© Wuhan University 2025

0 Introduction

For monotone twist maps on the annulus or two-dimensional cylinder, one important study object is the order-preserving orbits which are special orbits with rotation numbers. These orbits are usually called Birkhoff orbits^[1-4].

There are many results related to the existence of Birkhoff orbits for twist maps in the past few decades. Hall's result^[3] stated that if there is a $(p,q)$-periodic orbit, then there exists a Birkhoff $(p,q)$-periodic orbit. Furthermore, the Aubry-Mather theorem^[4-6] tells us that each $\omega$ in the twist interval of an area-preserving twist diffeomorphism can be realized by a Birkhoff orbit.

The main topic of this paper is to show the existence of Birkhoff orbits for twist homeomorphisms on the high-dimensional cylinder. For this purpose, we recall the definition of monotone recurrence relations of type $(k,l)$^[1] as follows:

$\mathrm{\Delta }(x_{n-k},\cdots ,x_n,\cdots ,x_{n+l})=\mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{Z},$(1)

where $k,l\in \mathbb{N}$ and $\mathrm{\Delta }\in C^{\mathrm{0}}({\mathbb{R}}^{k+l+\mathrm{1}})$ satisfies

(H1) $\mathrm{\Delta }(x_{-k},\cdots ,x_{\mathrm{0}},\cdots ,x_l)$ is a nondecreasing function of all the $x_j$ except $x_{\mathrm{0}}$. Moreover, it is strictly increasing in $x_{-k}$ and $x_l$,

(H2) $\mathrm{\Delta }(x_{-k}+\mathrm{1},\cdots ,x_l+\mathrm{1})=\mathrm{\Delta }(x_{-k},\cdots ,x_l)$,

(H3) $\underset{x_{-k}\to \pm \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{\Delta }(x_{-k},\cdots ,x_l)=\pm \mathrm{\infty }$ and $\underset{x_l\to \pm \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{\Delta }(x_{-k},\cdots ,x_l)=\pm \mathrm{\infty }.$

We say $\mathit{x}=(x_n)\in {\mathbb{R}}^{\mathbb{Z}}$ is a solution of (1) if it satisfies (1). Solutions of (1) will generate a dynamical system on the high-dimensional cylinder^[1]. In fact, monotone recurrence relation (1) can determine by (H1) and (H3) a homeomorphism $F_{\mathrm{\Delta }}:{\mathbb{R}}^{k+l}\to {\mathbb{R}}^{k+l}$ in the following way: $\mathit{x}=(x_n)$ is a solution of (1) if and only if

$F_{\mathrm{\Delta }}(x_{n-k},\cdots ,x_{n+l-\mathrm{1}})=(x_{n-k+\mathrm{1}},\cdots ,x_{n+l}),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbf{\mathbb{Z}}.$

Taking (H2) into account, (1) defines a homeomorphism $f_{\mathrm{\Delta }}:S^{\mathrm{1}}\times {\mathbb{R}}^{k+l-\mathrm{1}}\to S^{\mathrm{1}}\times {\mathbb{R}}^{k+l-\mathrm{1}}$. Therefore, we obtain the twist homeomorphism on the high-dimensional cylinder, a generalization of the classical monotone twist map on the two-dimensional cylinder. Because of the one-to-one correspondence between solutions of (1) and orbits of $F_{\mathrm{\Delta }}$, we may study orbits of $F_{\mathrm{\Delta }}$ according to solutions of (1).

For a subclass of monotone recurrence relations, we remark that the existence of Birkhoff solutions (see Definition 1) of (1), corresponding to the existence of Birkhoff orbits of $F_{\mathrm{\Delta }}$, has been very clear. We give a brief exposition of this kind of monotone recurrence relations. Let $r\in \mathbb{N}$ be a positive integer indicating the range of interactions between particles and $h:{\mathbb{R}}^{r+\mathrm{1}}\to \mathbb{R}$ be a $C^{\mathrm{2}}$ function satisfying the following conditions^[7-10]:

(C1) $h({\xi }_{\mathrm{1}}+\mathrm{1},\cdots ,{\xi }_{r+\mathrm{1}}+\mathrm{1})=h({\xi }_{\mathrm{1}},\cdots ,{\xi }_{r+\mathrm{1}})$;

(C2) $h$ is bounded from below and $h({\xi }_{\mathrm{1}},\cdots ,{\xi }_{r+\mathrm{1}})$$\to +\mathrm{\infty }$ if $|{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}|\to +\mathrm{\infty }$;

