Commuting circle diffeomorphisms with their derivatives having mixed moduli of continuity

TL;DR

This paper proves that commuting circle diffeomorphisms with derivatives having mixed moduli of continuity are topologically conjugate to rotations under certain conditions on their rotation numbers and moduli.

Contribution

It establishes a new conjugacy result for commuting diffeomorphisms with derivatives exhibiting mixed moduli of continuity, extending previous regularity conditions.

Findings

Diffeomorphisms with independent rotation numbers are conjugate to rotations.

The product of moduli of continuity conditions leads to conjugacy.

Results apply to a broad class of moduli including H"older continuity.

Abstract

Let $d\geq 2$ be an integer and let $\omega_1,\cdots ,\omega_d$ be moduli of continuity in a specified class which contains the moduli of H\"{o}lder continuity. Let $f_k$, $k\in\{1,\cdots,d\}$, be $C^{1+\omega_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $\omega_1(t)\cdots\omega_d(t)=t\omega(t)$ with $\lim_{t\rightarrow 0^+}\omega(t)=0$, then $f_1,\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations.

Equations

$$\underset{t\to 0^{+}}{\lim }\omega (t)=0\text{and}\omega (t_1{t}_2)\ll \omega (t_1)\omega (t_2),\forall t_1,t_2\in [0,1].$$

$${\omega }_f(t)=\underset{|x-y|\le t}{\sup }|f(x)-f(y)|.$$

$$\underset{x\ne y}{\sup }\frac{|f(x)-f(y)|}{\omega (|x-y|)}<+\infty .$$

$$x_n\le C{y}_n,\forall n\ge 1.$$

$${\omega }_k\left(\frac{1}{{\prod }_{j=1}^n{X}_{j,m}}\right)\ll \prod _{j=1}^n{\omega }_j\left(\frac{1}{X_{k,m}}\right).$$

$$X_{1,m}=\left[2^{{\alpha }_{1}m}\right],\cdots ,X_{1,m}=\left[2^{{\alpha }_{d}m}\right],$$

$${\omega }_j(x_{1,m}\cdots x_{d,m})$$

$${\omega }_1(x_{j,m})\cdots {\omega }_d(x_{j,m})$$

$${\omega }_j(x_{1,m}\cdots x_{d,m})\ll {\omega }_1(x_{j,m})\cdots {\omega }_d(x_{j,m}),\forall m\in \mathbb{N}.$$

$${\omega }_j\left(\frac{1}{X_{1,m}\cdots X_{d,m}}\right)\ll {\omega }_1\left(\frac{1}{X_{j,m}}\right)\cdots {\omega }_d\left(\frac{1}{X_{j,m}}\right),m\in \mathbb{N}.$$

$$\underset{x\ne y}{\sup }\frac{|\phi \circ \psi (x)-\phi \circ \psi (y)|}{\omega (|x-y|)}$$

$$\sum _{n=0}^{n_0-1}{\omega }_{k_{n+1}}(g_{k_n}\cdots g_{k_1}(I))\le S.$$

$$\left|\log \frac{D(g_{k_j}\cdots g_{k_1})(x)}{D(g_{k_j}\cdots g_{k_1})(y)}\right|$$

$$C\sum _{i=0}^{j-1}{\omega }_{k_{i+1}}\left(2|D(g_{k_j}\cdots g_{k_1})(I)|\right)$$

$$|D(g_{k_j}\cdots g_{k_1})(I_1)|=D(g_{k_j}\cdots g_{k_1})(x_j)|I_1|,$$

$$|D(g_{k_j}\cdots g_{k_1})(I)|=D(g_{k_j}\cdots g_{k_1})(y_j)|I|.$$

$$\frac{|D(g_{k_j}\cdots g_{k_1})(I_1)|}{|D(g_{k_j}\cdots g_{k_1})(I)|}=\frac{D(g_{k_j}\cdots g_{k_1})(x_j)|I_1|}{D(g_{k_j}\cdots g_{k_1})(y_j)|I|}\le \exp (CS)\frac{|I_1|}{|I|}\le 1.$$

$$\sum _{i=1}^m\omega (l_{i,j})\le m\omega \left(\frac{1}{mn}\right).$$

$$\sum _{i=1}^m\omega (l_{i,k})\le Am\omega \left(\frac{1}{mn}\right).$$

$$\sum _{i=1}^m\omega (l_{i,j})=m\sum _{i=1}^m\frac{1}{m}\omega (l_{i,j})\le m\omega \left(\sum _{i=1}^m\frac{l_{i,j}}{m}\right).$$

