# Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

TANGUE NDAWA Bertuel
<EMAIL_ADDRESS>
Institute of Mathematics and Physical Sciences
National Advanced School of Engineering

To the TIEMGNI DEFFO’s family: the mother Carine and the children Abigail,
Joakim, and Helza after the death of Richard the father of the family.

## Abstract

We consider order preserving $C^{3}$ circle maps with a flat piece, irrational
rotation number and critical exponents $(\ell_{1},\ell_{2})$.

We detect a change in the geometry of the system. For
$(\ell_{1},\ell_{2})\in[1,2]^{2}$ the geometry is degenerate and it becomes
bounded for $(\ell_{1},\ell_{2})\in[2,\infty)^{2}\setminus\\{(2,2)\\}$. When
the rotation number is of the form $[abab\cdots]$; for some
$a,b\in{\mathbb{N}}^{*}$, the geometry is bounded for $(\ell_{1},\ell_{2})$
belonging above a curve defined on $]1,+\infty[^{2}$. As a consequence we
estimate the Hausdorff dimension of the non-wandering set
$K_{f}={\mathcal{S}}^{1}\setminus\bigcup_{i=0}^{\infty}f^{-i}(U)$. Precisely,
the Hausdorff dimension of this set is equal to zero when the geometry is
degenerate and it is strictly positive when the geometry is bounded.

#####

Key words: Circle map, Irrational rotation number, Flat piece, Critical
exponent, Geometry and Hausdorff Dimension.

MSC-2010: 37E10.

Acknowledgements: The author was partially supported by the Centre
d’Excellence Africain en Science Mathématique set Applications (CEA-SMA).

I would sincerely thank Prof. M. Martens and Prof. Dr. L. Palmisano for
introducing me to the subject of this paper, his valuable advice, continuous
encouragement and helpful discussions. I thank Prof. Dr. Carlos Ogouyandjou
for having participated presentations related to this work.

##  1 Introduction

We study a certain class of weakly order preserving, non-injective (on an
interval exactly; called flat piece) circle maps which appear naturally in the
study of Cherry flows on the two dimensions torus (see [8, 11, 14, 15]), non-
invertible continuous circle map (see [7]) and of the dependence of the
rotation interval on the parameter value for one-parameter families of
continuous circle maps (see [16]). The dynamics of circle maps with a flat
interval has been intensively explored in the past years, see [4, 5, 7, 14,
17, 18].

We discuss the geometry of the non-wandering set (set obtained by removing
from the circle all pre-images of the flat piece). Where the geometry is
concerned, we discover a dichotomy; which generalize the one found in [5].
Some of our maps show a "degenerate geometry", while others seem to be subject
to the "bounded geometry".

Before we can explain more precisely our results, we introduce our class,
adopt some notations and present basic lemmas.

###  1.1 The class of functions

We fix $\ell_{1},\ell_{2}\geq 1$ and we consider the class $\mathscr{L}$ of
continuous circle maps $f$ of degree one for which an arc $U$ exists so that
the following properties hold:

  1. 1.

The image of $U$ is one point.

  2. 2.

The restriction of $f$ to ${\mathcal{S}}^{1}\setminus\overline{U}$ is a
$C^{3}$-diffeomorphism onto its image.

  3. 3.

Let $(a,b)$ be a preimage of $U$ under the projection of the real line to
${\mathcal{S}}^{1}$. On some right-sided neighborhood of $b$, $f$ can be
represented as

$h_{r}((x-b)^{\ell_{2}});$

where, $h_{r}$ is $C^{3}$-diffeomorphism on a two-sided neighbourhood of $b$.
Analogously, on a left-sided neighborhood of $a$, $f$ equals

$h_{l}((x-a)^{\ell_{1}});$

In the following, we assume that $h_{l}(x)=h_{r}(x)=x$. In fact, it is
possible to effect $C^{3}$ coordinate changes near $a$ and $b$ that will allow
us to replace both $h_{l}$ and $h_{r}$ by the identity function.

Let $F$ be a lift of $f$ on the real line. The rotation number $\rho(f)$ of
$f$ is defined (independently of $x$ and $F$) by

$\rho(f):=\lim_{n\rightarrow\infty}\dfrac{F^{n}(x)-x}{n}(mod\>1).$

Let $(q_{n})$ be the sequence of denominators of the convergents of $\rho(f)$
(irrational) defined recursively by $q_{1}=1$, $q_{2}=a_{1}$ and
$q_{n+1}=a_{n}q_{n}+q_{n-1}$ for all $n\geq 3$; with,

$\rho(f)=[a_{0}a_{1}\cdots]:=a_{0}+\dfrac{1}{a_{1}+\dfrac{1}{a_{2}+\dfrac{1}{\ddots}}}$

#### Additional Assumption

Let $f\in\mathscr{L}$. We say that $Sf$ (the Schwarzian derivative of $f$) is
negative if,

$Sf(x):=\dfrac{D^{3}f(x)}{Df(x)}-\frac{3}{2}\left(\dfrac{D^{2}f(x)}{Df(x)}\right)^{2}<0;\quad\forall\>x,\;Df(x)\neq
0;$ (A1)

With, $D^{n}f$ the $n^{th}$ derivative of $f$; for $n\in{\mathbb{N}}$.

We will assume in the proof of the first part of Theorem 1 that $f$ has a
negative negative Schwarzian derivative.

###  1.2 Notations and Definitions

The fundamental notations are established in [5] (p.2-3). Let
$f\in\mathscr{L}$.

  1. 1.

For every $i\in{\mathbb{Z}}$, the writing $\underline{i}$ means $f^{i}(U)$.

  2. 2.

Let $I$ and $J$ be two intervals. $(I,J)$ is the interval between $I$ and $J$.
$[I,J):=I\cup(I,J)$ and $(I,J]:=(I,J)\cup J$. $|I|$ is the length of the
interval $I$, $|[I,J)|:=|I|+|(I,J)|$ and $|(I,J]|:=|J|+|(I,J)|$. We say that
$I$ and $J$ are comparable when $|I|$ and $|J|$ are comparable. That means,
there is $k>0$ such that, $\dfrac{1}{k}|I|<|J|<k|I|$.

  3. 3.

For any sequence $\Gamma_{n}$ and for any real $d$, we adopt the writings:

$\Gamma_{n}^{d(\ell_{1},\ell_{2})}:=\begin{cases}\begin{array}[]{lcl}\Gamma_{n}^{d\ell_{1}}&\mbox{if}&n\equiv
0[2]\\\ \Gamma_{n}^{d\ell_{2}}&\mbox{if}&n\equiv
1[2]\end{array}\end{cases}\Gamma_{n}^{d(\frac{1}{\ell_{1}},\frac{1}{\ell_{2}})}:=\begin{cases}\begin{array}[]{lcl}\Gamma_{n}^{d\frac{1}{\ell_{1}}}&\mbox{if}&n\equiv
0[2]\\\ \Gamma_{n}^{d\frac{1}{\ell_{2}}}&\mbox{if}&n\equiv
1[2]\end{array}\end{cases}$

###  1.3  Discussion and statement of the results

#### Scaling ratios

The sequence

$\alpha_{n}:=\dfrac{|(\underline{-q_{n}},\underline{0})|}{|[\underline{-q_{n}},\underline{0})|}=\dfrac{|(f^{-q_{n}}(U),U)|}{|(f^{-q_{n}}(U),U)|+|f^{-q_{n}}(U)|}.$

measure the geometry near a critical point. In fact, it serves as scaling
relating the geometries of successive dynamic partitions.

The geometry is said degenerate when $\alpha_{n}$ goes to zero and the
geometry is bounded, when $\alpha_{n}$ is bounded away from zero.

The study of this geometry is parametrised by rotation number and critical
exponents. In [5], for pairs $(\ell,\ell)$; $\ell>1$ and $\rho$ irrational
number of bounded type (i.e. $\max_{n}a_{n}<\infty$), the authors found a
transition between degenerate geometry and bounded geometry. In fact, they
show under (A1) that, if $1<\ell\leq 2$ and
$\rho\in{\mathbb{R}}\setminus{\mathbb{Q}}$, then the geometry is degenerate
and it is bounded (independently of (A1)) if $\ell>2$ and $\rho$ is irrational
number of bounded type. In [4], for $\ell>1$, the author proved that the class
of function $f$ of critical exponents $(1,\ell)$ or $(\ell,1)$ have a
degenerate geometry. In [15], the authors show that, for the maps in
$\mathscr{L}$ with Fibonacci rotation number, when the critical exponents
$(\ell_{1},\ell_{2})$ belong in $(1,2)^{2}$, the geometry is degenerate. Let
us note that, Differently from other previous works, information on the
geometry of the system is obtained by the study of the asymptotic behaviour of
the renormalization operator.

In the present paper, we consider the cases where the critical exponents
$(\ell_{1},\ell_{2})$ belong in a subdomain (containing the previous domains)
of $[1,\infty)^{2}$; also, the rotation number is not necessarily Fibonacci
type and the results do not depend on the renormalization operator as in [9].
We use the formalism presented in [5] which is based on recursive inequalities
analysis of $\alpha_{n}$. For technical reason, in the case of bounded
geometry, we introduce the vector sequence
$v_{n}:=(-\ln\alpha_{n},-\ln\alpha_{n-1})$ and, the new recursive inequality
is controlled by a $2\times 2$ matrix. When the rotation number is bi-periodic
($\rho=[abab\cdots];a,b\in{\mathbb{N}}$), the $2\times 2$ matrix has two
eigenvalues (depending on rotation number and critical exponents)
$\lambda_{s}\in(0,1)$ and $\lambda_{u}>0$. The equation
$\lambda_{u}=\lambda_{u}((a,b);(\ell_{1},\ell_{2}))=1$ defines a curve
$\mathcal{C}_{\lambda_{u}=1}$ (presented above) which separates the
$(\ell_{1},\ell_{2})$ plan into two components ${C}_{\lambda_{u}>1}$ (below
the curve) and ${C}_{\lambda_{u}<1}$.