(C3) Twist condition:

${\partial }_{\mathrm{1},j}h({\xi }_{\mathrm{1}},\cdots ,{\xi }_{r+\mathrm{1}})\le -\lambda <\mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{2}\le j\le r+\mathrm{1},$

and

${\partial }_{i,j}h({\xi }_{\mathrm{1}},\cdots ,{\xi }_{r+\mathrm{1}})\le \mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}j\ne i.$

Let the Lagrangian $W$ denote the energy of a system of particles, that is,

$W(\mathit{x})=\sum _{i\in \mathbb{Z}}h(x_i,\cdots ,x_{i+r}).$

Then a variational monotone recurrence relation is defined by finding a stationary point of $W$:

$\begin{array}{l}{\partial }_{n}W(\mathit{x})={\partial }_{r+\mathrm{1}}h(x_{n-r},\cdots ,x_n)+\cdots +{\partial }_{\mathrm{1}}h(x_n,\cdots ,x_{n+r})\\ =\mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{Z}.\end{array}$(2)

We denote $\mathrm{\Delta }(x_{n-r},\cdots ,x_n,\cdots ,x_{n+r})=-{\partial }_{n}W(\mathit{x}),$ then $\mathrm{\Delta }\in C^{\mathrm{0}}({\mathbb{R}}^{\mathrm{2}r+\mathrm{1}})$ satisfies (H1)-(H3) due to (C1)-(C3), and (2) is equivalent to

$\mathrm{\Delta }(x_{n-r},\cdots ,x_n,\cdots ,x_{n+r})=\mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{Z}.$(3)

We call (3) the monotone recurrence relation generated by $h$ or the variational monotone recurrence relation. It follows from the Aubry-Mather theory^[11,12] that for each $\omega \in \mathbb{R}$, there exist Birkhoff minimizers (Birkhoff minimal solutions) with rotation number $\omega .$ We emphasize that (3) corresponds to a conservative system. However, there exist a large number of non-conservative systems which could not be generated by such type of $h.$ For instance, we consider the dissipative system:

$\lambda x_{n-\mathrm{1}}-(\mathrm{1}+\lambda )x_n+x_{n+\mathrm{1}}+K\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}\mathrm{\pi }{x}_n=\mathrm{0},\mathrm{f}\mathrm{o}\mathrm{r}n\in \mathbb{Z},$(4)

$\mathrm{i}\mathrm{n}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\lambda \in (\mathrm{0,1})\mathrm{a}\mathrm{n}\mathrm{d}K\in \mathbb{R}.$ It is easy to check that (4) is a monotone recurrence relation satisfying (H1)-(H3), but it could not be generated by any $C^{\mathrm{2}}$ function $h$ satisfying (C1)-(C3) (see Section 4).

For general monotone recurrence relation (1), we can not utilize the Aubry-Mather theory because of the loss of the variational structure. Therefore, it would be interesting to investigate by a new method the existence of Birkhoff orbits for general $f_{\mathrm{\Delta }}$ induced by (1).

We review relevant results for general systems (not necessarily conservative) obtained in low-dimensional cases. Angenent^[1] proved that a circle endomorphism with degree one must have a Birkhoff orbit, and for the two-dimensional case, he^[1] demonstrated that if there exists $a<b$ such that the twist homeomorphism (without the area-preserving assumption) maps $S^{\mathrm{1}}\times [a,b]$ into itself, then the twist homeomorphism has a Birkhoff orbit.

In this paper, we present the following criterion for the existence of Birkhoff orbits, which is a high-dimensional extension of previous results.

Theorem 1   If there exists a compact forward-invariant set $A\subset S^{\mathrm{1}}\times {\mathbb{R}}^{k+l-\mathrm{1}}$ for $f_{\mathrm{\Delta }}$, then $f_{\mathrm{\Delta }}$ has a Birkhoff orbit.

We stress that we are dealing with high-dimensional cases and our approach here is quite different. To obtain Theorem 1, one crucial step is to construct a compact invariant set for $f_{\mathrm{\Delta }}$, which helps us to obtain the orbits with bounded action in "two directions".

1 Preliminaries

We denote ${\mathbb{R}}^{\mathbb{Z}}$ equipped with the product topology by $X$ and $Y=X/<\mathrm{1}>$, where $\mathrm{1}$ denotes the configuration with all components being $\mathrm{1}$. Because of the periodicity hypothesis (H2), we often consider solutions of (1) in $Y$.