$$\frac{1}{n}\sum _{k=1}^n\left(\sum _{i=1}^m\omega (l_{i,j})\right)$$

$$\sum _{i=1}^m\omega (l_{i,j})\le m\omega \left(\frac{1}{mn}\right).$$

$$E:=\left\{k\in \{1,\cdots ,n\}:\sum _{i=1}^m\omega (l_{i,k})\ge Am\omega \left(\frac{1}{mn}\right)\right\}.$$

$$|E|\ge \left(1-\frac{1}{A}\right)n.$$

$$m\omega \left(\frac{1}{mn}\right)$$

$$R_{2m+1}=[i_m,i_{m+1}]\times [j_m,j_{m+2}],R_{2m+2}=[i_m,i_{m+2}]\times [j_{m+1},j_{m+2}].$$

$$\sum _{j=j_m}^{j_{m+2}}{\omega }_2(l_{r(2m+1),j})$$

$$\sum _{i=i_m}^{i_{m+2}}{\omega }_1(l_{i,r(2m+2)})$$

$$(i(n+1),j(n+1))-(i(n),j(n))=(0,1)\text{or}(1,0).$$

$$\sum _{m\ge 0}\left(\frac{{\omega }_2\left(\frac{1}{X_{2m+1}{Y}_{2m+1}}\right)}{{\omega }_1\left(\frac{1}{Y_{2m+1}}\right){\omega }_2\left(\frac{1}{Y_{2m+1}}\right)}\omega \left(\frac{1}{Y_{2m+1}}\right)+\frac{{\omega }_1\left(\frac{1}{X_{2m+2}{Y}_{2m+2}}\right)}{{\omega }_1\left(\frac{1}{X_{2m+2}}\right){\omega }_2\left(\frac{1}{X_{2m+2}}\right)}\omega \left(\frac{1}{X_{2m+2}}\right)\right),$$

Full text

1. Introduction

The classical Poincaré’s classification theorem points out that every orientation preserving homeomorphism of the circle with irrational rotation number is topologically conjugate or semiconjugate to a rigid rotation. Furthermore, Denjoy [1, 2] showed that it must be conjugate by adding differentiability of the homeomorphism. Precisely, for a circle diffeomorphism $f$ with irrational rotation number, if $Df$ has bounded variation, then $f$ is conjugate to a rigid rotation. However, for any $0<\tau<1$, there exists a circle diffeomorphism $f$ such that $Df$ is $\tau$-Hölder continuous and $f$ has wandering intervals.

The smoothness of the conjugacy between a circle diffeomorphism $f$ and a rigid rotation was intensively studied. If $f$ is analytic, Arnold [3] showed the analyticity of the conjugacy under the assumption that $f$ is sufficiently close to a rotation and its rotation number satisfies certain Diophantine condition. Moser [4] obtained the local result in the smooth case. Later Herman [6] proved the global version: if $f$ is $k$ times differentiable and its rotation number lies in some set $A$ of full measure, then the conjugacy is $k-1-\varepsilon$ times differentiable for any $\varepsilon>0$, and if $f$ is analytic then the conjugacy is analytic. Yoccoz [7] showed the global result for all Diophantine numbers and the result was sharpen by Katznelson and Ornstein [8].

Moser [5] considered the smoothness of the simultaneous conjugacy between a finite family of commuting circle diffeomorphisms and rigid rotations and obtained the local results by KAM method. Fayad and Khanin [9] established the global version: if the rotation numbers of several commuting $C^{\infty}$ circle diffeomorphisms satisfy the Diophantine condition, then they are smoothly conjugate to rigid rotations simultaneously.

Similar to the classical Denjoy Theorem, Kleptsyn and Navas [10] determined when several commuting circle diffeomorphisms are simultaneously topologically conjugate to rigid rotations. They showed the following theorem.

Theorem 1.1**.**

Let $d\geq 2$ be an integer. Let $f_{1},\cdots,f_{d}$ be commuting circle diffeomorphisms such that $f_{k}$ is $C^{1+\tau_{k}}$ for each $k\in\{1,\cdots,d\}$, where $0<\tau_{k}<1$. If $\tau_{1}+\cdots+\tau_{d}>1$ and the rotation numbers of $f_{k}$’s are independent over the rationals, then they are simultaneously topologically conjugate to rotations.

