for $y\in I^{0}$, we have
$F_{0,2}(I^{0}\times\\{y\\})\subset\mathbb{P}^{v}_{-\lambda y}$.

Finally, let

$\Phi_{0,3}(x,y)=(\phi_{0,3}(x,y),y)$

be the map which straightens the vertical foliation given by
$\\{F_{0,2}^{-1}(I^{1}\times\\{w\\})\\}_{w\in I^{1}}$ to the genuine vertical
foliation. Since $\Phi_{0,3}$ preserves the genuine horizontal foliation, the
image of a genuine horizontal line under
$F_{c_{0}}:=F_{0,2}\circ\Phi_{0,3}^{-1}$ is the same as $F_{0,2}$. It is now
straightforward to check that

$\Phi_{c_{0}}:=\Phi_{0,3}\circ\Phi_{0,0}\circ\Psi_{c_{0}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\Phi_{c_{1}}:=\Phi_{1,2}\circ\Phi_{1,1}\circ\Phi_{1,0}\circ\Psi_{c_{1}}$

provide the desired regular charts near $c_{1}$ and $c_{0}$ respectively. ∎

###  5.3. Closest returns for regular critical orbits

Let $c_{1}\in\Lambda$ be an $\varepsilon$-regular critical value. Let
$\delta\in({\bar{\varepsilon}},1)$ with ${\bar{\delta}}<1$.

######  Proposition 5.9.

Let $\\{\Phi_{c_{m}}:\mathcal{U}_{c_{m}}\to U_{c_{m}}\\}_{m\in\mathbb{Z}}$ be
a uniformization of the orbit of $c_{0}$ given in Theorem 5.8. Then the
regular radii can be chosen uniformly small so that
$c_{0}\not\in\mathcal{U}_{c_{m}}\setminus\\{c_{m}\\}$ for $m\in\mathbb{Z}$.

###### Proof.

Let
$\\{\Phi_{c_{m}}:\hat{\mathcal{U}}_{c_{m}}\to\hat{U}_{c_{m}}\\}_{m\in\mathbb{Z}}$
be a starting uniformization of the orbit of $c_{0}$. Recall that
$\hat{U}_{c_{m}}:=\mathbb{B}(k\lambda^{-{\bar{\varepsilon}}|m|})$ for some
uniform constant $k\in(0,1)$. Let $l\in(0,k)$, and denote

$U_{c_{m}}:=\mathbb{B}(l\lambda^{{\bar{\varepsilon}}|m|})\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{U}_{c_{m}}:=\Phi_{c_{m}}^{-1}(U_{c_{m}}).$

Suppose towards a contradiction that
$c_{0}\in\mathcal{U}_{c_{m}}\setminus\\{c_{m}\\}$. Note that by choosing $l$
small, we can assume that $|m|$ is uniformly large. Let $1\leq M,N\leq|m|$ be
the largest values for which we have $c_{i}\in\hat{\mathcal{U}}_{c_{m+i}}$ for
$-M<i<N$. Arguing in a similar way as in the proof of Proposition 5.5, we
obtain

$M>\min\\{(1+{\bar{\varepsilon}})^{-1}\log_{\lambda}(l/k),|m|\\},$

and

$N>\min\\{{\bar{\varepsilon}}^{-1}\log_{\lambda}(l/k),|m|\\}.$

Thus, by choosing $l$ sufficiently smaller than $k$, the result follows from
Proposition 5.4. ∎

######  Proposition 5.10 (Slow recurrence of the critical orbit).

There exists a uniformly small constant $r>0$ such that the following holds.
If

$\mathbb{D}_{c_{m}}(r\lambda^{\delta|m|})\cap\mathbb{D}_{c_{0}}(r)\neq\varnothing$

for some $m\in\mathbb{Z}$, then

$|m|>\frac{1}{{\bar{\varepsilon}}}\log_{\lambda}\left(\operatorname{dist}(c_{m},c_{0})\right).$

###### Proof.

For $m\in\mathbb{Z}$, let $\Phi_{c_{m}}$, $\hat{\mathcal{U}}_{c_{m}}$ and
$\mathcal{U}_{c_{m}}$ be the objects defined in the proof of Proposition 5.9.
Then

$\operatorname{dist}(c_{m},c_{0})>\operatorname{radius}(\mathcal{U}_{c_{m}})>l_{c_{0}}\lambda^{{\bar{\varepsilon}}|m|}.$

Let $r:=\underline{l_{c_{0}}}$. Then we have

$\operatorname{dist}(c_{m},c_{0})<r(1+\lambda^{\delta|m|})<\underline{l_{c_{0}}}.$

Thus,

$|m|>{\bar{\varepsilon}}^{-1}\left(\log_{\lambda}\left(\operatorname{dist}(c_{m},c_{0})\right)-\log_{\lambda}l_{c_{0}}\right)={\bar{\varepsilon}}^{-1}\log_{\lambda}\left(\operatorname{dist}(c_{m},c_{0})\right).$

∎

######  Lemma 5.11.

There exists a uniform constant $r>0$ such that the following holds. Let
$0\leq n<m$. If $\operatorname{dist}(c_{\pm m},c_{0})<r\lambda^{\delta n}$,
then we have $m>{\bar{\varepsilon}}^{-1}n$. Moreover, for all $k\in\mathbb{Z}$
such that $|k|\leq{\bar{\varepsilon}}^{-1}n$, the value $\pm m+k$ is not a
Pliss moment in the forward/backward orbit of $c_{1/0}$.

###### Proof.

We prove the claim for $c_{m}$. The proof of the statement for $c_{-m}$ is
identical.

The first claim follows immediately from Proposition 5.10. Denote
$p_{0}:=c_{m}$, and let

$\\{\hat{\Phi}_{p_{l}}:\hat{\mathcal{U}}_{p_{l}}\to\hat{U}_{p_{l}}\\}_{-m<l<\infty}$

be a linearization of the $(m-1,\infty)$-orbit of $p_{0}$. Then for
$k\in\mathbb{Z}$ such that $|k|\leq n$, we have

$\operatorname{diam}(\hat{\mathcal{U}}_{p_{k}})\asymp\lambda^{{\bar{\varepsilon}}k}>r\lambda^{\varepsilon^{\prime}n}.$

Hence, $c_{0}\in\hat{\mathcal{U}}_{p_{k}}$. Arguing as in the proof of
Proposition 5.9, we obtain a contradiction. ∎

##  6\. Regularity Near a Critical Orbit

Let $F$ be the diffeomorphism considered in Section 3. Let
$\eta,\varepsilon,\delta\in(0,1)$ be small constants satisfying
$\bar{\eta}<\varepsilon$, ${\bar{\varepsilon}}<\delta$ and ${\bar{\delta}}<1$.
Suppose that $F$ is $\eta$-homogeneous on $\Lambda$. Moreover, suppose $F$ has
an $\varepsilon$-regular critical value $c_{1}\in\Lambda$ of quadratic type
with the critical direction $E_{c_{1}}^{*}\in\mathbb{P}^{2}_{c_{0}}$. Let

$\\{\Phi_{c_{m}}:(\mathcal{U}_{c_{m}},c_{m})\to(U_{c_{m}},0)\\}_{m\in\mathbb{Z}}$

be the uniformization of the orbit of $c_{0}$ given in Theorem 5.8.