Let $\mathit{y}=(y_n)\in Y$ and $\mathit{x}=(x_n)\in X$ be a lift of $\mathit{y}$. We define

$y_i-y_j=x_i-x_j,\mathrm{f}\mathrm{o}\mathrm{r}i,j\in \mathbb{Z},$

which is independent of the lift $\mathit{x}$.

The space $X$ is partially ordered by

1) $\mathit{x}\le {\mathit{x}}^{\text{'}}$ if and only if $x_n\le x_n^{\text{'}}$ for all $n\in \mathbb{Z}$;

2) $\mathit{x}<{\mathit{x}}^{\text{'}}$ if and only if $\mathit{x}\le {\mathit{x}}^{\text{'}}$ and $\mathit{x}\cancel{=}{\mathit{x}}^{\text{'}}$;

3) $\mathit{x}\ll {\mathit{x}}^{\text{'}}$ if and only if $x_n<x_n^{\text{'}}$ for all $n\in \mathbb{Z}$.

We say a configuration $\mathit{x}=(x_n)$ has rotation number $\omega$, if the limit $\underset{n\to \pm \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{x}_n/n$ exists and it equals to $\omega$. We call $\mathit{x}=(x_n)$ a configuration with bounded action^[8] if there is a constant $L>\mathrm{0}$ such that $|x_{n+\mathrm{1}}-x_n|\le L$ for all $n\in \mathbb{Z}$.

We remark that a configuration with bounded action and a configuration with rotation number are two independent concepts. To be precise, on the one hand, a configuration with bounded action is not necessarily a configuration with rotation number. For example, $\mathit{x}=(x_n),x_n=|n\omega |$. On the other hand, a configuration with rotation number is not necessarily a configuration with bounded action. For example, $\mathit{x}=(x_n),x_n=n\omega +{(-\mathrm{1})}^n\sqrt[\mathrm{3}]{n}.$

We define a family of order-preserving homeomorphisms $\{{\tau }_{m,n}:(m,n)\in {\mathbb{Z}}^{\mathrm{2}}\}$ on $X$:

$({\tau }_{m,n}{\mathit{x})}_i=x_{i-m}+n,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$

The action of ${\tau }_{m,n}$ on $\mathit{x}$ can be described as shifting $\mathit{x}$ to the right by $m$ and shifting it up by $n$.

The left-shift operator $\sigma :Y\to Y$ is defined by $\sigma \mathit{y}=({\tau }_{-\mathrm{1,0}}\mathit{x})/<\mathrm{1}>$, where $\mathit{x}\in X$ is a lift of $\mathit{y}\in Y$. If we denote ${S}\subseteq Y$ to be the set of solutions of (1), then the system generated by $\sigma$ on ${S}$ is equivalent to that by $f_{\mathrm{\Delta }}$ on the high-dimensional cylinder.

Let $\mathit{z}\in Y$ be a solution of (1) with bounded action. We consider the closure of the set of translates of $\mathit{z}$:

$\mathrm{\Gamma }:={\overline{\{{\sigma }^m\mathit{z}\}}}_{m\in \mathbb{Z}}.$(5)

Lemma 1   The set $\mathrm{\Gamma }\subseteq Y$ is compact and $\sigma$-invariant.

Proof   It is clear that $\mathrm{\Gamma }$ is $\sigma$-invariant. It suffices to show $\mathrm{\Gamma }\subseteq Y$ is compact. Let ${\mathit{x}}^m=(x_i^m)$ with $x_{\mathrm{0}}^m\in [\mathrm{0,1}]$ be a lift of ${\sigma }^m\mathit{z}$ and $P$ be the projection from $X$ to $Y.$ Then ${\sigma }^m\mathit{z}=P({\mathit{x}}^m)$. Since $\mathit{z}$ is a solution of (1) with bounded action, there exists $L>\mathrm{0}$ independent of $m$ such that

$|x_{i+\mathrm{1}}^m-x_i^m|=|({\sigma }^m{\mathit{z})}_{i+\mathrm{1}}-({\sigma }^m{\mathit{z})}_i|\le L\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$

A simple calculation gives that $x_i^m\in [a_i,b_i]$, where $a_i=-|i|L$ and $b_i=\mathrm{1}+|i|L.$ Thus we have $\{{\mathit{x}}^m{\}}_{m\in \mathbb{Z}}{}_{}$$\subseteq$$\prod _{i\in \mathbf{\mathbb{Z}}}[a_i,b_i]=:\tilde{K}$. It follows from Tychonoff's theorem that $\tilde{K}\subseteq X$ is compact, and therefore $K:=P(\tilde{K})\subseteq Y$ is compact. Moreover,

$\mathrm{\Gamma }={\overline{\{{\sigma }^m\mathit{z}\}}}_{m\in \mathbb{Z}}\subseteq K.$

We come to the conclusion that $\mathrm{\Gamma }\subseteq Y$ is compact.