2211$\leftarrow$${C}_{\lambda_{u}<1}$$\mathcal{C}_{\lambda_{u}=1}$

##### Main result

Let $f\in\mathscr{L}$ with critical exponents $(\ell_{1},\ell_{2})$. Then,

  1. 1.

the scaling ratio $\alpha_{n}$ goes to zero when
$(\ell_{1},\ell_{2})\in[1,2]^{2}$, $\rho\in{\mathbb{R}}\setminus{\mathbb{Q}}$
and (A1) holds.

  2. 2.

the scaling ratio $\alpha_{n}$ bounded away from zero when
$(\ell_{1},\ell_{2})\in[2,\infty)^{2}\setminus\\{(2,2)\\}$ and $\rho$ is
bounded type.

  3. 3.

the scaling ratio $\alpha_{n}$ bounded away from zero when
$(\ell_{1},\ell_{2})\in{C}_{\lambda_{u}<1}$ and $\rho$ is bi-periodic.

##### Estimation of Hausdorff Dimension of the non-wandering set.

In the symmetric case $(\ell_{1}=\ell_{2}=\ell)$, in [8], for
$f\in\mathscr{L}$ with bounded type rotation number, the author shows that, if
$\ell\in(1,2]$, then the Hausdorff dimension of the non-wandering set is equal
to zero and that, if $\ell>2$, then the Hausdorff dimension of the non-
wandering set is strictly greater than zero. This result generalizes the one
in [19] where the author treats the maps in $\mathscr{L}$ with critical
exponents $(1,1)$. Let us note that, the results in [8] are more general (they
only depend on geometry); precisely, if the rotation number is the bounded
type, then, the Hausdorff Dimension of the non-wandering set is equal to zero
when the geometry is degenerate and it is strictly greater than zero when the
geometry is bounded; so we have the following result which is proved at the
end of the paper.

######  Corollary 1.1.

Let $f\in\mathscr{L}$ with critical exponent $(\ell_{1},\ell_{2})$ with
bounded type rotation number. Then,

  1. 1.

the Hausdorff dimension of the non-wandering set is equal to zero when
$(\ell_{1},\ell_{2})\in[1,2]^{2}$ and (A1) holds and for the pairs $(\ell,1)$;
$\ell>1$;

  2. 2.

the Hausdorff dimension is strictly bigger than zero when
$(\ell_{1},\ell_{2})\in[2,\infty)^{2}\setminus\\{(2,2)\\}$;

  3. 3.

the Hausdorff dimension is strictly bigger than zero when
$(\ell_{1},\ell_{2})\in{C}_{\lambda_{u}<1}$ and $\rho$ is bi-periodic.

The following remark will simplify statements and proofs of results.

######  Remark 1.2.

Let us note that, our setting has some inherent symmetry. This will simplify
statements and proof of our results.

##  2 Tools

###  2.1 Cross-Ratio Inequalities

######  Notation 2.1.

We denote by ${\mathbb{R}}_{<}^{4}$, the subset of ${\mathbb{R}}^{4}$ defined
by

${\mathbb{R}}_{<}^{4}:=\\{(x_{1},x_{2},x_{3},x_{4})\in{\mathbb{R}}^{4},\;such\;that\;x_{1}<x_{2}<x_{3}<x_{4}\\}.$

The following result can be found in [1] (Theorem 2)

######  Proposition 2.2.

The Cross-Ratio Inequality (CRI).

Let $f\in\mathscr{L}$. Let $(a,b,c,d)\in{\mathbb{R}}_{<}^{4}$. The cross-ratio
Cr is defined by

${\textbf{Cr}}\>(a,b,c,d):=\dfrac{|b-a||d-c|}{|c-a||d-b|}$

and the cross-ratio Poin is defined by

${\textbf{Poin}}\>(a,b,c,d):=\dfrac{|d-a||b-c|}{|c-a||d-b|}.$

The distortion of the cross-ratio Cr and cross-ratio Poin are given
respectively by

${\mathcal{D}}{\textbf{Cr}}\>(a,b,c,d):=\dfrac{{\textbf{Cr}}\>(f(a),f(b),f(c),f(d))}{{\textbf{Cr}}\>(a,b,c,d)}$

and

${\mathcal{D}}{\textbf{Poin}}\>(a,b,c,d):=\dfrac{{\textbf{Poin}}\>(f(a),f(b),f(c),f(d))}{{\textbf{Poin}}\>(a,b,c,d)}.$

Let us consider a set of $n+1$ quadruples $\\{a_{i},b_{i},c_{i},d_{i}\\}$ with
the following properties:

  1. 1.

Each point of the circle belongs to at most $k$ intervals $(a_{i},d_{i})$;

  2. 2.

The intervals $(b_{i},c_{i})$ do not intersect $U$.

Then there is $C_{k},C^{\prime}_{k}>0$ such that the following inequalities
hold

$\prod_{i=0}^{n}{\mathcal{D}}{\textbf{Cr}}\>(a_{i},b_{i},c_{i},d_{i})\leq
C_{k}$

and

$\prod_{i=0}^{n}{\mathcal{D}}{\textbf{Poin}}\>(a_{i},b_{i},c_{i},d_{i})\geq
C^{\prime}_{k}.$

Observe that, ${\textbf{Poin}}+{\textbf{Cr}}=1.$ Thus, by the 1.5 Lemma in [2]
we have the following result.

######  Proposition 2.3.

Let $f$ be a $C^{3}$ function such that the Schwarzian derivative is negative.
Then, $C^{\prime}_{k}>1$; that is, $C_{k}<1$.

######  Remark 2.4.

Let $I$ and $J$ be two intervals finite and non-zero length such that,
$\bar{I}\cap\bar{J}=\emptyset$. We assume that, $J$ is on the right of $I$ and
we put $I:=[a,b]$ and $J:=[c,d]$, then

${\textbf{Cr}}\>(I,J):=\dfrac{|I||J|}{|[I,J)||(I,J]|}={\textbf{Cr}}\>(a,b,c,d)$

and

${\textbf{Poin}}\>(I,J):=\dfrac{|(I,J)||[I,J]|}{|[I,J)||(I,J]|}={\textbf{Poin}}\>(a,b,c,d).$

######  Fact 2.5.

Let $f\in\mathscr{L}$. Let $l(U)$ and $r(U)$ be the left and right endpoints
of $U$ (the flat piece of $f$) respectively. There are a left-sided
neighborhood $I^{l}$ of $l(U)$, a right-sided neighborhood $I^{r}$ of $r(U)$
and three positive constants $K_{1},K_{2},K_{3}$ such that the following holds

  1. 1.

If $y\in I^{l_{i}}$ with $l_{1}:=l,\;l_{2}:=r$, then

$\begin{array}[]{c}K_{1}|l_{i}(U)-y|^{l_{i}}\leq|f(l_{i}(U))-f(y)|\leq
K_{2}|l_{i}(U)-y|^{l_{i}},\\\
K_{1}|l_{i}(U)-y|^{l_{i}-1}\leq\dfrac{df}{dx}(y)\leq
K_{2}|l_{i}(U)-y|^{l_{i}-1}.\end{array}$

  2. 2.

If $y\in(x,z)\subset I^{l_{i}}$, with $z$ the closest point to the flat
interval $U$ then

$\dfrac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq K_{3}\dfrac{|x-y|}{|x-z|}.$

.

The first part of Fact 2.5 implies that,

$f_{|I^{l_{i}}}\approx k_{i}x^{\ell_{i}};\;i=1,2.\quad\mbox{We assume that,
}\quad k_{1}=k_{2}.$ (A2)

We need this assumption to prove that $\alpha_{n}$ goes to zero (Lemma 2.9 and
Lemma 3.2).

###  2.2 Basic Results

######  Proposition 2.6.

Let $n\geq 1$.

  1. $\bullet$

The set of “long” intervals consists of the intervals

$\mathcal{A}_{n}:=\\{(\underline{i},\underline{q_{n}+i});\;0\leq i\leq
q_{n+1}-1\\}.$

  2. $\bullet$

The set of “short” intervals consists of the intervals

$\mathcal{B}_{n}:=\\{(\underline{q_{n+1}+i},\underline{i});\;0\leq i\leq
q_{n}-1\\}.$

The set $\mathcal{P}_{n}:=\mathcal{A}_{n}\cup\mathcal{B}_{n}$ covers the
circle modulo the end points and the flat piece and it is called the $n^{th}$
dynamical partition.

The dynamical partition produced by the first $\underline{q_{n+1}+q_{n}}$ pre-
images of $U$ is denoted $\mathcal{P}^{n}$. It consists of

$\wp_{n}:=\\{\underline{-i};\;0\leq i\leq q_{n+1}+q_{n}-1\\}$

together with the gaps between these sets. As in the case of $\mathcal{P}_{n}$
there are two kinds of gaps, “long” and “short”:

  1. $\bullet$

The set of “long” intervals consists of the intervals

$\mathcal{A}^{n}:=\\{(\underline{-q_{n}-i},\underline{-i})=:I^{n}_{i};\;0\leq
i\leq q_{n+1}-1\\}.$

  2. $\bullet$

The set of “short” intervals consists of the intervals

$\mathcal{B}^{n}:=\\{(\underline{-i},\underline{-q_{n+1}-i})=:I^{n+1}_{i};\;0\leq
i\leq q_{n}-1\\}.$

######  Proposition 2.7.

The sequence $|(\underline{0},\underline{q_{n}})|$ tends to zero at least
exponentially fast.