######  Proposition 6.1 (Recovery of forward regularity).

Let $p_{0}\in\mathcal{U}_{c_{0}}$ and
$\tilde{E}_{p_{0}}^{v}\in\mathbb{P}^{2}_{p_{0}}$. Write
$z_{0}=(x_{0},y_{0}):=\Phi_{c_{0}}(p_{0})$. Let $-n\leq 0$ be a Pliss moment
in the backward orbit of $c_{0}$. If $p_{-i}\in\mathcal{U}_{c_{-i}}$ for
$0\leq i\leq n$, and we have

$\measuredangle(\tilde{E}_{p_{0}}^{v},E^{v}_{p_{0}})>k|x_{0}|\hskip
14.22636pt\text{and}\hskip
14.22636ptn>(1+{\bar{\varepsilon}})\delta^{-1}\log_{\lambda}|x_{0}|,$

for some uniform constant $k\in(0,1)$, then $p_{-n}$ is $n$-times forward
$(O(1),\delta)_{v}$-regular along $\tilde{E}_{p_{-n}}^{v}$.

###### Proof.

Denote

$\tilde{E}^{v}_{z_{0}}:=D\Phi_{c_{0}}(\tilde{E}^{v}_{p_{0}})\hskip
14.22636pt\text{and}\hskip
14.22636pt\theta_{-i}:=\measuredangle(\tilde{E}^{v}_{z_{-i}},E_{z_{-i}}^{gh})\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq i\leq n.$

By Theorem 3.6 ii) and Proposition 3.9, there exist uniform constants $C,c>0$
such that if $\theta_{-i}<C$, then

$\theta_{-i}>c\lambda^{-(1-{\bar{\varepsilon}})i}\theta_{0},$

and if $\theta_{-i}>C$, then

$|\theta_{-i-1}-\pi/2|<\lambda^{(1-{\bar{\varepsilon}})}|\theta_{-i}-\pi/2|.$

Let $-j\leq 0$ be the last moment such that $\theta_{-j}<C$. Observe

$C>\theta_{-j}>ck\lambda^{-(1-{\bar{\varepsilon}})j}|x_{0}|.$

This implies

$j<(1+{\bar{\varepsilon}})\log_{\lambda}|x_{0}|<(1-{\bar{\varepsilon}})\delta
n.$

Observe that for $-n\leq m\leq-j$, we have

$\measuredangle\left(\tilde{E}^{v}_{p_{m}},E^{v}_{p_{m}}\right)<\nu$

for some sufficiently small $\nu>0$. Thus, by Proposition 3.9 and the fact
that $-n$ is Pliss, we obtain

$\|DF^{l}|_{\tilde{E}^{v}_{p_{-n}}}\|<K\lambda^{(1-{\bar{\varepsilon}})l}\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq l\leq n-j,$

where $K>0$ is a uniform constant. The result now follows from the fact that

$\lambda^{(1-{\bar{\varepsilon}})(n-j)}<\lambda^{(1-\delta)n}.$

∎

######  Proposition 6.2 (Obstruction to forward regularity).

There exists a uniform constant $L_{0}\geq 1$ such that the following holds.
For $L\geq L_{0}$, denote $L^{\prime}:=L^{2+{\bar{\delta}}}$. Let
$p_{1}\in\mathbb{D}_{c_{1}}(1/L^{\prime})$;
$\tilde{E}_{p_{1}}^{v}\in\mathbb{P}^{2}_{p_{1}}$ and $n\in\mathbb{N}$. If
$p_{-i}\in\mathcal{U}_{c_{-i}}$ for $0\leq i\leq n$, and we have

$\measuredangle(\tilde{E}_{p_{1}}^{v},E^{v}_{p_{1}})<1/L^{\prime}\hskip
14.22636pt\text{and}\hskip
14.22636pt-(1+{\bar{\delta}})\log_{\lambda}L<n<-\frac{1-{\bar{\delta}}}{\delta}\log_{\lambda}L,$

then $p_{-n}$ is not $n$-times forward $(L,\delta)_{v}$-regular along
$\tilde{E}_{p_{-n}}^{v}$.

###### Proof.

Denote

$z_{-i}:=\Phi_{c_{-i}}(p_{-i}),\hskip
14.22636pt\tilde{E}_{z_{-i}}^{v}:=D\Phi_{c_{-i}}(\tilde{E}_{p_{-i}}^{v})\hskip
14.22636pt\text{and}\hskip
14.22636pt\theta_{-i}:=\measuredangle(\tilde{E}_{z_{-i}}^{v},E_{z_{-i}}^{gh}).$

By Theorems 5.8 and 3.6 ii), and Proposition 3.9, we have

$\theta_{-i}<K\lambda^{-(1+{\bar{\varepsilon}})i}/L^{\prime}$

for some uniform constant $K\geq 1$.

If $\theta_{-i}<\lambda^{{\bar{\varepsilon}}i}\nu$, where $\nu>0$ is a
sufficiently small uniform constant, then

$\measuredangle(\tilde{E}^{v}_{p_{-i}},E^{h}_{p_{-i}})<\bar{\nu}.$

Thus, by Proposition 3.9, we obtain

$\|DF^{-i}|_{\tilde{E}^{v}_{p_{0}}}\|<\lambda^{-{\bar{\varepsilon}}i}.$

Let $j>0$ be the smallest integer such that

$K\lambda^{-(1+{\bar{\varepsilon}})j}/L^{\prime}>\lambda^{{\bar{\varepsilon}}j}\nu.$

Simplifying, we obtain

$j>-(1-{\bar{\varepsilon}})\log_{\lambda}L^{\prime}.$ (6.1)

Suppose $n<j$. By the assumed lower bound on $n$, we have
$L^{-1}>\lambda^{(1-{\bar{\delta}})n}.$ It follows that

$\|DF^{n}|_{\tilde{E}^{v}_{p_{-n}}}\|>\lambda^{{\bar{\varepsilon}}n}>L\lambda^{(1-\delta)n}.$

Hence, $p_{-n}$ is not $n$-times forward $(L,\delta)_{v}$-regular along
$\tilde{E}^{v}_{p_{-n}}$.