Conversely, given $\alpha_{1},\cdots,\alpha_{d}\in\mathbb{T}$ which are independent over the rationals and $\tau_{1},\cdots,\tau_{d}\in(0,1)$. If $\tau_{1}+\cdots+\tau_{d}<1$, then there exist commuting circle diffeomorphisms $f_{1},\cdots,f_{d}$ such that $f_{k}$ is $C^{1+\tau_{k}}$ and the rotation number of it is $\alpha_{k},k\in\{1,\cdots,d\}$.

The main purpose of this paper is to generalize their results by replacing the Hölder condition by general modulus of continuity. For technical reasons, we introduce the notion of consistency for a family of moduli of continuity in the next section and obtain the following result.

Theorem 1.2**.**

Let $d\geq 2$ be an integer and let $\omega_{1},\cdots,\omega_{d}$ be concave moduli of continuity with $\omega_{1}(t)\cdots\omega_{d}(t)=t\omega(t)$ for some function $\omega(t)$ satisfying that $\lim_{t\rightarrow 0^{+}}\omega(t)=0$. Suppose that $\omega_{1},\cdots,\omega_{d}$ satisfy the consistency condition. If $f_{k},k\in\{1,\cdots,d\}$, are respectively $C^{1+\omega_{k}}$ commuting circle diffeomorphisms and the rotation numbers of which are independent over the rationals, then they are simultaneously topologically conjugate to rotations.

Here we should remark that the proof idea of the above theorem comes from [10]. The consistency condition can be implied by some more simpler conditions, and we get the following corollary.

Corollary 1.3**.**

Let $d\geq 2$ be an integer and $\omega_{1},\cdots,\omega_{d}$ be concave moduli of continuity satisfying that $\omega_{1}(t)\cdots\omega_{d}(t)=t\omega(t)$ with

[TABLE]

Suppose that $\omega_{1},\cdots,\omega_{d}$ and the moduli of Hölder continuity compose a comparable family of moduli of continuity. If $f_{k},k\in\{1,\cdots,d\}$, are respectively $C^{1+\omega_{k}}$ circle diffeomorphisms which do commute and the rotation numbers of which are independent over the rationals, then they are simultaneously topologically conjugate to rotations.

The paper is organized as follows. In section 2, we show some definitions and lemmas. In section 3, we prove the main theorem. In the last section, we proved Corollary 1.3.

2. Some Definitions and Lemmas

For a continuous function $f$ of the circle, the* modulus of continuity* of $f$ is defined by

[TABLE]

Then $\omega_{f}(t)$ is a continuous function on $[0,+\infty)$ and has the following properties:

• (1)

$\omega_{f}$ is monotonic nondecreasing;

• (2)

$\omega_{f}(0)=0$ and $\omega_{f}(t)=1$ for any $t\geq 1$;

• (3)

$\omega_{f}(t_{1}+t_{2})\leq\omega_{f}(t_{1})+\omega_{f}(t_{2})$;

• (4)

$\omega_{f}(kt)\leq k\omega_{f}(t)$ for $k\in\mathbb{N}$ and $t>0$;

• (5)

$\omega_{f}(rt)\leq([r]+1)\omega_{f}(t)$ for $r\geq 0$.

Note that if $\omega_{f}(t)=O(t^{\alpha})$, then $f$ is called $\alpha$-Hölder continuous. In order to study the modulus of continuity of a function $f$, we often choose some standard moduli to measure the continuity of $f$. The modulus of Hölder continuity are often chosen.

Definition 2.1**.**

A modulus of continuity is a continuous function $w:[0,1]\rightarrow[0,+\infty)$ which is strictly increasing and $w(0)=0$. We say a continuous function $f$ of the circle is $\omega$-continuous if

[TABLE]

In this case, we say that $\sup_{x\neq y}\frac{|f(x)-f(y)|}{\omega(|x-y|)}$ is the $\omega$-constant of $f$. We say $f$ is of class $C^{1+\omega}$ if $f$ is $C^{1}$ and the derivative of $f$ is $\omega$-continuous.

In order to compare the moduli of continuity of distinct functions, we often choose some comparable moduli of continuity.

Definition 2.2**.**

Let $\omega_{1}$ and $\omega_{2}$ be moduli of continuity. We say that $\omega_{1}$* is stronger than $\omega_{2}$*, if there exists $\delta>0$ such that $\omega_{1}(x)\leq\omega_{2}(x),\forall x\in(0,\delta)$. In this case, we call they are comparable. We say a family $\Omega$ of moduli of continuity is comparable, if there exists $\delta>$ such that for any $\omega,\omega^{\prime}\in\Omega$, $\omega(x)\geq\omega^{\prime}(x),\forall x\in(0,\delta)$ or $\omega(x)\leq\omega^{\prime}(x),\forall x\in(0,\delta)$.