######  Proposition 2.8.

If $A$ is a pre-image of $U$ belonging to $\mathcal{P}^{n}$ and if $B$ is one
of the gaps adjacent to $A$, then $|A|/|B|$ is bounded away from zero by a
constant that does not depend on $n$, $A$ or $B$.

######  Lemma 2.9.

The sequence

$f(\sigma_{n})=\dfrac{|(\underline{1},\underline{q_{n}+1})|}{|(\underline{q_{n-1}+1},\underline{1})|}$

is bounded.

The proofs of these results can be found in [5] (proof of the Proposition 1,
Proposition 2 and Lemma 1.3).

A proof of the following Proposition can be found in [3] theorem:3.1 p.285).

######  Proposition 2.10 (Koebe principle).

Let $f\in\mathscr{L}$. For every $\varsigma,\,\alpha>0$, there exist a
constant $\zeta(\varsigma,\alpha)>0$, such that, the following holds. Let $T$
and $M\subset T$ be two intervals and let $S,\;D$ be the left and the right
component of $T\setminus M$ and $n\in{\mathbb{N}}$. Suppose that:

  1. 1.

$\sum_{i=0}^{n-1}f^{i}(T)<\varsigma$,

  2. 2.

$f^{n}:T\longrightarrow f^{n}(T)$ is a diffeomorphism,

  3. 3.

$\dfrac{|f^{n}(M)|}{|f^{n}(S)|},\;\dfrac{|f^{n}(M)|}{|f^{n}(D)|}<\alpha.$

Then,

$\dfrac{1}{\zeta(\varsigma,\alpha)}\leq\dfrac{Df^{n}(x)}{Df^{n}(y)}\leq\zeta(\varsigma,\alpha),\quad\forall
x,y\in M;$

that is,

$\dfrac{1}{\zeta(\varsigma,\alpha)}\cdot\dfrac{|A|}{|B|}\leq\dfrac{f^{n}(A)}{f^{n}(B)}\leq\zeta(\varsigma,\alpha)\cdot\dfrac{|A|}{|B|},\quad\forall
A,B\,\mbox{(intervals)}\subseteq M;$

where,

$\zeta(\varsigma,\alpha)=\dfrac{1+\alpha}{\alpha}e^{C\varsigma}$

and $C\geq 0$ only depends on $f$.

######  Remark 2.11.

Let $f\in\mathscr{L}$. Given $n>1$, $T$ and $M$ as before.
$f^{n}:T\longrightarrow f^{n}(T)$ is diffeomorphism if only if, for all $0\leq
i\leq n-1$, $f^{i}(T)\cap\overline{U}=\emptyset$; where, $\overline{U}$
designates the closure of $U$.

##  3  Proof of results

Let us put together (parameter) sequences which are frequently used in this
section.

$\alpha_{n}=\dfrac{|(\underline{-q_{n}},\underline{0})|}{|[\underline{-q_{n}},\underline{0})|},\quad\sigma_{n}=\dfrac{|(\underline{0},\underline{q_{n}})|}{|(\underline{q_{n-1}},\underline{0}|},\quad
s_{n}:=\dfrac{|[\underline{-q_{n-2}},\underline{0}]|}{|\underline{0}|},\quad\tau_{n}:=\dfrac{|(\underline{0},\underline{q_{n}})|}{|(\underline{0},\underline{q_{n-2}})|}$

and

$\beta_{n}(k)=\dfrac{|(\underline{-q_{n}+kq_{n-1}},\underline{0})|}{|[\underline{-q_{n}+kq_{n-1}},\underline{0})|};\;k=0,1,\cdots
a_{n-1}.$

###  3.1  Proof of the first part of main result

####  3.1.1 A priori Bounds of $\alpha_{n}$

######  Proposition 3.1.

Let $n\in{\mathbb{N}}$ and $(\ell_{1},\ell_{2})\in\Omega_{0}=[1,2]^{2}$.

For all $\alpha_{n}$,

$\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}<0.55;$

for at least every other $\alpha_{n}$

$\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}<0.3.$

If

$\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}>0.3,$

then either,

$\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}<0.44\quad\mbox{or}\quad\alpha_{n+1}^{\frac{\ell_{1},\ell_{2}}{2}}<0.16.$

###### Proof.

For every $n\in{\mathbb{N}}$ and $k=0,1,\cdots a_{n-1}$, we define the
parameter sequences

$\gamma_{1,n}(k):=|(\underline{-q_{n}+kq_{n-1}},\underline{0})|,\quad\gamma_{1,n-1}:=\gamma_{1,n-1}(0),\quad\gamma_{n}(k)=\dfrac{\gamma_{1,n}(k)}{\gamma_{1,n-1}}$
$\gamma_{n}^{(\ell_{1}|\ell_{2})}(k):=\dfrac{\gamma^{{{\ell_{1}}}}_{1,n}(k)}{\gamma^{{{\ell_{2}}}}_{1,n-1}}\hskip
2.56073pt\mbox{ if }\hskip 2.56073ptn\in 2{\mathbb{Z}}\hskip 2.84526pt\mbox{
and }\hskip
2.84526pt\gamma_{n}^{(\ell_{1}|\ell_{2})}(k):=\dfrac{\gamma^{{{\ell_{2}}}}_{1,n}(k)}{\gamma^{{{\ell_{1}}}}_{1,n-1}}\hskip
2.56073pt\mbox{ if }\hskip 2.56073ptn\in 2{\mathbb{Z}}+1.$

These notations simplify the formalization of the following lemma which will
play an important (essential) role in the proof of Proposition 3.1.

######  Lemma 3.2.

For $n$ large enough and for every $k=0,1,\cdots a_{n-1}-1$, the following
inequality holds

$\dfrac{(\beta_{n}(k)^{\ell_{1},\ell_{2}}+\alpha^{\ell_{1},\ell_{2}}_{n-1}\gamma_{n}^{(\ell_{1}|\ell_{2})}(k))(1+\gamma_{n}^{(\ell_{1}|\ell_{2})}(k))}{(1+\alpha^{\ell_{1},\ell_{2}}_{n-1}\gamma_{n}^{(\ell_{1}|\ell_{2})}(k))(\beta_{n}(k)^{\ell_{1},\ell_{2}}+\gamma_{n}^{(\ell_{1}|\ell_{2})}(k))}\leq
s_{n}\beta_{n}(k+1).$ (3.1)

###### Proof.

Let $n$ be an even non negative integer large enough. For fixed $k=0,1,\cdots
a_{n-1}-1,$ according to the assumption (A2), the left hand side of (3.1) is
equal to the cross-ratio

${\textbf{Poin}}\,(\underline{-q_{n}+kq_{n-1}+1},\underline{-q_{n-1}+1}).$

Applying $f^{q_{n-1}-1}$, by expanding cross-ratio property, we get the
inequality. ∎

The left hand side is a function of the three variables
$\beta_{n}(k)^{\ell_{1},\ell_{2}}$, $\alpha^{\ell_{1},\ell_{2}}_{n-1}$ and
$\gamma_{n}^{(\ell_{1}|\ell_{2})}.$ Observe that the function increases
monotonically with each of the first two variables. However, relatively to the
third variable, the function reaches a minimum. To see this, take the
logarithm of the function and check that the first derivative is equal to zero
only when

$(\gamma_{n}^{(\ell_{1}|\ell_{2})})^{2}=\dfrac{\beta_{n}(k)^{\ell_{1},\ell_{2}}}{\alpha^{\ell_{1},\ell_{2}}_{n-1}}.$

By substituting this value for $\gamma_{n}^{(\ell_{1}|\ell_{2})}$ in (3.1), we
get that

$\left(\dfrac{\beta_{n}(k)^{\frac{\ell_{1},\ell_{2}}{2}}+\alpha^{\frac{\ell_{1},\ell_{2}}{2}}_{n-1}}{1+\beta_{n}(k)^{\frac{\ell_{1},\ell_{2}}{2}}\alpha^{\frac{\ell_{1},\ell_{2}}{2}}_{n-1}}\right)^{2}\leq
s_{n}\beta_{n}(k+1).$ (3.2)

Put:

$x_{n}(k):=\min\\{\beta_{n}(k)^{\frac{\ell_{1},\ell_{2}}{2}},\alpha^{\frac{\ell_{1},\ell_{2}}{2}}_{n-1}\\}\quad\mbox{
and }\quad y_{n}(k):=\beta_{n}(k)^{\frac{\ell_{1},\ell_{2}}{2}}.$

Since $\beta_{n}(k+1)\leq y_{n}(k+1)$, substituting the above variable into
(3.2) gives rise to a quadratic inequality in $x_{n}(k)$ whose only root in
the interval $(0,1)$ is given by

$\dfrac{\sqrt{s_{n}y_{n}(k+1)}}{1+\sqrt{1-s_{n}y_{n}(k+1)}};$

that is,

$x_{n}(k)=\min\\{\beta_{n}(k)^{\frac{\ell_{1},\ell_{2}}{2}},\alpha^{\frac{\ell_{1},\ell_{2}}{2}}_{n-1}\\}\leq\dfrac{\sqrt{s_{n}y_{n}(k+1)}}{1+\sqrt{1-s_{n}y_{n}(k+1)}}.$
(3.3)

######  Lemma 3.3.

There is a subsequence of $\alpha_{n}$ including at least every other
$\alpha_{n}$, such that

$\limsup\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\leq 0.3.$

###### Proof.

We use the following elementary lemma in [5].