Suppose $n>j$. Observe that

$\|DF^{n}|_{\tilde{E}^{v}_{p_{-n}}}\|>\lambda^{{\bar{\varepsilon}}j}\lambda^{(1+\eta)(n-j)}=\lambda^{-(1-{\bar{\varepsilon}})j}\lambda^{(\delta+\eta)n}\lambda^{(1-\delta)n}.$

By (6.1) and the assumed upper bound on $n$, we have

$\lambda^{-(1-{\bar{\varepsilon}})j}\lambda^{(\delta+\eta)n}>(L^{\prime})^{(1-{\bar{\varepsilon}})^{2}}L^{-\frac{(1-{\bar{\delta}})(\delta+\eta)}{\delta}}>L.$

Hence, we again see that $p_{-n}$ is not $n$-times forward
$(L,\delta)_{v}$-regular along $\tilde{E}^{v}_{p_{-n}}$. ∎

######  Proposition 6.3 (Recovery of backward regularity).

Let $p_{1}\in\mathcal{U}_{c_{1}}$ and
$\tilde{E}_{p_{1}}^{h}\in\mathbb{P}^{2}_{p_{1}}$. Write
$z_{1}=(x_{1},y_{1}):=\Phi_{c_{1}}(p_{1}).$ Let $n\geq 1$ be a Pliss moment in
the forward orbit of $c_{1}$. If $p_{i}\in\mathcal{U}_{c_{i}}$ for $1\leq
i\leq n$, and we have

$\measuredangle(\tilde{E}_{p_{1}}^{h},E^{v}_{p_{1}})>k|x_{1}|\hskip
14.22636pt\text{and}\hskip
14.22636ptn>(1+{\bar{\varepsilon}})\delta^{-1}\log_{\lambda}|x_{1}|$

for some uniform constant $k\in(0,1)$, then $p_{n}$ is $n$-times backward
$(O(1),\delta)_{h}$-regular along $\tilde{E}_{p_{n}}^{h}$.

###### Proof.

Denote

$\tilde{E}_{z_{1}}^{h}:=D\Phi_{c_{1}}(\tilde{E}_{p_{1}}^{h})\hskip
14.22636pt\text{and}\hskip
14.22636pt\theta_{i}:=\measuredangle(\tilde{E}_{z_{i}}^{h},E_{z_{i}}^{gv})\hskip
14.22636pt\text{for}\hskip 14.22636pt1\leq i\leq n.$

By Theorem 3.6 ii), Proposition 3.9 and Lemma 3.10, there exist uniform
constants $C,c>0$ such that if $\theta_{i}<C$, then

$\theta_{i}>c\lambda^{-(1-{\bar{\varepsilon}})i}\theta_{1},$

and if $\theta_{i}>C$, then

$|\theta_{i+1}-\pi/2|<\lambda^{(1-{\bar{\varepsilon}})}|\theta_{i}-\pi/2|+\lambda^{(1-{\bar{\varepsilon}})i}.$

Let $j\geq 1$ be the last moment such that $\theta_{j}<C$. Observe

$C>\theta_{j}>ck\lambda^{-(1-{\bar{\varepsilon}})j}|x_{1}|.$

This implies

$j<(1+{\bar{\varepsilon}})\log_{\lambda}|x_{1}|<(1-{\bar{\varepsilon}})\delta
n.$

Observe that for $j\leq m\leq n$, we have

$\measuredangle\left(\tilde{E}^{h}_{p_{m}},E^{h}_{p_{m}}\right)<\nu$

for some sufficiently small $\nu>0$. Thus, by Proposition 3.9 and the fact
that $n$ is Pliss, we obtain

$\|DF^{-l}|_{\tilde{E}^{h}_{p_{n}}}\|<K\lambda^{-{\bar{\varepsilon}}l}\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq l<n-j,$

where $K>0$ is a uniform constant. Thus,

$\|DF^{-n+j-l}|_{\tilde{E}^{h}_{p_{n}}}\|<K\lambda^{-{\bar{\varepsilon}}(n-j)}\lambda^{-(1+\eta)l}\hskip
14.22636pt\text{for}\hskip 14.22636pt0\leq l<j.$

The result now follows from the fact that

$\lambda^{-(1+\eta)j}<\lambda^{-(1-{\bar{\varepsilon}})\delta n}.$

∎

######  Proposition 6.4 (Obstruction to backward regularity).

There exists a uniform constant $L_{0}\geq 1$ such that the following holds.
For $L\geq L_{0}$, denote $L^{\prime}:=L^{2+{\bar{\delta}}}$. Let
$p_{1}\in\mathbb{D}_{c_{1}}(1/L^{\prime})$;
$\tilde{E}_{p_{1}}^{h}\in\mathbb{P}^{2}_{p_{1}}$ and $n\in\mathbb{N}$. If
$p_{i}\in\mathcal{U}_{c_{i}}$ for $1\leq i\leq n$, and we have

$\measuredangle(\tilde{E}_{p_{1}}^{h},E^{v}_{p_{1}})<1/L^{\prime}\hskip
14.22636pt\text{and}\hskip
14.22636pt-(1+{\bar{\delta}})\log_{\lambda}L<n<-\frac{1-{\bar{\delta}}}{\delta}\log_{\lambda}L,$

then $p_{n}$ is not $n$-times backward $(L,\delta)_{h}$-regular along
$\tilde{E}_{p_{n}}^{h}$.

For $p\in\mathbb{R}^{2}$ and $t>0$, let

$\mathbb{D}_{p}(t):=\\{q\in\mathbb{R}^{2}\;|\;\operatorname{dist}(q,p)<t\\}.$

######  Proposition 6.5.

For $t>0$, denote

$\Lambda_{c_{0}}^{t}:=\Lambda\cap\bigcup_{n=0}^{\infty}\mathbb{D}_{c_{-n}}(t\lambda^{\delta
n}).$

Then we have $\mu\left(\Lambda_{c_{0}}^{t}\right)\to 0$ as $t\to 0.$

###### Proof.

Fix a sufficiently small constant $t_{0}>0$ such that

$\mathbb{D}_{c_{-n}}(t_{0}\lambda^{\delta n})\subset\mathcal{U}_{c_{-n}}\hskip
14.22636pt\text{for all}\hskip 14.22636ptn\geq 0.$

Let $K\geq 1$. We claim that there exists $N=N(K)\in\mathbb{N}$ such that if
$p_{0}\in\Lambda\cap\mathbb{D}_{c_{-n}}(t_{0}\lambda^{\delta n})$ for some
$n\geq N$, then $p_{0}$ is not infinitely forward
$(K,\underline{\delta})$-regular.

Suppose towards a contradiction that $p_{0}$ is forward regular. Then by
Proposition 3.4, $p_{n+1}$ is infinitely forward $(K\lambda^{-\delta
n},\underline{\delta})$-regular. Write
$z_{0}=(x_{0},y_{0}):=\Phi_{c_{0}}(p_{0})$. By Theorem 3.6, Proposition 3.9
and $\eta$-homogeneity, we have

$|x_{0}|<t_{0}\lambda^{(\delta-\eta)n}\hskip 14.22636pt\text{and}\hskip
14.22636pt|y_{0}|<t_{0}\lambda^{n}.$

By Theorem 5.8,

$\operatorname{dist}(p_{n+1},c_{1})<\lambda^{(\delta-\bar{\eta})n}.$

It follows that $p_{n+m}\in\mathcal{U}_{c_{m}}$ for all $1\leq m\leq M$, where
$M\in\mathbb{N}$ is the smallest integer such that

$\lambda^{(\delta-\bar{\eta})n}\lambda^{-\eta
M}<\lambda^{{\bar{\varepsilon}}M}.$

This implies that $M>n/{\bar{\varepsilon}}$. By Proposition 3.11, we have

$\measuredangle(E^{ss}_{p_{n+1}},E^{v}_{p_{n+1}})<\lambda^{n/{\bar{\varepsilon}}}<\operatorname{dist}(p_{n+1},c_{1})$

for all $n$ sufficiently large. Observe that

$\frac{1-{\bar{\delta}}}{(2+{\bar{\delta}})\underline{\delta}n}\log_{\lambda}(\operatorname{dist}(p_{n+1},c_{1}))\to\infty\hskip
14.22636pt\text{as}\hskip 14.22636ptn\to\infty.$

Thus, by Proposition 6.2, $p_{0}$ is not infinitely forward
$(K,\underline{\delta})$-regular for any finite constant $K$.