Definition 1   We say a configuration $\mathit{x}=(x_i)$ is Birkhoff, if for any $m,n\in \mathbb{Z}$, ${\tau }_{m,n}\mathit{x}\le \mathit{x}$, or ${\tau }_{m,n}\mathit{x}\ge \mathit{x}$.

Let $\mathrm{ℬ}$ denote the set of Birkhoff configurations. It is easy to see that $\mathrm{ℬ}$ is closed in the product topology and ${\tau }_{m,n}\mathrm{ℬ}=\mathrm{ℬ}$ for all $m,n\in \mathbb{Z}$. Moreover, Birkhoff configurations possess the following property.

Proposition 1^[1,2,13] If $\mathit{x}=(x_i)$ is Birkhoff, then $\mathit{x}$ has a rotation number $\omega$. Moreover,

$|x_i-x_{\mathrm{0}}-i\omega |\le \mathrm{1},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$

We know from Proposition 1 that each Birkhoff configuration is a configuration both with rotation number and bounded action.

We recall definitions of subsolutions and supersolutions for (1).

Definition 2^[1,13-15] It is said that $\underline{\mathit{x}}=({\underline{x}}_i)$ and $\overline{\mathit{x}}=\left({\overline{x}}_i\right)$ are a subsolution and a supersolution of (1) respectively, if $\mathrm{\Delta }({\underline{x}}_{n-k},\cdots ,{\underline{x}}_{n+l})\ge \mathrm{0},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Delta }({\overline{x}}_{n-k},\cdots ,{\overline{x}}_{n+l})\le \mathrm{0},$ for all $n\in \mathbf{\mathbb{Z}}.$

Lemma 2^[1,13] If $\{{\mathit{x}}^{\alpha }{\}}_{\alpha \in {A}}$ is a family of subsolutions which is bounded from above (w.r.t. the partial ordering on $X$), then $\underset{\alpha \in {A}}{\mathrm{s}\mathrm{u}\mathrm{p}}\{{\mathit{x}}^{\alpha }\}$ defined by

$(\underset{\alpha \in {A}}{\mathrm{s}\mathrm{u}\mathrm{p}}\{{\mathit{x}}^{\alpha }{\})}_i=\underset{\alpha \in {A}}{\mathrm{s}\mathrm{u}\mathrm{p}}\{x_i^{\alpha }\}$

is a subsolution. Analogously, a family of supersolutions $\{{\mathit{x}}^{\alpha }{\}}_{\alpha \in {A}}$ which is bounded from below has an infimum, and $\underset{\alpha \in {A}}{\mathrm{i}\mathrm{n}\mathrm{f}}\{{\mathit{x}}^{\alpha }\}$ is also a supersolution.

Lemma 3^[1,13] Let $\underline{\mathit{x}},\overline{\mathit{x}}\in X$ be a subsolution and a supersolution, respectively, satisfying: $\underline{\mathit{x}}\le \overline{\mathit{x}}$. If $\underline{\mathit{x}}$ or $\overline{\mathit{x}}$ is Birkhoff, then there is a Birkhoff solution $\mathit{x}$ satisfying $\underline{\mathit{x}}\le \mathit{x}\le \overline{\mathit{x}}.$

2 Main Lemmas

In this section, we propose the following theorem, which is a nontrivial generalization of Theorem 5.1 in Ref. [1].

Theorem 2   If (1) has a solution with bounded action, then (1) has a Birkhoff solution.

We should mention that Theorem A in Ref. [13] or Theorem 3.10 in Ref. [14] can also lead to Theorem 2, which relies on a gluing technique. Now we give a direct proof in this section. We would divide the process into two steps. First, we shall use the ergodic theorem to show that (1) has a solution both with bounded action and rotation number provided it has a solution with bounded action (see Lemma 4). Second, we prove further that (1) has a Birkhoff solution with rotation number $\omega$ if and only if it has a solution with bounded action and rotation number $\omega$ (see Lemma 7). It should be pointed out that Angenent's generalization (Theorem 5.1 in Ref. [1]) of Hall's result (Theorem 1 in Ref. [3]) can be derived by Lemma 7.