######  Lemma 3.4.

The function

$h_{n}(z)=\dfrac{\sqrt{s_{n}z}}{1+\sqrt{1-s_{n}z}}$

moves points to the left, $h(z)<z$, if $z\geq 0.3$ and $n$ is large enough.

We select the subsequence.

  1. 1.

The initial term: there exists $n-2\in{\mathbb{N}}$, such that
$\alpha_{n-2}^{\frac{\ell_{1},\ell_{2}}{2}}\leq 0.3$. This comes directly from
the properties of the function $h_{n}$ (Lemma 3.4) and from (3.3).

  2. 2.

The next element: suppose that $\alpha_{n-2}$ has been selected. If

$x_{n}(k)=\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}\quad\mbox{or
}\quad\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}\leq 0.3,$

for some $k=0,1,\cdots a_{n-1}-1,$ then, we select $\alpha_{n-1}$ as the next
term. Otherwise, $\alpha_{n}$ is the next term. Thus, the sequence is
constructed.

∎

######  Corollary 3.5.

For the whole sequence $(\alpha_{n})$ we have

$\limsup\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\leq 0.3$

Moreover,

-

if $\alpha_{n-1}$ does not belong to the subsequence $(\alpha_{n})$ defined by
the Lemma 3.3 then either

$\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}<0.44\quad\mbox{or}\quad\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}<0.16$

###### Proof.

Observe that the function

$H:(s,t)\in{\mathbb{R}}_{+}^{2}\mapsto F(s,t)=\dfrac{s+t}{1+st}$

is symmetric and for fixed $s$, the function $F(s,\cdot)$ reaches its minimum
in zero by taking the value $s$. Therefore, for every $s,t\geq 0$,

$s,t\leq\dfrac{s+t}{1+st}.$

So,

$\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}},\;\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}\leq\dfrac{\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}+\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}}{1+\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}}.$
(3.4)

Thus, according to that $\alpha_{n-2}$ is an element of the sequence and
suppose that $\alpha_{n-1}$ do not belong to the previous subsequence of the
Lemma 3.3, then, it follows from (3.2) that the right member of (3.4) is
estimated as following

$\dfrac{\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}+\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}}{1+\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}}\leq\sqrt{s_{n}\beta_{n}(1)^{\frac{\ell_{1},\ell_{2}}{2}}}\approx\sqrt{\beta_{n}(1)^{\frac{\ell_{1},\ell_{2}}{2}}}\leq\sqrt{0.3}.$
(3.5)

Also,

$\min\\{\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}},\alpha_{n-1}^{\frac{\ell_{1},\ell_{2}}{2}}\\}=\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\leq
0.3.$ (3.6)

Thus, if $\alpha_{n}^{\frac{\ell_{1},\ell_{2}}{2}}\geq 0.16,$ then by
combining this with (3.5) and (3.6), we obtain the desired estimate. ∎

The Proposition 3.1 is proved. ∎

####  3.1.2 Recursive formula of $\alpha_{n}$

######  Proposition 3.6.

Let $n$ be integer large enough,

  1. 1.

if $\ell_{1},\,\ell_{2}>1$, we have

$\alpha_{2n}^{\ell_{1}}\leq
M_{2n}(\ell_{1})\alpha^{2}_{2n-2}\quad\mbox{and}\quad\alpha_{2n+1}^{\ell_{2}}\leq
M_{2n+1}(\ell_{2})\alpha^{2}_{2n-1};$ (3.7)

where,

$M_{n}(\ell)=s^{2}_{n-1}\cdot\dfrac{2}{\ell}\cdot\dfrac{1}{1+\sqrt{1-\dfrac{2(\ell-1)}{\ell}s_{n-1}\alpha_{n-1}}}\cdot\dfrac{1}{1-\alpha_{n-2}}\cdot\frac{\sigma_{n}}{\sigma_{n-2}}.$

  2. 2.

if $\ell_{1}=\ell_{2}=1$, then

$\alpha_{n}\leq W^{1}_{n}\cdot\frac{\sigma_{n}}{\sigma_{n-2}}\alpha_{n-2}.$
(3.8)

###### Proof.

We treat the case $n$ even and the case $n$ odd is treated in a similar way.
Recall that,

$\alpha_{n}=\dfrac{|(\underline{-q_{n}},\underline{0})|}{|[\underline{-q_{n}},\underline{0})|}.$

For every $n$ even large enough, by Proposition 2.7 and point 2 of Fact 2.5,
applying $f$ to the equality, we get

$\alpha^{\ell_{1}}_{n}=\dfrac{|(\underline{-q_{n}+1},\underline{1})|}{|[\underline{-q_{n}+1},\underline{1})|}.$

which is certainly less than the cross-ratio

${\textbf{Poin}}(\underline{-q_{n}+1},(\underline{1},\underline{-q_{n-1}+1}]).$

Since the cross-ratio Poin is expanded by $f^{q_{n-1}-1}$, then,

$\alpha^{\ell_{1}}_{n}<\delta_{n}(1)s_{n}(1);$ (3.9)

with,

$\delta_{n}(k):=\dfrac{|(\underline{-q_{n}+kq_{n-1}},\underline{kq_{n-1}})|}{|[\underline{-q_{n}+kq_{n-1}},\underline{kq_{n-1}})|}.$

and

$s_{n}(k):=\dfrac{|[\underline{-q_{n}+kq_{n-1}},\underline{0}]|}{|(\underline{-q_{n}+kq_{n-1}},\underline{0}]|}.$

If $a_{n-1}=1$, by multiplying and dividing the right member of (3.9) by
$\alpha^{2}_{n-2},$ we obtain directly (3.11).

Suppose that $a_{n-1}>1$ and estimate $\delta_{n}(k)$. By the Mean Value
Theorem (Lagrange), $f$ transforms the intervals defining the ratio
$\delta_{n}(k)$ into a pair whose ratio is

$\dfrac{u_{k}}{v_{k}}\delta_{n}(k)$

with $u_{k}$ being the derivative of $f\,(x^{\ell_{1}})$ at a point in the
interval

$U_{k}:=(\underline{-q_{n}+kq_{n-1}},\underline{kq_{n-1}}),$

and $v_{k}$ being the derivative of $f\,(x^{\ell_{1}})$ at a point in the
interval

$V_{k}:=[\underline{-q_{n}+kq_{n-1}},\underline{kq_{n-1}}).$

Note that, for $n$ sufficiently large,

$u_{1}<v_{1}<u_{2}<v_{2}\cdots<v_{a_{n-1}}.$

We see that the image of $\delta_{n}(k)$ by $f$ is smaller than:

${\textbf{Poin}}(\underline{-q_{n}+kq_{n-1}+1},(\underline{kq_{n-1}+1},\underline{-q_{n-1}+1}]).$

Once again, by expanding cross-ratio property $(f^{q_{n-1}-1})$, it follows
that,

$\dfrac{u_{k}}{v_{k}}\delta_{n}(k)\leq s_{n}(k+1)\cdot\delta_{n}(k+1).$ (3.10)

Multiplying (3.10) for $k=0,...,a_{n-1}-1$, and substituting the resulting
estimate of $\delta_{n}(1)$ into (3.9) , we obtain:

$\alpha^{\ell_{1}}_{n}\leq\delta_{n}(a_{n-1})\dfrac{v_{a_{n-1}}}{u_{1}}s_{n}(1)\cdots
s_{n}(a_{n-1}).$

Observe that, $s_{n}(1)\cdots s_{n}(a_{n-1})<s_{n}$ and

$\dfrac{v_{a_{n-1}}}{u_{1}}\leq\left(\dfrac{|(\underline{-q_{n-2}},\underline{0})|}{|(\underline{q_{n-1}},\underline{0})|}\right)^{\ell_{1}-1}\leq\dfrac{|(\underline{-q_{n-2}},\underline{0})|}{|(\underline{q_{n-1}},\underline{0})|}.$

Thus,

$\alpha^{\ell_{1}}_{n}\leq
s_{n}\cdot\dfrac{|(\underline{-q_{n-2}},\underline{0})|}{|(\underline{q_{n-1}},\underline{0})|}\cdot\delta_{n}(a_{n-1});$

which can be rewritten in the form

$\alpha^{\ell_{1}}_{n}\leq s_{n}\nu_{n-2}\mu_{n-2}\alpha^{2}_{n-2};$ (3.11)

with,

$\nu_{n-2}:=\dfrac{|[\underline{-q_{n-2}},\underline{0})|}{|(\underline{q_{n-1}},\underline{0})|}\cdot\dfrac{|[\underline{-q_{n-2}},\underline{0})|}{|[\underline{-q_{n-2}},\underline{a_{n-1}q_{n-1}})|}$

and

$\mu_{n-2}:=\dfrac{|(\underline{-q_{n-2}},\underline{a_{n-1}q_{n-1}})|}{|(\underline{-q_{n-2}},\underline{0})|}.$

It remains to estimate $\nu_{n-2}$ and $\mu_{n-2}$ to end this part. For
$\nu_{n-2}$, observe that,

$|(\underline{-q_{n-2}},\underline{0})|\leq|(\underline{q_{n-3}},\underline{0})|$

so that,

$\nu_{n-2}\leq\dfrac{1}{\sigma_{n-1}\sigma_{n-2}}\cdot\dfrac{1}{1-\alpha_{n-2}}$
(3.12)

The estimation of $\mu_{n-2}$ is facilitated by the following elementary lemma
in [5].