For $t\in(0,t_{0})$, let $N(t)\in\mathbb{N}\cup\\{\infty\\}$ be the largest
integer such that for all $q\in\Lambda_{c_{0}}^{t}$, there exists
$n=n(t,q)\geq N(t)$ such that $q\in\mathbb{D}_{c_{-n}}(t_{0}\lambda^{\delta
n})$. Clearly, $N(t)\to\infty$ as $t\to 0$. Thus, by above, we have

$\Lambda_{c_{0}}^{t}\cap\Lambda^{+}_{K(t),\underline{\delta}}=\varnothing,$

with $K(t)\to\infty$ as $t\to 0$. The result now follows immediately from
Proposition 3.24. ∎

###  6.1. Critical tunnels and valuable crescents

Let ${t}\in(0,l_{c_{0}}]$. Denote

$\delta>\delta^{\prime}:=(1-{\bar{\delta}})\delta>\delta^{\prime\prime}:=(1-{\bar{\delta}})\delta^{\prime}.$

For $n\geq 0$, let $j_{n}\geq 0$ be the largest integer less than $n$ such
that $-j_{n}$ is a Pliss moment in the backward orbit of $c_{0}$. Define the
($1/\delta^{\prime\prime}$-pinched, $\delta^{\prime}$-truncated) critical
tunnel at $c_{-n}$ (of length ${t}$) as

$\mathcal{T}_{c_{-n}}=\mathcal{T}_{c_{-n}}({t}):=F^{-n}\left(\mathcal{T}_{c_{0}}^{1/\delta^{\prime\prime},\delta^{\prime}j_{n}}({t})\right)$

(see Subsection 3.4). Note that
$F(\mathcal{T}_{c_{-n-1}})\subseteq\mathcal{T}_{c_{-n}}$, and the equality
holds unless

$n=j_{n}>\frac{\log_{\lambda}{t}}{\delta^{\prime}}.$

Similarly, for $n\geq 1$, let $i_{n}\geq 1$ be the largest integer less than
$n$ such that $i_{n}$ is a Pliss moment in the forward orbit of $c_{1}$.
Define the ($1/\delta^{\prime\prime}$-pinched, $\delta^{\prime}$-truncated)
valuable crescent at $c_{n}$ (of length ${t}$) by

$\mathcal{T}_{c_{n}}=\mathcal{T}_{c_{n}}({t}):=F^{n-1}\left(\mathcal{T}_{c_{1}}({t})\cap\mathcal{U}_{c_{1}}^{\delta^{\prime}i_{n}}\right)\hskip
14.22636pt\text{where}\hskip
14.22636pt\mathcal{T}_{c_{1}}({t}):=F(\mathcal{T}_{c_{0}}({t})).$

Note that $F^{-1}(\mathcal{T}_{c_{n}})\subseteq\mathcal{T}_{c_{n-1}}$, and the
equality holds unless

$n=i_{n}>\frac{2\log_{\lambda}{t}}{\delta^{\prime}}.$

######  Lemma 6.6.

For ${t}\in(0,l_{c_{0}}]$, let

$N:=\frac{(1+{\bar{\varepsilon}})\log_{\lambda}{t}}{\delta^{\prime}}.$

Then

$\mathcal{T}_{c_{-n}}({t})=\mathcal{T}_{c_{-n}}(l_{c_{0}})\hskip
14.22636pt\text{for}\hskip 14.22636ptn>N$

and

$\mathcal{T}_{c_{1+n}}({t})=\mathcal{T}_{c_{1+n}}(l_{c_{0}})\hskip
14.22636pt\text{for}\hskip 14.22636ptn>2N.$

######  Proposition 6.7.

For $m\in\mathbb{Z}$, we have
$\mathcal{T}_{c_{m}}\subset\mathcal{U}_{c_{m}}\setminus\\{c_{m}\\}.$

###### Proof.

The result follows immediately from Lemma 3.15 (for $m\geq 1$) and Lemma 3.16
(for $m\leq 0$). ∎

######  Proposition 6.8.

For $m\in\mathbb{Z}$, we have
$\operatorname{diam}(\mathcal{T}_{c_{m}})<\lambda^{\delta|m|}.$

###### Proof.

Suppose $m=-n\leq 0$. By Proposition 6.7, we have

$\mathcal{T}_{c_{-n}}=\Phi_{c_{-n}}^{-1}\circ
F_{c_{0}}^{-n}\left(\mathcal{T}_{c_{0}}^{1/\delta^{\prime\prime},\delta^{\prime}j_{n}}({t})\right).$

Let $q_{-n}\in\mathcal{T}_{c_{-n}}$. Write
$z_{-n}=(x_{-n},y_{-n}):=\Phi_{c_{-n}}(q_{-n}).$ By Theorem 3.6 ii) and
Proposition 3.9, we have

$|x_{-n}|<\lambda^{-{\bar{\varepsilon}}n}|x_{0}|<\lambda^{-{\bar{\varepsilon}}n}\lambda^{\delta^{\prime}n}<\lambda^{(1-{\bar{\varepsilon}})\delta^{\prime}n}.$

Moreover,

$|y_{-n}|<\lambda^{-(1+{\bar{\varepsilon}})n}|y_{0}|<\lambda^{-(1+{\bar{\varepsilon}})n}|x_{0}|^{1/\delta^{\prime\prime}}<\lambda^{-(1+{\bar{\varepsilon}})n}\lambda^{(1+{\bar{\delta}})n}<\lambda^{{\bar{\delta}}n}.$

The result now follows from the fact that

$\|\Phi_{c_{-n}}^{-1}\|_{C^{r}}=O(\lambda^{-{\bar{\varepsilon}}n}).$

A similar argument proves the case when $m\geq 1$. ∎

######  Proposition 6.9 (Invariance of tunnels and crescents).

Let $m\in\mathbb{Z}$, and $n>|m|$. Then the following statements hold.