Let $(x_{n})_{n\geq 1}$ and $(y_{n})_{n\geq 1}$ be two sequences of positive real numbers. We write $x_{n}\ll y_{n}$ if there exists a constant $C>0$ such that

[TABLE]

Definition 2.3**.**

Let $\omega_{1},\cdots,\omega_{n}$ be moduli of continuity. We call them consistent if there exist strictly increasing sequences $(X_{1,m})_{m\geq 1},\cdots$, $(X_{n,m})_{m\geq 1}$ of positive integers such that for each $k\in\{1,\cdots,n\}$,

[TABLE]

Remark 2.4**.**

The consistency is a technical condition. It is easy to check that the usual moduli of continuity satisfy the consistency. Taking $\omega_{1}(x)=x^{\alpha_{1}},\cdots,\omega_{d}(x)=x^{\alpha_{d}}$ as example, we can choose

[TABLE]

for $m$ large enough.

Proposition 2.5**.**

Let $d\geq 2$ be an integer and $\alpha_{1},\cdots,\alpha_{d}\in(0,1),\varepsilon_{1},\cdots$, $\varepsilon_{d}\geq 0$. The moduli of continuity $\omega_{1}(x)=x^{\alpha_{1}}\left(\log\frac{1}{x}\right)^{\varepsilon_{1}},\cdots,\omega_{d}(x)=x^{\alpha_{d}}\left(\log\frac{1}{x}\right)^{\varepsilon_{d}}$ for $x$ small are consistent.

Proof.

Let $a>1$ and $x_{j,m}=\frac{1}{a^{\alpha_{j}m}},j\in\{1,\cdots,d\}$. Then for any $j\in\{1,\cdots,d\}$, we have

[TABLE]

and

[TABLE]

It is easy to see that

[TABLE]

Set $X_{j,m}=[a^{\alpha_{j}m}]$ for $m$ large enough. Then, by the continuity, we also have

[TABLE]

This shows that $\omega_{1},\cdots,\omega_{d}$ are consistent. ∎

Lemma 2.6**.**

Let $\varphi$ and $\psi$ be continuous functions of the circle. If $\varphi$ is Lipschitz-continuous and $\psi$ is $\omega$-continuous, then $\varphi\circ\psi$ is also $\omega$-continuous.

Proof.

The $\omega$-continuity of $\varphi\circ\psi$ can be seen as follows:

[TABLE]

∎

Remark 2.7**.**

If $g$ is an orientation preserving circle diffeomorphism, then there exist constants $C_{1},C_{2}>0$ such that $C_{1}\leq Dg_{k}(x)\leq C_{2},\forall x\in\mathbb{T}^{1}$. Note that the function $\log(x),x\in[C_{1},C_{2}]$ is Lipschitz. Then, by Lemma 2.6, we know that $\log Dg$ has the same modulus of continuity with $Dg$.

Lemma 2.8**.**

Let $\omega_{1},\cdots,\omega_{l}$ be concave moduli of continuity and $g_{k},k\in\{1,\cdots,l\}$ be orientation preserving circle diffeomorphisms which are respectively of class $C^{1+\omega_{k}}$. Let $C_{k}$ denote the $\omega_{k}$-constant of $\log Dg_{k}$ and $C=\max\{C_{1},\cdots,C_{l}\}$. Given $n_{0}\in\mathbb{N}$, for each $n\leq n_{0}$, let us choose $k_{n}\in\{1,\cdots l\}$ and for a fixed interval $I$, let us choose a constant $S>0$ such that

[TABLE]

If there exists $n\leq n_{0}$ such that $g_{k_{n}}\cdots g_{k_{1}}(I)$ is contained in the $L$-neighborhood of $I$ but does not intersect $I$, then $g_{k_{n}}\cdots g_{k_{1}}$ has a fixed point, where $L:=\frac{|I|}{2\exp(2CS)}$.

Proof.

Let $J$ be the closed $2L$-neighborhood of $I$ and $I_{1},I_{2}$ the connected components of $J\setminus I_{1}$ to the complement of $I$. By induction on $j\in\{0,\cdots,n_{0}\}$, we will prove the following properties:

• ($A_{j}$)

$|g_{k_{j}}\cdots g_{k_{1}}(I_{1})|\leq|g_{k_{j}}\cdots g_{k_{1}}(I)|$;

• ($B_{j}$)

$\sup_{x,y\in I\cup I_{1}}\frac{D(g_{k_{j}}\cdots g_{k_{1}})(x)}{D(g_{k_{j}}\cdots g_{k_{1}})(y)}\leq\exp(2CS)$.