######  Lemma 3.7.

Let $\ell\in(1,2)$. For all numbers $x>y$, we have the following inequality:

$\dfrac{x^{\ell}-y^{\ell}}{x^{\ell}}\geq\left(\dfrac{x-y}{x}\right)\left[\ell-\dfrac{\ell(\ell-1)}{2}\left(\dfrac{x-y}{x}\right)\right].$

Now, apply $f$ into the intervals defining the ratio $\mu_{n-2}$. By Lemma
3.7, the resulting ratio is larger than

$\mu_{n-2}(\ell_{1}-\dfrac{\ell_{1}(\ell_{1}-1)}{2}\mu_{n-2}).$

The cross-ratio Poin

${\textbf{Poin}}\>(-q_{n-2}+1,(\underline{q_{n-1}+1},\underline{1}));$

that is,

$\dfrac{|(\underline{-q_{n-2}+1},\underline{q_{n-1}+1})||[\underline{-q_{n-2}+1},\underline{1})|}{|[\underline{-q_{n-2}+1},\underline{q_{n-1}+1})||(\underline{-q_{n-2}+1},\underline{1})|}$

is larger again. Thus, by expanding cross-ratio property on $f^{q_{n-2}}$, we
obtain:

$\mu_{n-2}(\ell_{1}-\dfrac{\ell_{1}(\ell_{1}-1)}{2}\mu_{n-2})\leq
s_{n-1}\sigma_{n}\sigma_{n-1}.$ (3.13)

By solving this quadratic inequality, we obtain

$\mu_{n-2}\geq\dfrac{1+\sqrt{1-\dfrac{2}{\ell_{1}}(\ell_{1}-1)s_{n-1}\sigma_{n}\sigma_{n-1}}}{\ell_{1}-1}.$
(3.14)

Thus, by combining the (3.13) and (3.14), we obtain

$\mu_{n-2}<\dfrac{2}{\ell_{1}}\cdot\dfrac{1}{1+\sqrt{1-\dfrac{2(\ell_{1}-1)}{\ell_{1}}s_{n-1}\sigma_{n}\sigma_{n-1}}}s_{n-1}\sigma_{n}\sigma_{n-1}.$
(3.15)

Since, $\sigma_{n}\sigma_{n-1}<\alpha_{n-1}$, the first inequality in (3.7)
follows by combining the inequalities (3.11), (3.12) and (3.15). Likewise, the
second inequality in (3.7) is obtained by following suitably the same
reasoning as previously.

∎

####  3.1.3  $\alpha_{n}$ go to zero

If $\ell_{1}=\ell_{2}=1$, then by Proposition 3.1,
$\prod_{k=2}^{k=n}W^{1}_{k}$ goes to zero; thus, by composing the inequality
obtained by $f$, since $f(\sigma_{n})$ is bounded ( Lemma 2.9 ), then the
result follows.

Note that, the cases where the critical exponents are of the form $(1,\ell)$
or $(\ell,1)$; with $\ell>1$, are treated in [4].

Now, Let us suppose that $\ell_{1},\ell_{2}>1$.

Technical reformulation of the Proposition 3.6.

Let $W_{n}$ be a sequence defined by

$M_{n}(\ell)=W_{n}(\ell)\dfrac{\sigma_{n}}{\sigma_{n-2}}.$

Let

$M^{\prime}_{n}(\ell):=M_{n}(\ell)\alpha_{n-2}^{2-\ell}\quad\mbox{and}\quad
W^{\prime}_{n}(\ell):=W_{n}(\ell)\alpha_{n-2}^{2-\ell}$

The recursive formula (3.7) can be written for $n$ even in the form:

$\alpha_{n}^{\ell_{1}}\leq
W^{\prime}_{n}(\ell_{1})\dfrac{\sigma_{n}}{\sigma_{n-2}}\alpha_{n-2}^{\ell_{1}}.$

so,

$\alpha_{n}^{\ell_{1}}\leq\prod_{k=2}^{k=n}W^{\prime}_{k}(\ell_{1})\dfrac{\sigma_{n}}{\sigma_{0}}\alpha_{0}^{\ell_{1}}.$

$\prod_{k=2}^{k=n}W^{\prime}_{k}(\ell_{1})$ goes to zero

Observe that, the size of $W^{\prime}_{n}({{\ell_{1}}})$ is given by the study
of the function

$W^{\prime}_{n}(x,y,{{\ell_{1}}})=\dfrac{1}{\dfrac{{{\ell_{1}}}}{2}+\dfrac{{{\ell_{1}}}}{2}\sqrt{1-\dfrac{2({{\ell_{1}}}-1)}{{{\ell_{1}}}}x^{\frac{2}{{{\ell_{2}}}}}}}\cdot\frac{y^{\frac{4}{{{\ell_{1}}}}-2}}{1-y^{\frac{2}{{{\ell_{1}}}}}}$

The meaning of variation of $W^{\prime}_{n}(x,y,{{\ell_{1}}})$ relative to the
third variable is given by the following lemma in [5]).

######  Lemma 3.8.

For any $0<y<\frac{1}{\sqrt{e}}$, $x\in(0,1)$ and ${{\ell_{1}}}\in(1,2]$ the
function $W^{\prime}_{n}(x,y,{{\ell_{1}}})$ is increasing with respect to
${{\ell_{1}}}$.

#### Analyse the asymptotic size of $W^{\prime}_{i}(2)$

Since the hypotheses of the Lemma 3.8 are satisfied (Proposition 3.1), the
only remaining point is the verification of the convergence of
$\prod_{i=1}^{n}W^{\prime}_{i}(2)$.

-

If $\alpha_{n-2}<(0.3)^{{{\ell_{1}}}}$, then
$W^{\prime}(2)<W^{\prime}(0.55,0.16,2)<0,9$.

-

If not, then by the Proposition 3.1,
$W^{\prime}(2)<W^{\prime}(0.3,0.44,2)<0,98$ or else,
$W_{n+1}^{\prime}(2)W_{n}^{\prime}(2)<W^{\prime}(0.55,0.16,2)W^{\prime}(0.16,0.55,2)<0,85$

######  Corollary 3.9.

Let $\ell_{1},\ell_{2}\in[1,2]$. If $1<\ell_{1}<2$ respectively
$1<\ell_{2}<2$, then $\alpha_{2n}$ respectively $\alpha_{2n+1}$ goes to zero
least double exponentially fast. And, if $\ell_{1}=2$ or $1$ respectively
$\ell_{2}=2$ or $1$, then $\alpha_{2n}$ respectively $\alpha_{2n+1}$ goes to
zero least exponentially fast

###### Proof.

Let $n:=2p_{n}\in{\mathbb{N}}$. From the analysis of the asymptotic size of
$W^{\prime}_{i}(2)$, it follows that, when $n$ goes to infinity,
$\prod_{i=0}^{n}W^{\prime}_{i}(\ell_{1})$ goes to zero and $\alpha_{n}$ does
so. Therefore,

$\prod_{i=0}^{n}M_{i}(\ell_{1})$

goes to zero when $n$ goes to infinity. Thus, by the Proposition 3.6, for $n$
even, there is $\lambda_{0}$ such that

  1. -

if $1<\ell_{1}<2$,

$\alpha_{n}\leq\lambda_{0}^{\left(\frac{2}{\ell_{1}}\right)^{p_{n}}},$

  2. -

and if $\ell_{1}=1,2$,

$\alpha_{n}\leq\lambda_{0}^{n}.$

The case $n$ odd is treated the same way. ∎

###  3.2  Proof of the second part of main result

In this section we find a bounded geometry domain.

####  3.2.1 Recursive Affine Inequality of order two on $\alpha_{n}$

Let

$\kappa_{n}:=\dfrac{|(\underline{0},\underline{q_{n}})|}{|(\underline{0},\underline{-q_{n-1}})|}.$

######  Remark 3.10.

Since the point $\underline{q_{n-2}}$ lies in the gap between
$\underline{-q_{n-1}}$ and
$\underline{-q_{n-1}+q_{n-2}}$ of the dynamical partition $\mathcal{P}_{n-2}$,
then by the
Proposition 2.8, $\tau_{n}/{\alpha_{n-1}}$ and $\kappa_{n}$ are comparable.

######  Proposition 3.11.

For any bounded type rotation number, there is a uniform constant $K$ so that

$\kappa_{2n}>K\left(\alpha_{2n-1}\right)^{\dfrac{1-\ell_{2}^{a_{2n}+1}}{\ell_{2}-1}}\quad\mbox{and}\quad\kappa_{2n+1}>K\left(\alpha_{2n}\right)^{\dfrac{1-\ell_{1}^{a_{2n+1}+1}}{\ell_{1}-1}}.$

##### Proof of the Proposition:

If $a_{n}=1$, it comes down to showing that the sequence $\kappa_{n}$ is
bounded away from zero; which becomes relatively very simple. In fact, suppose
that,
$|(\underline{q_{n}},\underline{-q_{n-1}})|\leq|(\underline{0},\underline{q_{n}})|,$
then

$\kappa_{n}=\dfrac{|(\underline{0},\underline{q_{n}})|}{|(\underline{0},\underline{-q_{n-1}})|}\geq\dfrac{1}{2},$

else, $\kappa_{n}$ is greater than

$\dfrac{|\underline{-q_{n+1}}|}{|[\underline{-q_{n+1}},\underline{-q_{n-1}})|};$

which by the Proposition 2.8 is bounded away from zero.