  1. i)

If

$F^{-1}(\mathcal{T}_{c_{-n+1}})\cap\mathcal{T}_{c_{m}}\neq\varnothing,$

then

$n>{\bar{\varepsilon}}^{-1}|m|\hskip 14.22636pt\text{and}\hskip
14.22636pt\mathcal{T}_{c_{-n}}=F^{-1}(\mathcal{T}_{c_{-n+1}}).$

  2. ii)

If

$F(\mathcal{T}_{c_{n-1}})\cap\mathcal{T}_{c_{m}}\neq\varnothing,$

then

$n>{\bar{\varepsilon}}^{-1}|m|\hskip 14.22636pt\text{and}\hskip
14.22636pt\mathcal{T}_{c_{n}}=F(\mathcal{T}_{c_{n-1}}).$

###### Proof.

Suppose

$F^{-1}(\mathcal{T}_{c_{-n+1}})\cap\mathcal{T}_{c_{-m}}=\varnothing$

for some $0\leq m<n$.

Note that

$\operatorname{dist}(c_{-n+m},c_{0})<\operatorname{diam}(\mathcal{T}_{c_{0}}\cap\mathcal{U}_{c_{0}}^{{\bar{\delta}}m})+\lambda^{-\eta
m}\operatorname{diam}(F^{-1}(\mathcal{T}_{c_{-n+1}}))<\lambda^{{\bar{\delta}}m}.$

The result now follows immediately from Lemma 5.11.

The other cases can be proved in a similar way. ∎

If $p\in\mathcal{T}_{c_{-n}}({t})$ for some $n\geq 0$, then the ${t}$-critical
depth of $p$ is defined as the largest integer $0\leq
d_{-}:=\operatorname{cd}_{t}(p)\leq\infty$ such that
$p\in\mathcal{T}_{c_{-d_{-}(p)}}({t}).$ Similarly, if
$q\in\mathcal{T}_{c_{n}}({t})$ for some $n\geq 1$, then the ${t}$-valuable
depth of $p$ is defined as the largest integer $1\leq
d_{+}:=\operatorname{vd}_{t}(p)\leq\infty$ such that
$p\in\mathcal{T}_{c_{d_{+}(p)}}({t}).$

Define the critical dust of $c_{0}$ as

$\Delta^{c}_{c_{0}}:=\bigcap_{n=0}^{\infty}\bigcup_{i=n}^{\infty}\mathcal{T}_{c_{-i}}({t})=\\{p\in\Lambda\;|\;\operatorname{cd}_{t}(p)=\infty\\}.$

Similarly, define the valuable dust of $c_{1}$ as

$\Delta^{v}_{c_{1}}:=\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}\mathcal{T}_{c_{i}}({t})=\\{p\in\Lambda\;|\;\operatorname{vd}_{t}(p)=\infty\\}.$

Lastly, define the full critical dust of $c_{0}$ as

$\Delta^{fc}_{c_{0}}:=\Delta^{c}_{c_{0}}\cup\Delta^{v}_{c_{1}}.$

######  Proposition 6.10.

We have

$\Delta^{c}_{c_{0}}=\bigcap_{{t}>0}\\{p\in\Lambda\;|\;\operatorname{cd}_{t}(p)=\infty\\}$

and

$\Delta^{v}_{c_{1}}=\bigcap_{{t}>0}\\{p\in\Lambda\;|\;\operatorname{vd}_{t}(p)=\infty\\}.$

###### Proof.

The result follows immediately from Lemma 6.6. ∎

######  Proposition 6.11.

The sets $\Delta^{c/fc}_{c_{0}}$ and $\Delta_{c_{1}}^{v}$ are totally
invariant. Consequently, $\mu(\Delta_{c_{0}}^{fc})=0$.

###### Proof.

Let $p\in\Delta^{c}_{c_{0}}$. If $p\in\mathcal{T}_{c_{-n}}$ with $n>0$, then
$p\in\mathcal{T}_{c_{-n+1}}$. Thus,
$F(\Delta^{c}_{c_{0}})\subset\Delta^{c}_{c_{0}}.$ Let $k\geq 0$ be the
smallest integer such that $p\in\mathcal{T}_{c_{-k}}$. Then by Proposition
6.9, we have $F^{-1}(\mathcal{T}_{c_{-n}})=\mathcal{T}_{c_{-n-1}}.$ Thus,
$F^{-1}(\Delta^{c}_{c_{0}})\subset\Delta^{c}_{c_{0}}.$ The total invariance of
$\Delta^{v}_{c_{1}}$ can be proved in a similar way. ∎

The tunnel escape moment of $p_{0}\in\mathcal{T}_{c_{0}}$ is the unique
integer $-j<0$ such that

$p_{-j+1}\in\mathcal{T}_{c_{-j+1}}\hskip 14.22636pt\text{and}\hskip
14.22636ptp_{-j}\not\in\mathcal{T}_{c_{-j}}.$

Similarly, the crescent escape moment of $p_{1}\in\mathcal{T}_{c_{1}}$ is the
unique integer $i>1$ such that

$p_{i-1}\in\mathcal{T}_{c_{i-1}}\hskip 14.22636pt\text{and}\hskip
14.22636ptp_{i}\not\in\mathcal{T}_{c_{i}}.$

Write $(x_{1},y_{1}):=\Phi_{c_{1}}(p_{1}).$ If
$\tilde{E}^{v}_{p_{1}}\in\mathbb{P}^{2}_{p_{1}}$ is $k|x_{1}|^{1/2}$-vertical
in $\mathcal{U}_{c_{1}}$ for some sufficiently small uniform constant $k>0$,
then we say that $\tilde{E}^{v}_{p_{1}}$ is sufficiently stable-aligned in
$\mathcal{U}_{c_{1}}$. Similarly, if
$\tilde{E}^{h}_{p_{0}}\in\mathbb{P}^{2}_{p_{0}}$ is $k|x_{0}|^{2}$-horizontal
in $\mathcal{U}_{c_{0}}$, then we say that $\tilde{E}^{h}_{p_{1}}$ is
sufficiently center-aligned in $\mathcal{U}_{c_{0}}$.

######  Proposition 6.12 (Forward/backward regularity implies stable/center
alignment).

Let $p_{0}\in\mathcal{T}_{c_{0}}$ and
$\tilde{E}^{v/h}_{p_{0}}\in\mathbb{P}^{2}_{p_{0}}$. Write

$z_{0}=(x_{0},y_{0}):=\Phi_{c_{0}}(p_{0})\hskip 14.22636pt\text{and}\hskip
14.22636pt\tilde{E}^{v/h}_{z_{0}}:=D\Phi_{c_{0}}(\tilde{E}^{v/h}_{p_{0}}).$

Let

$N>K\log_{\lambda}|x_{0}|$

for some sufficiently large uniform constant $K\geq 1$. If
$p_{1}\in\mathcal{U}_{c_{1}}$ is $N$-times forward $(1,\delta)_{v}$-regular
along $\tilde{E}^{v}_{p_{1}}$, then $\tilde{E}^{v}_{p_{1}}$ is sufficiently
stable-aligned in $\mathcal{U}_{c_{1}}$. Similarly, if $p_{0}$ is $N$-times
backward $(1,\delta)_{h}$-regular along $\tilde{E}^{h}_{p_{0}}$, then
$E^{h}_{p_{0}}$ is sufficiently center-aligned in $\mathcal{U}_{c_{0}}$.