Lemma 4   If (1) has a solution with bounded action, then (1) has a solution with bounded action and rotation number.

Proof   Assume that $\mathit{z}\in Y$ is a solution of (1) with bounded action and $\mathrm{\Gamma }={\overline{\{{\sigma }^m\mathit{z}\}}}_{m\in \mathbb{Z}}$ as we defined in (5). We see $\mathrm{\Gamma }$ is metrizable and $\sigma :\mathrm{\Gamma }\to \mathrm{\Gamma }$ is a homeomorphism on the compact metric space $\mathrm{\Gamma }$. Hence, there exists a measure $\mu$, which is not only $\sigma$-ergodic, but also ${\sigma }^{-\mathrm{1}}$-ergodic.

We construct $f:\mathrm{\Gamma }\to \mathbb{R},\overline{\mathit{y}}=({\overline{y}}_i)\mapsto {\overline{y}}_{\mathrm{1}}-{\overline{y}}_{\mathrm{0}}$, then $f$ is continuous and thus $\mu$-integrable. Applying the Birkhoff ergodic theorem^[16,17], we can find $\mathit{y}=(y_i)\in \mathrm{\Gamma }$ such that

$\begin{array}{l}\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{y_n-y_{\mathrm{0}}}{n}=\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{1}}{n}{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}f({\sigma }^i\mathit{y})={\int }_{\mathrm{\Gamma }}f\mathrm{d}\mu \\ =\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{1}}{n}{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}f({\sigma }^{-i}\mathit{y})=\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{y_n-y_{\mathrm{0}}}{n}.\end{array}$

Therefore, $\mathit{y}$ is a solution of (1) with bounded action and rotation number ${\int }_{\mathrm{\Gamma }}f\mathrm{d}\mu$.

Next, we shall construct Birkhoff solutions with certain rotation numbers.

Lemma 5   Let $N>\mathrm{0}$. If (1) has a subsolution $\underline{\mathit{x}}=({\underline{x}}_i)$ and a supersolution $\overline{\mathit{x}}=\left({\overline{x}}_i\right)$ satisfying

${\underline{x}}_i\le i\omega +N\mathrm{a}\mathrm{n}\mathrm{d}{\overline{x}}_i\ge i\omega -N\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z},$(6)

then (1) has a Birkhoff solution with rotation number $\omega$.

Proof   We define as in Lemma 2 that $\underline{\mathit{w}}=\mathrm{s}\mathrm{u}\mathrm{p}\{{\tau }_{m,n}\underline{\mathit{x}}-N\cdot \mathrm{1}:n\le m\omega \}$ and $\overline{\mathit{w}}=\mathrm{i}\mathrm{n}\mathrm{f}\{{\tau }_{m,n}\overline{\mathit{x}}+N\cdot \mathrm{1}:n\ge m\omega \}.$ Due to (6), it is evident that $\underline{\mathit{w}}$ and $\overline{\mathit{w}}$ are well-defined and ${\underline{w}}_i\le i\omega \le {\overline{w}}_i\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$ Moreover, it follows from Lemma 2 that $\underline{\mathit{w}}$ is a subsolution of (1) and $\overline{\mathit{w}}$ is a supersolution of (1).

We shall prove that $\underline{\mathit{w}}$ and $\overline{\mathit{w}}$ are Birkhoff configurations. Indeed, let $r,s\in \mathbb{Z}$ be given and consider ${\tau }_{r,s}\underline{\mathit{w}}$.

$\begin{array}{l}{\tau }_{r,s}\underline{\mathit{w}}=\mathrm{s}\mathrm{u}\mathrm{p}\{{\tau }_{m+r,n+s}\underline{\mathit{w}}-N\cdot \mathrm{1}:n\le m\omega \}\\ =\mathrm{s}\mathrm{u}\mathrm{p}\{{\tau }_{m,n}\underline{\mathit{w}}-N\cdot \mathrm{1}:n\le m\omega +s-r\omega \}\\ \le \underline{\mathit{w}},\mathrm{i}\mathrm{f}s\le r\omega ,\\ \ge \underline{\mathit{w}},\mathrm{i}\mathrm{f}s\ge r\omega .\end{array}$

Then $\underline{\mathit{w}}\in \mathrm{ℬ}$, and $\overline{\mathit{w}}\in \mathrm{ℬ}$ can be proved similarly. It is easy to see that they have the same rotation number $\omega$. Applying Lemma 3, there is a Birkhoff solution with rotation number $\omega$.