The properties $(A_{0})$ and $(B_{0})$ are trivial. Suppose that $(A_{i})$ and $(B_{i})$ hold for every $i\in\{0,\cdots,j-1\}$. Then for any $x,y\in I\cup I_{1}$, we have

[TABLE]

Since $\omega(t)$ is increasing and $|(D(g_{k_{j}}\cdots g_{k_{1}})(I_{1})|\leq|(D(g_{k_{j}}\cdots g_{k_{1}})(I)|$ by induction hypothesis, the righthand side of the above inequality is bounded by

[TABLE]

The first inequality above is followed that $\omega(t)\geq\frac{1}{2}\omega(0)+\frac{1}{2}\omega(2t)\Longrightarrow\omega(2t)\leq 2\omega(t)$, since $\omega(t)$ is concave. Thus $(B_{j})$ follows. By the Mean Value Theorem, there exist $x_{j}\in I_{1}$ and $y_{j}\in I$ such that

[TABLE]

and

[TABLE]

Then by $(B_{j})$, we obtain $(A_{j})$, since

[TABLE]

Similarly, we have the analogous arguments for $I_{2}$.

Suppose that $g_{k_{n}}\cdots g_{k_{1}}(I)$ is contained in the $L$-neighborhood of $I$ but does not intersect $I$. Then $(A_{n})$ implies that $g_{k_{n}}\cdots g_{k_{1}}(J)\subset J$. Hence $g_{k_{n}}\cdots g_{k_{1}}$ has a fixed point in $J$. ∎

Lemma 2.9**.**

Let $l_{i,j}$ be positive real numbers with $i\in\{1,\cdots,m\}$ and $j\in\{1,\cdots,n\}$. Suppose that $\sum_{i=1}^{m}\sum_{j=1}^{n}l_{i,j}\leq 1$. Let $\omega:[0,1]\rightarrow\mathbb{R}_{+}$ be an increasing and concave function. Then there exists $k\in\{1,\cdots,n\}$ such that

[TABLE]

Moreover, for any $A>0$, there is a proportion of indices $k\in\{1,\cdots,n\}$ no less than $(1-1/A)$ such that

[TABLE]

Proof.

For any $k\in\{1,\cdots,n\}$ , by the concavity of $\omega$, we have

[TABLE]

Thus

[TABLE]

Therefore, there exists $k\in\{1,\cdots,n\}$ such that

[TABLE]

In order to prove the second part, we define

[TABLE]

In other words, we need to show that

[TABLE]

Note that

[TABLE]

Thus $|E|\geq\left(1-\frac{1}{A}\right)n.$ ∎

3. Proof of Theorem1.2

Suppose that the theorem does not hold. Then there is a wandering interval with nonempty interior. Let $I$ be a maximal wandering interval. We will search for a sequence $h_{n}:=f_{k_{n}}\cdots f_{k_{1}}$ satisfying the hypothesis of Lemma 2.8. Then we conclude that $h_{n}$ has a fixed point. This implies that the rotation number of $h_{n}$ is zero. Recall that for a commutative group $G$ of circle homeomorphisms, there is a $G$-invariant probability measure. Then the rotation number is a homomorphism from $G$ to $\mathbb{R}/\mathbb{Z}$. But this contradicts to the hypothesis that the rotation numbers of $f_{1},\cdots,f_{d}$ are independent over the rationals.

3.1. The case $d=2$

Let $l_{i,j}=\left|f_{1}^{i}f_{2}^{j}(I)\right|$ . Since $I$ is a wandering interval, $\sum_{i,j\geq 0}l_{i,j}\leq 1$.

In the subsequent, we use $[a,b]$ to denote the integers between $a$ and $b$ including the end if it is an integer.

Let $(i_{n})_{n\geq 1}$ and $(j_{n})_{n\geq 1}$ be two sequences of non negative integers satisfying $i_{0}=i_{1}=j_{0}=j_{1}=0$. Consider a sequence of rectangles $R_{m}\subset\mathbb{N}\times\mathbb{N}$ with $R_{0}=\{(0,0)\}$ and

[TABLE]

Denote by $X_{m}$ and $Y_{m}$ the number of integers on the horizontal and vertical sides of $R_{m}$ respectively .