######  Proposition 6.13.

Let $p_{0}\in\mathcal{T}_{c_{0}}$. Write
$z_{0}=(x_{0},y_{0}):=\Phi_{c_{0}}(p_{0}).$ If $\tilde{E}^{v}_{p_{1}}$ is
sufficiently stable-aligned in $\mathcal{U}_{c_{1}}$, then the following
statements hold.

  1. i)

If $-j<0$ is the tunnel escape moment of $p_{0}$, then $p_{-j}$ is $j$-times
forward $(O(1),\delta)_{v}$-regular along $\tilde{E}_{p_{-j}}^{v}$.

  2. ii)

The direction $\tilde{E}^{v}_{p_{-n}}$ is sufficiently vertical in
$\mathcal{U}_{c_{-n}}$ for all
$n>(1+{\bar{\varepsilon}})\log_{\lambda}|x_{0}|$ such that
$p_{-i}\in\mathcal{U}_{c_{-i}}$ for $i\leq n$.

  3. iii)

We have $\|DF^{n}|_{\tilde{E}^{v}_{p_{-n}}}\|>\lambda^{{\bar{\varepsilon}}n}$
for all $0\leq n<(1-{\bar{\varepsilon}})\log_{\lambda}|x_{0}|$.

###### Proof.

By Lemma 4.3, we have

$j>(1-{\bar{\varepsilon}})\frac{\log_{\lambda}|x_{0}|}{\delta^{\prime}}>(1+{\bar{\delta}})\delta^{-1}\log_{\lambda}|x_{0}|.$

The first claim now follows from Lemma 6.7 and Proposition 6.1.

Write

$\tilde{E}^{v}_{z_{0}}:=D\Phi_{c_{0}}(\tilde{E}^{v}_{p_{0}})\hskip
14.22636pt\text{and}\hskip
14.22636pt\theta^{h/v}_{0}:=\measuredangle(\tilde{E}^{v}_{z_{0}},E^{gh/gv}_{z_{0}}).$

By Theorem 3.6 and Proposition 3.9, there exists a uniform constant $k>0$ such
that for $0\leq n<j$, if we have $\theta^{h/v}_{-n}<k,$ then

$\lambda^{\mp(1\pm{\bar{\varepsilon}})}<\frac{\theta^{h/v}_{-n-1}}{\theta^{h/v}_{-n}}<\lambda^{\mp(1\pm{\bar{\varepsilon}})}.$

Note that $\tilde{E}^{v}_{p_{-n}}$ is sufficiently horizontal/vertical in
$\mathcal{U}_{c_{-n}}$ if $\theta^{h/v}_{-n}<l\lambda^{{\bar{\varepsilon}}n}$
for some uniform constant $l>0$. Since $\theta^{h}_{0}\asymp|x_{0}|$ we see
that for some uniform constant $K\geq 1$, we have

$\theta^{h}_{-n}<K\lambda^{-(1+{\bar{\varepsilon}})n}|x_{0}|<l\lambda^{{\bar{\varepsilon}}n}{\bar{K}}\lambda^{-(1+{\bar{\varepsilon}})n}|x_{0}|<l\lambda^{{\bar{\varepsilon}}n}$

for

$0\leq n<(1-{\bar{\varepsilon}})\log_{\lambda}|x_{0}|.$

The third claim follows from Proposition 3.9.

Observe that if $\theta^{h}_{-n}>k,$ then $\theta^{v}_{-n-N}<k$ for some
uniform constant $N\geq 0$. Arguing similarly as above, we conclude the second
claim. ∎

######  Proposition 6.14.

Let $p_{1}\in\mathcal{T}_{c_{1}}$. Write
$z_{1}=(x_{1},y_{1}):=\Phi_{c_{1}}(p_{1}).$ If $\tilde{E}^{h}_{p_{0}}$ is
sufficiently center-aligned in $\mathcal{U}_{c_{0}}$, then the following
statements hold.

  1. i)

If $i>1$ be the crescent escape moment of $p_{1}$, then $p_{i}$ is $i$-times
backward $(O(1),\delta)_{h}$-regular along $\tilde{E}^{h}_{p_{i}}$.

  2. ii)

The direction $\hat{E}^{h}_{p_{n}}$ is sufficiently horizontal in
$\mathcal{U}_{c_{n}}$ for all $n>(1+{\bar{\varepsilon}})\log_{\lambda}|x_{1}|$
such that $p_{j}\in\mathcal{U}_{c_{j}}$ for $j\leq n$.

  3. iii)

We have
$\|DF^{-n}|_{\hat{E}^{h}_{p_{n}}}\|>\lambda^{-(1-{\bar{\varepsilon}})n}$ for
all $0\leq n<(1-{\bar{\varepsilon}})\log_{\lambda}|x_{1}|$.

For $i\in\\{0,1\\}$, let $\Gamma_{i}\subset\mathcal{U}_{c_{i}}$ be a
$C^{1}$-curve parameterized by its arclength $|\Gamma_{i}|$. We say that
$\Gamma_{1}$ is sufficiently stable-aligned in $\mathcal{U}_{c_{1}}$ if the
direction of $\Gamma_{1}^{\prime}$ is sufficiently stable-aligned in
$\mathcal{U}_{c_{1}}$. Similarly, $\Gamma_{0}$ is sufficiently center-aligned
in $\mathcal{U}_{c_{0}}$ if the direction of $\Gamma_{0}^{\prime}$ is
sufficiently center-aligned in $\mathcal{U}_{c_{0}}$.

##  7\. Mild Dissipativity

Let $F$ the diffeomorphism as considered in Section 3. Suppose additionally
that $F$ is mildly dissipative. Let $L\geq 1$ and $\varepsilon\in(0,1)$ with
${\bar{\varepsilon}}<1$.

A point $p\in\Lambda^{+}_{L,\varepsilon}$ is said to be proper forward regular
if $W^{ss}(p)$ is proper in $\Omega$. In this case, let $W^{ss}_{\Omega}(p)$
be the connected component of $W^{ss}(p)\cap\Omega$ containing $p$. Lastly,
denote the set of properly forward regular points in
$\Lambda^{+}_{L,\varepsilon}$ and $\Lambda^{P}_{L,\varepsilon}$ by
$\hat{\Lambda}^{+}_{L,\varepsilon}$ and $\hat{\Lambda}^{P}_{L,\varepsilon}$
respectively.

######  Lemma 7.1.

The sets $\hat{\Lambda}^{+}_{L,\varepsilon}$ and
$\hat{\Lambda}^{P}_{L,\varepsilon}$ are compact.