In the following part of the proof, we suppose that $a_{n}>1$ and the
following lemma in [5] (Lemma 4.1.) is used.

######  Lemma 3.12.

The ratio

$1-\beta_{n}(i)=\dfrac{|\underline{-q_{n}+iq_{n-1}}|}{|[\underline{-q_{n}+iq_{n-1}}),\underline{0})|}$

is bounded away from zero by a uniform constant for all $i=0,\cdots,a_{n-1}.$

######  Lemma 3.13.

For every $n\in{\mathbb{N}}$ and for all $i=0,\cdots,a_{n-1}$, there is a
uniform constant $K$ such that

$\beta_{n}(i)^{\ell_{1},\ell_{2}}\geq\beta_{n}(i+1).$

###### Proof.

For $n$ large enough and for fixed $i=0,\cdots,a_{n-1}$, by the Proposition
2.7 and the Fact 2.5, we have:

$\beta_{n}(i)^{\ell_{1},\ell_{2}}=\dfrac{|(\underline{-q_{n}+kq_{n-1}+1},\underline{1})|}{|[\underline{-q_{n}+iq_{n-1}+1},\underline{1})|};$

which is greater than,

${\textbf{Cr}}([\underline{-q_{n-2}+1},\underline{-q_{n}+iq_{n-1}+1}),(\underline{-q_{n}+iq_{n-1}+1},\underline{1})).$

By applying the cross-ratio inequality under $f^{q_{n-2}-1}$, by the Fact 2.5
the resulting ratio is greater than

$\begin{array}[]{r}{\textbf{Cr}}([\underline{-q_{n-2}+1},\underline{-q_{n}+iq_{n-1}+q_{n-2}+1}),\vspace{0.1cm}\\\
(\underline{-q_{n}+iq_{n-1}+q_{n-2}+1},q_{n-2}+\underline{1}))\end{array}$

times a uniform constant. We now repeat this sequence of steps $a_{n-2}-1$
times: Apply $f^{q_{n-2}-1}$, discard the interval containing $\underline{0}$
and use the point 2 of the Fact 2.5, and replace the result by a cross-ratio
spanning the intervals $\underline{-q_{n-2}+1}$. At the end, this will produce
the cross ratio

$\begin{array}[]{r}{\textbf{Cr}}([\underline{-q_{n-2}+1},\underline{-q_{n}+iq_{n-1}+a_{n-2}q_{n-2}+1}),\vspace{0.1cm}\\\
(\underline{-q_{n}+iq_{n-1}+a_{n-2}q_{n-2}+1},a_{n-2}q_{n-2}+\underline{1})).\end{array}$

And finally, by applying $f^{q_{n-3}-1}$, since by the Lemma 3.12 the interval
containing $\underline{-q_{n-2}+q_{n-3}}$ bounded away from zero, then
resulting ratio is

$\dfrac{|(\underline{-q_{n}+(i+1)q_{n-1}},\underline{q_{n-1}})|}{|[\underline{-q_{n}+(i+1)q_{n-1}},\underline{q_{n-1}})|}$

times a uniform constant. Thus, since $\underline{-q_{n}+q_{n-1}}$ lies
between
$\underline{-q_{n}+(i+1)q_{n-1}}$ and $\underline{q_{n-1}}$, then, by the
Lemma 3.12, this ratio is comparable to $\beta_{n}(i+1).$ ∎

#####  Back to the proof of the Proposition 3.11:

Observe by the Proposition 2.8 that

$|(\underline{-q_{n}+(i+1)q_{n-1}},\underline{0})|$

and

${|[\underline{-q_{n}+iq_{n-1}},\underline{0})|}$

are comparable. Therefore, $\kappa_{n-1}$ is comparable to the product

$\beta_{n}(1)\cdots\beta_{n}(a_{n-1}-1).$

By combining this with the Lemma 3.13, we have the Proposition 3.11.

##### Recursive Affine Inequality of order two on $\alpha_{n}$

######  Proposition 3.14.

If $\rho(f)$ is of bounded type, then there is a uniform constant $K$ so that,

$\alpha_{2n}\geq
K\left(\alpha_{2n-1}\right)^{\dfrac{\ell_{2}}{\ell_{1}}\cdot\dfrac{1-\ell_{2}^{-a_{2n}}}{\ell_{2}-1}}\left(\alpha_{2n-2}\right)^{\ell_{1}^{-a_{2n-1}}}$

and

$\alpha_{2n+1}\geq
K\left(\alpha_{2n}\right)^{\dfrac{\ell_{1}}{\ell_{2}}\cdot\dfrac{1-\ell_{1}^{-a_{2n+1}}}{\ell_{1}-1}}\left(\alpha_{2n-1}\right)^{\ell_{2}^{-a_{2n}}}.$

##### Proof of the Proposition

If $n$ is even and large enough, then

$\alpha^{{{\ell_{1}}}}_{n}=\dfrac{|(\underline{-q_{n}+1},\underline{1})|}{|[\underline{-q_{n}+1},\underline{1})|};$

which in turn is larger than the product of two ratios

$\xi_{1,n}=\dfrac{|(\underline{-q_{n}+1},\underline{1})|}{|(\underline{-q_{n}+1},\underline{-q_{n-1}+1})|}\quad\mbox{and}\quad\xi_{2,n}=\dfrac{|(\underline{-q_{n}+1},\underline{-q_{n-1}+1})|}{|[\underline{-q_{n}+1},\underline{-q_{n-1}+1})|}$

######  Lemma 3.15.

For all $n$ even large enough

$\xi_{1,n}\geq K\tau_{n}.$

###### Proof.

Observe that $\xi_{1,n}$ is greater than

${\textbf{Cr}}\,((\underline{-q_{n}+1},\underline{1}),\underline{-q_{n-1}+1}).$

By applying CRI on $f^{q_{n-1}-1}$ and discarding the intervals containing
$\underline{0}$. Repeat this $a_{n-1}-1$ times more: By the Fact 2.5, the
resulting ratio is large than

${\textbf{Cr}}\,((\underline{-q_{n-2}-q_{n-1}+1},\underline{(a_{n-1}-1)q_{n-1}+1}),(\underline{1},\underline{-q_{n-1}+1}])).$

Apply $f^{q_{n-1}-1}$, and discard the intervals containing the flat interval.
Apply $f$, replace the resulting by the cross-ratio

${\textbf{Cr}}\,((\underline{-q_{n-2}+1},\underline{q_{n-1}+1}),(\underline{1},\underline{-q_{n-1}+1}])).$

Thus, by CRI on $f^{q_{n-2}-1}$ and the inequalities above, we obtain

$\xi_{1,n}>\dfrac{|(\underline{q_{n-2}},\underline{-q_{n-3}}]|}{|(\underline{q_{n}},\underline{-q_{n-3}}]|}\tau_{n}$
(3.16)

The first factor on the right hand side of 3.16 is greater than

$\dfrac{|\underline{-q_{n-3}}|}{|(\underline{0},\underline{-q_{n-3}}]|};$

which by the Proposition 2.8 goes away from zero. The Lemma is proved. ∎

######  Lemma 3.16.

There is a uniform constant $K$ so that, for all $n$

$\xi_{2,2n}\geq
K\left(\alpha_{2n-2}\right)^{\ell_{1}^{-a_{2n-1}+1}}\quad\mbox{and
}\quad\xi_{2,2n+1}\geq K\left(\alpha_{2n-1}\right)^{\ell_{1}^{-a_{2n}+1}}.$

###### Proof.

If $a_{n-1}=1$, then $\xi_{2,n}$ is greater than

${\textbf{Cr}}\,([\underline{-q_{n-2}+1},\underline{-q_{n}+1}),(\underline{-q_{n}+1},\underline{-q_{n-1}+1})).$

By applying CRI $({q_{n-2}-1})$, the ratio resulting is greater than

${\textbf{Cr}}\,([\underline{-q_{n-2}+1},\underline{-q_{n-1}+1}),(\underline{-q_{n-1}+1},\underline{-q_{n-3}+1}))$

times a uniform constant. Thus, by applying CRI to this ratio with
$f^{q_{n-3}-1}$, inequalities above and the Proposition 2.8 the result
follows.

Now, suppose that $a_{n-1}>1$ then $\xi_{2,n}$ is greater than

${\textbf{Cr}}\,([\underline{-q_{n}+q_{n-1}+1},\underline{-q_{n}+1}),(\underline{-q_{n}+1},\underline{-q_{n-1}+1})).$

By applying $f^{q_{n-1}-1}$ to this ratio, it follows from the Lemma 3.12 that

$\xi_{2,n}\geq K^{\prime}\beta_{n}(1).$

And the lemma follows from this by using Lemma 3.13 modulo the fact that

$\beta_{n}(a_{n-1})=\alpha_{n-2}.$

∎

Combining the Lemma 3.15, the Lemma 3.16, the Proposition 3.11 and the Remark
3.10, the result of the Proposition 3.14 follows.