Applying Lemma 2.9 to $R_{m}$ gives us a sequence $(r(m))_{m\geq 0}$ of integers such that

• •

$r(2m+1)\in[i_{m},i_{m+1}]$ such that

[TABLE]

• •

$r(2m+2)\in[j_{m+1},j_{m+2}]$ such that

[TABLE]

Starting from the origin and following the corresponding horizontal line $y=r(2m+2)$ and vertical line $x=r(2m+1)$, we obtain a path $(i(n),j(n))_{n\geq 0}$ with $i(0)=j(0)=0$ and

[TABLE]

Moreover, the sum $\sum_{n\geq 0}\omega_{\xi(n)}\left(l_{i(n),j(n)}\right)$is bounded by

[TABLE]

where

[TABLE]

Since $\omega_{1}$ and $\omega_{2}$ are consistent, we can choose two strictly increasing sequences of integers $(i_{m})_{m\geq 0}$ and $(j_{m})_{m\geq 0}$ such that both $X_{m}$ and $Y_{m}$ tend to infinity as $m$ goes to infinity and

[TABLE]

and

[TABLE]

Since $\lim_{t\rightarrow 0^{+}}\omega(t)=0$, there exist a subsequence $m_{1}<m_{2}<\cdots$ of positive integers such that

[TABLE]

Thus we may assume that

[TABLE]

Therefore, there exists a constant $C_{1}>0$ such that

[TABLE]

For $n\geq 1$, let $k_{n}=\xi(n-1)\in\{1,2\}$. Then we obtain a sequence $h_{n}:=f_{k_{n}}\cdots f_{k_{1}}$ such that

[TABLE]

Hence For $n\geq 1$, let $k_{n}=\xi(n-1)\in\{1,2\}$. Then we obtain a sequence $h_{n}:=f_{k_{n}}\cdots f_{k_{1}}$ such that

[TABLE]

In order to apply Lemma2.8, it suffices to show that there exists some $n\geq 1$ such that $h_{n}(I)=f_{k_{n}}\cdots f_{k_{1}}(I)$ is contained in the $L$-neighborhood of $I$, since $I$ is a wandering interval.

Since $f_{1}$ and $f_{2}$ are semiconjugate to irrational rotations, if we collapse every connected component of the complement of the minimal invariant Cantor set, then we get a topological circle $\tilde{S^{1}}$ on which $f_{1}$ and $f_{2}$ induce minimal homeomorphisms $\tilde{f_{1}}$ and $\tilde{f_{2}}$ respectively. Now the $L$-neighborhood of $I$ becomes an interval $V$ with nonempty interior in $\tilde{S^{1}}$. Then there must exist $N\in\mathbb{N}$ such that both $\tilde{f_{1}}^{-1}(V),\cdots,\tilde{f_{1}}^{-N}(V)$ and $\tilde{f_{2}}^{-1}(V),\cdots,\tilde{f_{2}}^{-N}(V)$ cover the circle $\tilde{S^{1}}$. Since both $X_{m}$ and $Y_{m}$ go to infinity as $m$ tends to infinity, there exists an integer $r\in\mathbb{N}$ such that $k_{r}=k_{r+1}=\cdots=k_{r+N}$. Converting to the original standard circle, there exist $k,k^{\prime}\in\{1,\cdots,N\}$ such that both $f_{1}^{k}f_{k_{r}}\cdots f_{k_{1}}(I)$ and $f_{2}^{k^{\prime}}f_{k_{r}}\cdots f_{k_{1}}(I)$ are contained in the $L$-neighborhood of $I$. This implies that at least one of the intervals $h_{r+1}(I),\cdots,h_{r+N}(I)$ is contained in the $L$-neighborhood of $I$. Thus we complete the proof.

3.2. The general case

Let $(l_{i_{1},\cdots,i_{d}})_{i_{1},\cdots,i_{d}\geq 1}$ be a multi-indexed sequence of positive real numbers with $\sum_{i_{1},\cdots,i_{d}\geq 1}l_{i_{1},\cdots,i_{d}}\leq 1$. Let $(x_{1,m})_{m\geq 1}$, $\cdots,(x_{d,m})_{m\geq 1}$ be sequences of strict increasing nonnegative integers and let $R_{0}=\{(0,\cdots,0)\}$. Consider a sequence $(R_{m})_{m\geq 0}$ of rectangles of the form $R_{m}=[0.x_{1},m]\times\cdots\times[0,x_{d,m}]$ satisfying that for each $k\in\{1,\cdots,d\}$,

[TABLE]

Now for each $m\geq 1$, let $r(m)$ be the unique integer in $\{1,\cdots,d\}$ such that $m\equiv r(m)\mod d$. Denote by $F_{m}$ the face