###### Proof.

Note that since $(L,\varepsilon)$-regularity is a closed condition, the sets
$\Lambda^{+}_{L,\varepsilon}$ and $\Lambda^{P}_{L,\varepsilon}$ are compact.
The result now follows immediately from Proposition 3.22. ∎

For $p\in\Lambda^{P}_{L,\varepsilon}$, let $\Phi_{p}:\mathcal{U}_{p}\to U_{p}$
be a regular chart at $p$. Recall that a regular box at $p$ is given by

$\mathcal{B}_{p}:=\Phi_{p}^{-1}(\mathbb{B}(1/L_{1}))\hskip 14.22636pt\text{for
some}\hskip 14.22636ptL_{1}:={\bar{L}}$

(see Subsection 3.5). For $n\geq 0$, let $\mathcal{B}_{p}^{n}$ be the
connected component of $\mathcal{B}_{p}\cap F^{n}(\Omega)$ containing $p$.
Denote $B_{p}^{n}:=\Phi_{p}(\mathcal{B}_{p}^{n}).$

######  Proposition 7.2 (Iterated image of $\partial\Omega$ as top-bottom
boundary).

There exists a uniform constant $M\in\mathbb{N}$ with

$M>-\log_{\lambda}\overline{L_{1}}$

such that the following holds. Let
$p_{0}\in\hat{\Lambda}^{P}_{(L,\varepsilon)}$. Then for $n\geq M$, the set
$\partial B_{p_{0}}^{n}\cap B_{p_{0}}$ is the union of two $C^{r}$ Jordan arcs
$\gamma_{+}^{n}$ and $\gamma_{-}^{n}$. Moreover, $\gamma_{\pm}$ is the
horizontal graph of a $C^{r}$-function
$g^{n}_{\pm}:(-1/L_{1},1/L_{1})\to\mathbb{R}$ such that

$\lambda^{(1+{\bar{\varepsilon}})n}<|g^{n}_{\pm}(x)|<\lambda^{(1-{\bar{\varepsilon}})n}\hskip
14.22636pt\text{for}\hskip 14.22636pt-1/L_{1}<x<1/L_{1}$

and

$\|(g^{n}_{\pm})^{\prime}\|,\|(g^{n}_{\pm})^{\prime\prime}\|<\lambda^{(1-{\bar{\varepsilon}})n}.$

###### Proof.

Let $\kappa\in({\bar{\varepsilon}},1)$ be a uniformly small constant such that
$\bar{\kappa}<1$. Let $\\{-m_{i}\\}_{i=1}^{\infty}$ be the decreasing sequence
of all negative Pliss moments in the full orbit of $p_{0}$ such that
$p_{-m_{i}}\in\hat{\Lambda}^{P}_{1,\kappa}$. Choose $M=m_{i}$ with $i$
sufficiently large such that $M>-\log_{\lambda}{\bar{L}},$ and consider
$q_{0}:=p_{-M}\in\hat{\Lambda}^{P}_{1,\kappa}.$

Let $\\{\Phi_{q_{m}}:\mathcal{U}_{q_{m}}\to U_{q_{m}}\\}_{m\in\mathbb{Z}}$ be
a linearization of the full orbit of $q_{0}$. Then

$\operatorname{radius}(U_{q_{m}})=l_{q_{m}}\asymp\lambda^{\bar{\kappa}|m|}.$

For $a>0$ and $n\geq 0$, denote

$\mathcal{U}_{q_{m}}^{n}(a):=\Phi_{q_{m}}^{-1}(U_{q_{m}}^{n}(a))\hskip
14.22636pt\text{where}\hskip
14.22636ptU_{q_{m}}^{n}(a):=\mathbb{B}(al_{q_{m}}\lambda^{\bar{\kappa}n},l_{q_{m}}).$

By Lemma 7.1, there exists a uniform constant $N_{0}<M$ such that for $n\geq
N_{0}$, the curve

$\mathcal{C}_{n}:=F^{n}(W^{ss}_{\Omega}(q_{0}))$

is contained in $W^{ss}_{\operatorname{loc}}(q_{n})$. By choosing a new domain
$\tilde{\Omega}$ such that $\Omega\supset\tilde{\Omega}\supset F(\Omega)$ if
necessary, we may assume that for some small uniform constant $a_{0}\in(0,1)$,
the set $\partial F^{N_{0}}(\Omega)\cap\mathcal{U}^{0}_{q_{N_{0}}}(a_{0})$ is
the union of two $C^{r}$-Jordan arcs $\Gamma_{+}^{0}$ and $\Gamma_{-}^{0}$.
Moreover, by Theorem 3.6 and Proposition 3.9, there exists a uniform constant
$K\geq 1$ such that the following holds. For $n\geq 0$, the curve

$\gamma_{\pm}^{n}:=\Phi_{q_{N_{0}+n}}(\Gamma_{\pm}^{n})\hskip
14.22636pt\text{where}\hskip
14.22636pt\Gamma_{\pm}^{n}:=F^{n}(\Gamma_{\pm}^{0}\cap\mathcal{U}_{q_{N_{0}}}^{n}(a_{0}))$

is the horizontal graph of a $C^{r}$-map
$g_{\pm}^{n}:\mathbb{I}(a_{0}l_{0}\lambda^{\bar{\kappa}n})\to\mathbb{R}$
satisfying

$\|g_{\pm}\|_{C^{r}}<K\lambda^{(1-\bar{\kappa})n}$

and

$g_{\pm}^{n}(x)>K^{-1}\lambda^{(1+\bar{\kappa})n}\hskip
14.22636pt\text{for}\hskip
14.22636ptx\in\mathbb{I}(a_{0}l_{0}\lambda^{\bar{\kappa}n}).$

Observe that

$\lambda^{\bar{\kappa}M},\lambda^{(1\pm\bar{\kappa})M}<1/{\bar{L}},\hskip
14.22636pt\text{and}\hskip 14.22636pt\|\Phi_{q_{M}}^{-1}\|_{C^{r}}<{\bar{L}}.$

Let $\\{\Phi_{p_{m}}:\mathcal{U}_{p_{m}}\to U_{p_{m}}\\}_{m\in\mathbb{Z}}$ be
a linearization of the full orbit of $p_{0}$. Then
$\|\Phi_{p_{0}}\|_{C^{r}}<{\bar{L}}.$ Since

$\gamma_{\pm}=\Phi_{p_{0}}\circ\Phi_{q_{M}}^{-1}(\gamma_{\pm}^{M-N_{0}}),$

the result follows. ∎

By Proposition 7.2, we can form dynamically meaningful top-bottom boundaries
of a regular box by choosing them as subarcs of $F^{M}(\partial\Omega)$ for
some sufficiently large $M\in\mathbb{N}$. The following result states that if
$F$ is homogeneous, then there is a uniform bound on how large $M$ has to be
(so that there is a uniform lower bound on the resulting height of the regular
box).