######  Remark 3.17.

By the inequality obtained from the Proposition 3.14, the sequence $\nu_{n}$
defined by

$\nu_{n}=-\ln\alpha_{n}$

verifies the following Recursive Affine Inequalities of order two for every
$n>0$

$\nu_{2n}\leq\dfrac{\ell_{2}}{\ell_{1}}\cdot
t_{2}(a_{2n})\nu_{2n-1}+\ell_{1}^{-a_{2n-1}}\nu_{2n-2}+\widetilde{K^{\prime}}$
(3.17)

and

$\nu_{2n+1}\leq\dfrac{\ell_{1}}{\ell_{2}}\cdot
t_{1}(a_{2n+1})\nu_{2n}+\ell_{2}^{-a_{2n}}\nu_{2n-1}+\widetilde{K^{\prime}};$
(3.18)

with

$t_{i}(j)=\dfrac{1-\ell_{i}^{-j}}{\ell_{i}-1}.$

####  3.2.2 Analysis of Recursive Affine Inequality

We will prove that the sequence $\nu_{n}$ is bounded. Let us consider the
sequence of vectors $(v_{n})$ defined by

$v_{n}=\left(\begin{array}[]{c}\nu_{n}\\\ \nu_{n-1}\end{array}\right),$

the vector given by

$\overline{\kappa}=\left(\begin{array}[]{c}\widetilde{K^{\prime}}\\\
0\end{array}\right)$

and the sequence matrix

$A_{{{\ell_{1}}},{{\ell_{2}}}}(2n)=\left(\begin{array}[]{cc}\dfrac{\ell_{2}}{\ell_{1}}\cdot
t_{2}(a_{2n})&\ell_{1}^{-a_{2n-1}}\\\ 1&0\end{array}\right)$

and

$A_{{{\ell_{1}}},{{\ell_{2}}}}(2n+1)=\left(\begin{array}[]{cc}\dfrac{\ell_{1}}{\ell_{2}}\cdot
t_{1}(a_{2n+1})&\ell_{2}^{-a_{2n}}\\\ 1&0\end{array}\right);$

say associated matrix to the Recursive Affine Inequalities (3.17) and (3.18)
respectively; in this sense that (3.17) and (3.18) can be rewritten
respectively in the form:

$v_{2n}\leq
A_{{{\ell_{1}}},{{\ell_{2}}}}(2n)v_{2n-1}+\overline{\kappa}\quad\mbox{and}\quad
v_{2n+1}\leq A_{{{\ell_{2}}},{{\ell_{1}}}}(2n+1)v_{2n}+\overline{\kappa}.$
(3.19)

Therefore, for every $n:=2p_{n}+r_{n}$; with $p_{n}\in{\mathbb{N}}^{*}$ and
$r_{n}\in\\{0,1\\}$, we have

$\displaystyle v_{n}\leq$
$\displaystyle\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n-2)\cdots\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(2+r_{n})v_{2-r_{n}}+$
$\displaystyle\left(Id+\sum_{i=4+r_{n}}^{n}\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n-2)\cdots\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(i)\right)\overline{\kappa^{\prime}}.$

where,

$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)=A_{{{\ell_{1}}},{{\ell_{2}}}}(n)A_{{{\ell_{1}}},{{\ell_{2}}}}(n-1).$

Observe that, if $({{\ell_{1}}},{{\ell_{2}}})$ is very close to an element of
the set

$\\{(a,\infty),\;(\infty,b),\;(\infty,\infty),\;a,b\in{\mathbb{R}}\\}$

$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)$ is diagonalizable with
nonnegative eigenvalues and at most one is strictly positive; that is,
${1}/{\ell_{1}}$ or ${1}/{\ell_{2}}$ and as $\ell_{1},\ell_{2}>2$, then,
$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)$ contracts the Euclidean metric;
therefore, $v_{n}\mbox{ (also }\nu_{n}\mbox{ and }\alpha_{n}\mbox{)}$ is
bounded. In the following analysis, we suppose that
$1<\ell_{1},\ell_{2}<C<\infty$; for some $C$ in ${\mathbb{R}}_{+}$.

######  Lemma 3.18.

Fix $({{\ell_{1}}},{{\ell_{2}}})\in[2,\infty)^{2}$. The sequence

$\overline{A}^{\circ
n}_{{{\ell_{1}}},{{\ell_{2}}}}:=\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n-2)\cdots\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(4)$

is bounded (uniformly) by
$\max\\{{{{\ell_{1}}}}/{{{\ell_{2}}}},{{{\ell_{2}}}}/{{{\ell_{1}}}}\\}.$

###### Proof.

Observe that, for all $n\in{\mathbb{N}}$,

$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)\leq\left(\begin{array}[]{cc}(1-b_{n}(2))(1-b_{n-1}(2))&\frac{\ell_{1}}{\ell_{2}}(1-b_{n}(2))b_{n-2}(2)\\\
\frac{\ell_{2}}{\ell_{1}}(1-b_{n-1}(2))&b_{n-2}(2)\end{array}\right)=:\overline{B}_{{{\ell_{1}}},{{\ell_{2}}}}(n);$

with,

$b_{2n}(\ell)=b_{2n}:=\ell_{2}^{-a_{2n}}\quad\mbox{and}\quad
b_{2n+1}(\ell)=b_{2n+1}:=\ell_{1}^{-a_{2n+1}}.$

Let be a sequence $(x_{n})_{n\in{\mathbb{N}}}$ defined by:

$(x_{n})_{n\in{\mathbb{N}}}:=\\{1-b_{n}(2),b_{n}(2);n\in{\mathbb{N}}\\}.$

Remark that for every $n\in{\mathbb{N}}$, $x_{n}\in[2^{-a},1-2^{-a}]$; where,
$a:=\max\\{a_{n},n\in{\mathbb{N}}\\}$.

Thus, by setting

$\overline{B}^{\circ
n}_{{{\ell_{1}}},{{\ell_{2}}}}=\left(\begin{array}[]{cc}d^{1,n}&\frac{\ell_{2}}{\ell_{1}}d^{3,n}\\\
\frac{\ell_{1}}{\ell_{2}}d^{2,n}&d^{4,n}\end{array}\right),$

it follows that for every $i\in\\{1,2,3,4\\}$, there is $x_{k_{i,j}}$,
$j=p_{n-2},\cdots,n-2$ such that:

$\displaystyle d^{i,n}\leq\sum_{j=p_{n-2}}^{n-2}x_{k_{i,1}}\cdots
x_{k_{i,j}}\leq 1$

This proves the lemma.∎

######  Proposition 3.19.

When $(\ell_{1},\ell_{2})\in[2,\infty)^{2}\setminus\\{(2,2)\\}$, then
$\overline{A}^{\circ n}_{{{\ell_{1}}},{{\ell_{2}}}}$ contracts the Euclidean
metric provided $n$ is large enough. The scale of the contraction is bounded
away from 1 independently of $\ell_{1}$ and $\ell_{2}$ and the particular
sequence $b_{n}$, whereas the moment when the contraction starts depends on
the upper bound of $b_{n}$.

###### Proof.

For fixed $n\in{\mathbb{N}}$, putting

$\overline{A}^{\circ
n}_{{{\ell_{1}}},{{\ell_{2}}}}=\left(\begin{array}[]{cc}d^{1,n}(z_{1},z_{2})&\frac{\ell_{1}}{\ell_{2}}d^{3,n}(z_{1},z_{2})\\\
\frac{\ell_{2}}{\ell_{1}}d^{2,n}(z_{1},z_{2})&d^{4,n}(z_{1},z_{2})\end{array}\right);$

with $z_{1}=1/(\ell_{1}-1)$ and $z_{2}=1/(\ell_{2}-1)$ Thus,
$d^{i,n}(z_{1},z_{2})$, $i=1,2,3,4$ are polynomials of respective degree
$n-2$, $n-3$, $n-3$ and $n-4$; whose coefficients belong to the interval
$[\ell^{-a},1-\ell^{-a}]$; with, $\ell=\max\\{\ell_{1},\ell_{2}\\}$. For fixed
$i\in\\{1,2,3,4\\}$, we denote by $d^{i,n}_{j}$ the coefficients of
$d^{i,n}(z_{1},z_{2})$. Let us put:

$\begin{cases}\begin{array}[]{ccccc}d_{1}(2)=1&\mbox{ and }&d_{1}(i)=0;\quad
i=1,3,4\\\ d_{2}(3)=1&\mbox{ and }&d_{2}(i)=0;\quad
i=1,2,4.\end{array}\end{cases}$

Then by the Lemma 3.18, the sums

$\sum\limits_{j=0}^{p_{n-2}}d^{i,j}z_{1}^{j+d_{1}(i)}z_{2}^{j+d_{2}(i)};\quad
i=1,2,3,4$

are uniformly bounded. Therefore, for every $i\in\\{1,2,3,4\\}$

$\sum\limits_{j=k}^{\infty}d_{j}^{i,n}z_{1}^{j+d_{1}(i)}z_{2}^{j+d_{2}(i)}\longrightarrow
0,\quad\mbox{ when }k\longrightarrow\infty$

######  Lemma 3.20.

For every $i\in\\{1,2,3,4\\}$, the sequence $(d_{j}^{i,n})$ tends to zero at
least exponentially.