[TABLE]

of $R_{m}$. Let $A_{m}>1$ and denote by $E_{m}$ the set of $(i_{1},\cdots,i_{r(m)-1},0,i_{r(m)+1}$, $\cdots,i_{d})\in F_{m}$ such that

[TABLE]

where $X_{j,m}:=1+x_{j,m}$. Then by Lemma 2.9, we have

[TABLE]

Similar to the proof of the case $d=2$, we need to choose a path of points $(x_{1}(n),\cdots,x_{d}(n))_{n\geq 0}$ starting at the origin which is long enough and satisfies

[TABLE]

where $P$ denote the length of the path and $S$ is independent of $P$. In order to show the existence of such path, we recall the following lemma which is showed in [10].

Lemma 3.1**.**

Denote by $\mathcal{L}(m)$ the set of all lines inside the rectangle $R_{m}$ in the $r(m)$-direction for each $m\geq 1$. Let $\mathcal{L}^{\prime}(m)$ be a subset of $\mathcal{L}(m)$ such that$|\mathcal{L}^{\prime}(m)|\geq\left(1-\frac{1}{A_{m}}\right)|\mathcal{L}(m)|$. If there exist $M\in\mathbb{N}$ such that $\sum_{m=1}^{M}\frac{1}{A_{m}}<1$, then there exists a sequence of lines $L_{m}\in\mathcal{L}^{\prime}(m),m=1,\cdots,M$, such that $L_{m+1}$ intersects $L_{m}$ for every $0\leq m<M$, where $L_{0}$ denote the origin.

Since $\omega_{1},\cdots,\omega_{d}$ satisfy the consistency condition, we can choose $x_{k,m}$’s such that for each $m\geq 0$,

[TABLE]

and since $\lim_{t\rightarrow 0^{+}}\omega(t)=0$, by passing to a subsequence, we may assume that

[TABLE]

We choose $A_{m}=2S_{r(m)}/\left(\omega\left(\frac{1}{X_{r(m),m}}\right)\right)^{\frac{1}{2}}$. Then $\sum_{m\geq 0}\frac{1}{A_{m}}<1$. Now the previous Lemma 3.1 provides us a desired sequence of lines $L_{m},m\in\{0,\cdots,M\}$ for each $M\in\mathbb{N}$. Then the sequence of lines $L_{m}$ induces a finite paths of points $(x_{1}(n),\cdots,x_{d}(n))_{n\geq 0}$ starting at the origin satisfying

[TABLE]

Moreover, if we denote by $\ell(M)$ the length of the path, then

[TABLE]

where $\xi(n)$ is the unique index in $\{1,\cdots,d\}$ such that $|x_{\xi(n)}(n+1)-x_{\xi(n)}(n)|=1$, and $A$ is a constant independent of $M$.

Now we can prove the general case in a similar way as in the case $d=2$. Let $I$ be a maximal open wandering interval, and define

[TABLE]

Let $C:=\max\{C_{1},\cdots,C_{d},C_{1}^{\prime},\cdots,C_{d}^{\prime}\}$ with $C_{k}$ (resp. $C_{k}^{\prime}$) being the $\omega_{k}$-constant of $\log(Df_{k})$ (resp. $\log(Df_{k}^{-1})$) and take $C^{\prime}$ such that

[TABLE]

Now set $L:=\frac{1}{2\exp(2CS)}$. Let $V$ be the $L$-neighborhood of $I$. Let $\Gamma$ be the semigroup generated by $f_{1},\cdots,f_{d},f_{1}^{-1},\cdots,f_{d}^{-1}$.

By the same reason as in the proof of the case $d=2$, there exists $N\in\mathbb{N}$ such that for any $g\in\Gamma$, there exist $r_{1},\cdots,r_{d}\in\{1,\cdots,N\}$ such that $f_{1}^{r_{1}}(g(I)),\cdots,f_{d}^{r_{d}}(g(I))$ are all contained in the $L$-neighborhood of $I$.

Now choose $M\in\mathbb{N}$ large enough such that the number of points with integer coordinates in the $L_{M}$ in the $r(M)$-direction which are all contained in $R_{M}\setminus R_{M-1}$ exceeds N. Then we can complete the proof in the very way as in the case $d=2$.

4. Proof of Corollary 1.3

Proof.

Let $C>0$ be such that $\omega(t_{1}t_{2})\leq C\omega(t_{1})\omega(t_{2}),\forall t_{1},t_{2}\in[0,1]$. For simplicity, we firstly show the case of $d=2$.