######  Proposition 7.3.

If $F$ is $\eta$-homogeneous for some $\eta\in(0,\underline{\varepsilon})$,
then we can choose $M$ in Proposition 7.2 to be

$M=-\log_{\lambda}\overline{L_{1}}.$

###### Proof.

Consider the sequence of Pliss moments $\\{-m_{i}\\}_{i=1}^{\infty}$
considered in the proof of Proposition 7.2. By Proposition 4.4 and Proposition
4.3, we have

$m_{1}<-\kappa^{-1}\log_{\lambda}{\bar{L}}=-\log_{\lambda}{\bar{L}},$

and if

$m_{i}>-\log_{\lambda}{\bar{L}},$

then

$m_{i}>(1-{\bar{\varepsilon}})m_{i+1}+\log_{\lambda}{\bar{L}}.$

Hence, we may choose

$M=-\log_{\lambda}\overline{L_{1}}.$

∎

For $p\in\hat{\Lambda}^{P}_{L,\delta}$ and $a\in[0,1]$, define

$\bar{\mathcal{B}}_{p}(a):=\Phi_{p}^{-1}\left(B_{p}^{M}\cap U_{p}(a)\right),$

where $M$ is given in Proposition 7.2 (or Proposition 7.3 if $F$ is
homogeneous). Define an enclosed regular box at $p$ as

$\bar{\mathcal{B}}_{p}:=\bar{\mathcal{B}}_{p}(1/2),$ (7.1)

and the outer boundary of $\bar{\mathcal{B}}_{p}$ as

$\bar{\partial}\bar{\mathcal{B}}_{p}:=\partial\mathcal{B}^{M}_{p}(1)\cap\partial
F^{M}(\Omega).$ (7.2)

A Jordan curve $\Gamma:(0,1)\to\bar{\mathcal{B}}_{p}$ is said to be vertically
proper in $\bar{\mathcal{B}}_{p}$ if $\Gamma$ extends continuously to $[0,1]$,
and $\Gamma(0)$ and $\Gamma(1)$ are in distinct components of
$\bar{\partial}\bar{\mathcal{B}}_{p}$. More generally, a Jordan curve $\Gamma$
in $\Omega$ is said to be vertically proper in $\bar{\mathcal{B}}_{p}$ if
$\Gamma\cap\bar{\mathcal{B}}_{p}(1)$ is.

######  Proposition 7.4.

Let $p\in\hat{\Lambda}^{P}_{L,\varepsilon}$. If $q\in\bar{\mathcal{B}}_{p}$ is
$N$-times forward $(L,\varepsilon)$-regular for some

$N>-\log_{\lambda}{\bar{L}},$

then $W^{v}(q)$ is proper in $\bar{\mathcal{B}}_{p}$.

###### Proof.

By Proposition 3.18, $W^{v}(q)$ is sufficiently vertical in $\mathcal{U}_{p}$.
Moreover,

$\operatorname{diam}(W^{v}(q))>\overline{\operatorname{diam}(\bar{\mathcal{B}}_{p})}.$

The result follows. ∎

Define

$\Lambda^{\pm}_{\varepsilon}:=\bigcup_{L\geq
1}\Lambda^{\pm}_{L,\varepsilon}\hskip 14.22636pt\text{and}\hskip
14.22636pt\Lambda^{P}_{\varepsilon}:=\bigcup_{L\geq
1}\Lambda^{P}_{L,\varepsilon}.$

Let $\hat{\Lambda}^{+}_{\varepsilon}\subset\Lambda^{+}_{\varepsilon}$ be the
set of all mildly forward regular points in $\Lambda^{+}_{\varepsilon}$.

######  Proposition 7.5.

Suppose that $F|_{\Lambda}$ is uniquely ergodic. Then we have

$\hat{\Lambda}^{+}_{\varepsilon}=\Lambda^{+}_{\varepsilon}\hskip
14.22636pt\text{and}\hskip
14.22636pt\hat{\Lambda}^{P}_{\varepsilon}=\Lambda^{P}_{\varepsilon}.$

###### Proof.

By replacing $F$ by $F^{N}$ for some sufficiently large $N\in\mathbb{N}$, we
may assume that $F$ is $\eta$-homogeneous for some sufficiently small constant
$\eta\in(0,\underline{\varepsilon})$ (see Proposition 4.1).

Consider the following open set

${\mathbf{U}}:=\bigcup_{p\in\hat{\Lambda}^{P}_{1,{\bar{\varepsilon}}}}\bar{\mathcal{B}}_{p}.$

By Proposition 4.8,

$\alpha:=\mu({\mathbf{U}}\cap\Lambda)>\frac{\varepsilon-\eta}{\varepsilon+\eta}\simeq
1.$

Let $p_{0}\in\Lambda^{+}_{L,\varepsilon}$ for some $L\geq 1$. Let
$\\{n_{i}\\}_{i=1}^{\infty}$ and $\\{m_{i}\\}_{i=1}^{\infty}$ be the
increasing sequences of all moments in the forward orbit of $p_{0}$ such that
$p_{n_{i}}\in{\mathbf{U}}$ and
$p_{m_{i}}\in\Lambda^{+}_{1,{\bar{\varepsilon}}}$. By unique ergodicity, we
have

$\lim_{i\to\infty}\frac{i}{n_{i}}=\alpha.$

Thus, by Proposition 4.2, there exist $i,j\geq 1$ such that $n_{i}=m_{j}$. The
result now follows from Proposition 7.4. ∎

##  8\. Infinite Renormalizability

Let $F$ be the mildly dissipative diffeomorphism considered in Section 7.
Suppose additionally that $F$ is infinitely renormalizable. Then there exists
a nested sequence $\Omega=:\mathcal{D}^{0}\Supset\mathcal{D}^{1}\Supset\ldots$
of Jordan domains and a sequence of natural numbers $1=:R_{0}<R_{1}<\ldots$
such that $\mathcal{D}^{n}$ is $R_{n}$-periodic and

$r_{n-1}:=R_{n}/R_{n-1}\geq 2\hskip 14.22636pt\text{for}\hskip
14.22636ptn\in\mathbb{N}.$

Recall that the renormalization limit set is given by

$\Lambda:=\bigcap_{n=1}^{\infty}\bigcup_{i=0}^{r_{n}-1}F^{i}(D^{n}).$

Consider the standard odometer $S:\mathbb{O}\to\mathbb{O}$ defined by

$\mathbb{O}:=\prod_{n\in\mathbb{N}}(\mathbb{Z}/R_{n}\mathbb{Z}),$

and

$S(a_{1},\ldots,a_{n-1},a_{n},a_{n+1},\ldots):=(0,\ldots,0,a_{n}+1,a_{n+1},\ldots),$

where $n\geq 1$ is the smallest integer such that $a_{n}\neq R_{n}-1$. Denote

$\mathbb{O}_{0}^{n}:=\\{(0,\ldots,0,a_{n+1},a_{n+2},\ldots)\;|\;a_{i}\in\mathbb{Z}/R_{i}\mathbb{Z}\hskip
8.53581pt\text{for}\hskip 8.53581pti\geq n+1\\}.$

Lastly, let $\mu_{0}$ be the unique invariant probability measure for
$T|_{\mathbb{O}}$. Note that

$\mu_{0}(\mathbb{O}_{0}^{n})=\prod_{i=1}^{n}\frac{1}{R_{i}}.$

######  Proposition 8.1.

There exists a unique continuous semi-conjugacy $\pi:\Lambda\to\mathbb{O}$
such that

$\pi\circ F|_{\Lambda}=S\circ\pi$

and

$\pi(D^{n})=\mathbb{O}_{0}^{n}\hskip 14.22636pt\text{for}\hskip
14.22636ptn\in\mathbb{N}.$

Moreover, for every $x\in\mathbb{O}$, the fiber $\pi^{-1}(x)$ is a connected
component of $\Lambda$.