###### Proof.

By a simple calculation, we have

$d^{1,n}(0,0)=\prod_{i=1}^{p_{n-2}}b_{n-2i+1},\;d^{4,n}(0,0)=\prod_{i=1}^{p_{n-2}}b_{n-2i},\;d^{2,n}(0,0)=d^{3,n}(0,0)=0.$

Now, suppose that, for given $0<n-1$ and $0<j<n-1$ all the coefficients
$d_{j}^{i,n-1}$ $i\in\\{1,2,3,4\\}$, tend to zero at least exponentially fast.
Then, since

$\overline{A}^{\circ
n}_{{{\ell_{1}}},{{\ell_{2}}}}=\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)\overline{A}^{\circ{n-1}}_{{{\ell_{1}}},{{\ell_{2}}}}$

and by the form of the coefficients of
$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}(n)$, the Lemma is proved and
therefore, the Proposition 3.19. ∎

∎

####  3.2.3 Particular case of Bounded Geometry

######  Proposition 3.21.

Let $f\in\mathscr{L}$ with critical exponents $(\ell_{1},\ell_{2})$ and
rotation number $\rho(f)=[abab\cdots]$; for some $a,b\in{\mathbb{R}}$. If the
inequality

$\begin{array}[]{r}\sqrt{(\ell_{1}^{-b}-\ell_{2}^{-a})^{2}+(t_{1}(b)t_{2}(a)+2(\ell_{1}^{-b}+\ell_{2}^{-a}))t_{1}(b)t_{2}(a)}+\\\
+t_{1}(b)t_{2}(a)+\ell_{1}^{-b}+\ell_{2}^{-a}-2<0;\end{array}$

holds, then the geometry of $f$ is bounded.

###### Proof.

If $\rho(f)=[abab\cdots]$, then

$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}=\left(\begin{array}[]{cc}t_{1}(b)t_{2}(a)+\ell_{1}^{-b}&\dfrac{\ell_{2}}{\ell_{1}}\ell_{2}^{-a}t_{2}(a)\\\
\dfrac{\ell_{1}}{\ell_{2}}t_{1}(b)&\ell_{2}^{-a}\end{array}\right)$

and these eigenvalues $\lambda_{s}$ and $\lambda_{u}$ are defined as following

$\begin{array}[]{r}2\lambda_{s}=-\sqrt{(\ell_{1}^{-b}-\ell_{2}^{-a})^{2}+(t_{1}(b)t_{2}(a)+2(\ell_{1}^{-b}+\ell_{2}^{-a}))t_{1}(b)t_{2}(a)}+\\\
+t_{1}(b)t_{2}(a)+\ell_{1}^{-b}+\ell_{2}^{-a}\end{array}$

and

$\begin{array}[]{r}2\lambda_{u}=\sqrt{(\ell_{1}^{-b}-\ell_{2}^{-a})^{2}+(t_{1}(b)t_{2}(a)+2(\ell_{1}^{-b}+\ell_{2}^{-a}))t_{1}(b)t_{2}(a)}+\\\
+t_{1}(b)t_{2}(a)+\ell_{1}^{-b}+\ell_{2}^{-a};\end{array}$

Observe that, $\lambda_{s}\in(0,1)$. Thus, if $\lambda_{u}<1$, then
$\overline{A}_{{{\ell_{1}}},{{\ell_{2}}}}$ contracts the Euclidean metric.

This proves the proposition. ∎

###  3.3 Proof of Corollary

######  Lemma 3.22.

Let

$w_{n}(i)=\dfrac{|(\underline{-q_{n}+(i+1)q_{n-1}},\underline{-q_{n}+iq_{n-1}})|}{|\underline{-q_{n}+iq_{n-1}}|}$

be a parameter sequence with, $i=0\cdots a_{n-1}-1$. $w_{n}(i)$, $i=1\cdots
a_{n-1}-1$ and $w^{\ell_{1},\ell_{2}}_{n}(0)$ are comparable to
$\alpha_{n-1}$.

###### Proof.

Suppose that $a_{n-1}>1$ and let $i=1,\cdots,a_{n-1}-1$. We apply the
Proposition 2.10 to

  1. -

$T=[\underline{-q_{n}+(i+1)q_{n-1}},\underline{-q_{n}+(i-1)q_{n-1}}]$,

  2. -

$M=(\underline{-q_{n}+(i+1)q_{n-1}},\underline{-q_{n}+(i-1)q_{n-1}})$,

  3. -

$S=\underline{-q_{n}+(i+1)q_{n-1}}$,

  4. -

$D=\underline{-q_{n}+(i-1)q_{n-1}}$,

  5. -

$f^{q_{n}-(i-1)q_{n-1}}$.

  1. 1.

For every $j<q_{n}-(i-1)q_{n-1}$, $f^{j}(T)\cap\overline{U}=\emptyset$; so
$f^{q_{n}-(i-1)q_{n-1}}$ is diffeomorphism on $T$ (Remark 2.11);

  2. 2.

the set $\bigcup_{j=0}^{q_{n}-(i-1)q_{n-1}}(T)$ covers the circle at most two
times;

  3. 3.

for $n$ large enough, and by the Proposition 2.8, we have

$\dfrac{f^{q_{n}-(i-1)q_{n-1}}(M)}{f^{q_{n}-(i-1)q_{n-1}}(S)}<\dfrac{f^{q_{n}-(i-1)q_{n-1}}(M)}{f^{q_{n}-(i-1)q_{n-1}}(D)}=\dfrac{|(\underline{0},\underline{-2q_{n-1}})|}{|\underline{-2q_{n-1}}|}<K.$

Therefore, it follows from the Proposition 2.10 and Proposition 2.8
$\mbox{(}|\underline{-q_{n-1}}|\mbox{ and
}|(\underline{0},\underline{-q_{n-1}}]|$ are comparable) that $w_{n}(i)$ and
$\alpha_{n-1}$ are comparable.

For $i=0$ (which is the only case when $a=1$), we apply the Proposition 2.10
to

  1. -

$T=[\underline{-q_{n}+q_{n-1}+1},\underline{-q_{n}-q_{n-1}+1}]$,

  2. -

$M=(\underline{-q_{n}+q_{n-1}+1},\underline{-q_{n}-q_{n-1}+1})$,

  3. -

$S=\underline{-q_{n}+q_{n-1}+1}$,

  4. -

$D=\underline{-q_{n}-q_{n-1}+1}$,

  5. -

$f^{q_{n}-q_{n-1}-1}$.

As before, the hypotheses are satisfied. And for $n$ large enough,

$w^{\ell_{1},\ell_{2}}_{n}(0)=\dfrac{|(\underline{-q_{n}+q_{n-1}+1},\underline{-q_{n}+1})|}{|\underline{-q_{n}+1}|};$

which is also uniformly comparable to $\alpha_{n-1}$.

This concludes the proof. ∎

The rest of the proof of Corollary is as in [8] (Theorem 1.4 and Theorem 1.5).

## References

  * [1] de Melo, W. and Van Strien, S., One-Dimensional Dynamics: The Schwarzian Derivative And Beyond, Amer. J. Math., 1988, vol. 18, no. 2, pp. 159–162.
  * [2] de Melo, W. and Van Strien, S., One-Dimensional Dynamics: The Schwarzian Derivative and Beyond, Ann. of Math., 1989, vol. 129, pp. 519–546.
  * [3] de Melo, W. and Van Strien, S., One-Dimensional Dynamics, Springer–Verlag, 1993.
  * [4] 4 Graczyk, J., Dynamics of circle maps with flat spots, Fund. Math., 2010, vol. 209, no. 3, pp. 267–290.
  * [5] Graczyk, J., Jonker, L. B., Światek, G., Tangerman, F. M. and Veerman, J. J. P., Differentiable Circle Maps with a Flat Interval, Commun. Math. Phys., 1995, vol. 173, no. 3, pp. 599–622.
  * [6] Mendes, P., A metric property of Cherry vector fields on the torus, J. Differential Equations, 1991, vol. 89, no. 2, pp. 305–316.
  * [7] Martens, M., Palmisano, L., Invariant Manifolds for Non-differentiable Operators, arXiv:1704.06328, (20 Apr 2017).
  * [8] Martens, M., Strien, S., Melo, W. and Mendes, P., On Cherry flows, Ergod. Theory Dyn. Syst., 1990, vol. 10 , pp. 531–554.
  * [9] Misiurewicz, M., Rotation interval for a class of maps of the real line into itself, Erg. Th. and Dyn. Sys. 1986, vol. 6, no. 3, pp. 17–132.
  * [10] Moreira, P. C. and Ruas, A. A. Gaspar, Metric properties of Cherry flows, J. Differential Equations, 1992, vol. 97, no. 1, pp. 16–26.
  * [11] Palmisano, L., A Phase Transition for circle Maps and Cherry Flows, Commun. Math. Phys., 2013, vol. 321, no. 1, pp. 135–155.
  * [12] Palmisano, L., On physical measures for Cherry flows, Fund. Math., 2016, vol. 232, no. 2, pp. 167–179.
  * [13] Palmisano, L., Cherry Flows with non-trivial attractors, Fund. Math., 2019 vol. 244, no. 3, pp. 243–253.
  * [14] Palmisano, L., Quasi-symmetric conjugacy for circle maps with a flat interval, Ergodic Theory Dynam. Systems, 2019 vol. 39, no. 2, pp. 425–445.
  * [15] Palmisano, L. and Tangue, B., A Phase Transition for Circle Maps with a Flat Spot and Different Critical Exponents, arXiv: 1907.10909v1, (27 Jul. 2019).
  * [16] Światek, G. , Rational rotation numbers for maps of the circle, Comm. Math. Phys. 1988, vol. 119, no. 1, pp. 109–128.
  * [17] Tangerman, F. M. and Veerman, J. J. P., Scalings in circle maps. I, Comm. Math. Phys., 1990, vol. 134, no. 1, pp. 89–107.
  * [18] Tangerman, F. M. and Veerman, J. J. P., Scalings in circle maps. II, Comm. Math. Phys., 1991, vol. 141, no. 3, pp. 279–291.
  * [19] Veerman, J. J. P., Irrational Rotation Numbers, Nonlinearity, 1989, vol. 3, no. 3, pp. 419–428.