By the continuity, it suffices to show that for any neighborhood $V$ of $(0,0)$, there exist $(x,y)\in V$ such that

[TABLE]

and

[TABLE]

Define

[TABLE]

Now let $y=\phi(x)$. Then $\omega_{1}(xy)=\omega_{1}(x)\omega_{2}(y)$. Thus (4.2) is equivalent to

[TABLE]

Since $\omega_{1}(t)\cdots\omega_{d}(t)=t\omega(t)$, (4.3) is equivalent to (1.1). It is easy to see that $\lim_{x\rightarrow 0}\phi(x)=0$. Hence $(x,\phi(x))$ is such that both (4.1) and (4.2) hold.

For the general case, it suffices to show that for any any neighborhood $U$ of the origin, there exist $(x_{1},\cdots,x_{d})\in U$ such that

[TABLE]

Since $\omega_{1},\cdots,\omega_{d}$ are comparable and $\lim_{t\rightarrow 0^{+}}\omega(t)=0$, we may assume that there exists a constant $\delta\in(0,1)$ such that for any $x\in(0,\delta)$,

[TABLE]

We will show that for any $x_{1}\in(0,\delta)$, there exist $x_{2},\cdots,x_{d}\in(0,1)$ such that

[TABLE]

Note that

[TABLE]

Thus there exists a unique $t_{2}\in(0,1)$ such that

[TABLE]

since $\omega_{1}$ is strictly increasing and continuous. By Theorem 1.1, we may assume that $\omega_{1}(x)\geq x^{\frac{1}{2}},\forall x\in(0,\delta)$. Then we have

[TABLE]

Hence $t_{2}\leq x_{1}$. Then

[TABLE]

Thus, since $\omega_{2}$ is strictly increasing and continuous, there exists a unique $x_{2}\in[t_{2},1)$ such that

[TABLE]

In order to make the process be continued, we have to make subtler estimation. By Theorem 1.1, we may assume that $\omega_{k}(x)\geq x^{\tau_{k}},k\in\{1,\cdots,d\}$, with $1>\tau_{d}\geq\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{d-1}\geq 0$ and $\tau_{1}+\cdots+\tau_{d}\leq 1$. Provided that we have found $x_{1},x_{2},\cdots,x_{j}$ and $t_{2},\cdots,t_{j}$, for $2\leq j\leq d-2$ such that $t_{k}=x_{k}t_{k-1}$ and

[TABLE]

Next we need to show there exists a unique $t_{j+1}\in(t_{j},1)$ such that

[TABLE]

It can be guaranteed by

[TABLE]

So it suffices to show that

[TABLE]

Since $x_{j}=\frac{t_{j-1}}{t_{j}}$, (4.9) is equivalent to show that $x_{j}\geq\frac{t_{j-1}}{(x_{1}t_{2})^{\tau_{j+1}}}$ which can be implied by

[TABLE]

So it suffices to show that

[TABLE]

Therefore, the existence of $x_{j+1}$ and $t_{j+1}$ can be guaranteed by

[TABLE]

That is

[TABLE]

Note that

[TABLE]

Thus

[TABLE]

Since $\tau_{2}+\cdots\tau_{j+1}\leq\tau_{2}+\cdots+\tau_{d-1}\leq 1-\tau_{1}-\tau_{d}\leq 1-2\tau_{1}$, we have

[TABLE]

Then combining (4.10) and (4.12), we obtain (4.9). Thus we have showed (4.4). Since

[TABLE]

we have

[TABLE]

Note that for $j\in\{1,\cdots,d-1\}$,

[TABLE]

We have $\lim_{x_{1}\rightarrow 0^{+}}x_{j}=0$. By (4.5) and $\omega_{1}(x)\cdots\omega_{d}(x)=x\omega(x)$, we have $\omega_{d}(x)\leq x^{\frac{1}{d}}$ for $x$ small enough. Thus if

[TABLE]

then we still have

[TABLE]

Since $\lim_{x_{1}\rightarrow 0^{+}}(x_{1}\cdots x_{d-1})^{\frac{1}{d}}=0$, we can choose $x_{d}$ with $\lim_{x_{1}\rightarrow 0^{+}}x_{d}=0$ for which we have (4.13) hold and

[TABLE]

Hence $\omega_{1},\cdots,\omega_{d}$ satisfy the consistency condition. Therefore, we complete the proof.

∎

Acknowledgements

The work is supported by NSFC (No. 11771318, No. 11790274).