Lemma 6   If $\mathit{x}=(x_n)\in X$ is a solution with bounded action and

$I_{\mathit{x}}:=\left[\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{x_n}{n},\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{x_n}{n}\right]\bigcap \left[\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{x_n}{n},\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{x_n}{n}\right]$

is nonempty, then for any $\omega \in I_{\mathit{x}}$, there is a Birkhoff solution with rotation number $\omega$.

Proof   Let $\mathit{z}=\mathit{x}/<\mathrm{1}>\in Y$ and $\mathrm{\Gamma }={\overline{\{{\sigma }^m\mathit{z}\}}}_{m\in \mathbb{Z}}$ be the compact set as we defined in (5). Then $z_i-z_j=x_i-x_j$ for $i,j\in \mathbb{Z}$.

Since $\omega \in I_{\mathit{x}}$ yields $\mathrm{m}\mathrm{a}\mathrm{x}\{\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{x_n}{n},\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{x_n}{n}\}\le \omega$

$\le \mathrm{m}\mathrm{i}\mathrm{n}\{\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{x_n}{n},\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{x_n}{n}\},$ we claim that there exists$\underline{\mathit{y}}=({\underline{y}}_i)\in \mathrm{\Gamma }$, such that ${\underline{y}}_i-{\underline{y}}_{\mathrm{0}}-i\omega \le \mathrm{1}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}$.

We argue by contradiction. Assume that for arbitrary $\mathit{y}=(y_i)\in \mathrm{\Gamma }$, there exists $m(\mathit{y})\in \mathbb{Z}$, such that $y_{m(\mathit{y})}-y_{\mathrm{0}}-m(\mathit{y})\omega >\mathrm{1}$.

Let $U_{\mathit{y}}=\{\mathit{u}\in Y:u_{m(\mathit{y})}-u_{\mathrm{0}}-m(\mathit{y})\omega >\mathrm{1}\}\subseteq Y$, then $\{U_{\mathit{y}}{\}}_{\mathit{y}\in \mathrm{\Gamma }}$ is an open cover of $\mathrm{\Gamma }$. Due to the compactness of $\mathrm{\Gamma }$, we can find $U_{{\mathit{y}}^{\mathrm{1}}},U_{{\mathit{y}}^{\mathrm{2}}},\cdots ,U_{{\mathit{y}}^j}$, such that $\mathrm{\Gamma }\subseteq {\displaystyle \underset{i=\mathrm{1}}{\overset{j}{\cup }}}{U}_{{\mathit{y}}^i}.$

For $q>\mathrm{1}$, we have

$z_{n_{\mathrm{1}}}-z_{\mathrm{0}}-n_{\mathrm{1}}\omega >\mathrm{1},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}{n}_{\mathrm{1}}\in \{m({\mathit{y}}^{\mathrm{1}}),\cdots ,m({\mathit{y}}^j)\};$

$({\sigma }^{n_{\mathrm{1}}}{\mathit{z})}_{n_{\mathrm{2}}}-({\sigma }^{n_{\mathrm{1}}}{\mathit{z})}_{\mathrm{0}}-n_{\mathrm{2}}\omega >\mathrm{1},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}{n}_{\mathrm{2}}\in \{m({\mathit{y}}^{\mathrm{1}}),\cdots ,m({\mathit{y}}^j)\};$

$⋮$

$\begin{array}{l}({\sigma }^{n_{\mathrm{1}}+\cdots +n_{q-\mathrm{1}}}{\mathit{z})}_{n_q}-({\sigma }^{n_{\mathrm{1}}+\cdots +n_{q-\mathrm{1}}}{\mathit{z})}_{\mathrm{0}}-n_q\omega >\mathrm{1},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}{n}_q\in \{m({\mathit{y}}^{\mathrm{1}}),\\ \cdots ,m({\mathit{y}}^j)\}.\end{array}$

Thus we have

$z_{n_{\mathrm{1}}}-z_{\mathrm{0}}-n_{\mathrm{1}}\omega >\mathrm{1};$

$z_{n_{\mathrm{1}}+n_{\mathrm{2}}}-z_{n_{\mathrm{1}}}-n_{\mathrm{2}}\omega >\mathrm{1};$

$⋮$

$z_{n_{\mathrm{1}}+\cdots +n_q}-z_{n_{\mathrm{1}}+\cdots +n_{q-\mathrm{1}}}-n_q\omega >\mathrm{1}$

Adding up the above inequalities, we obtain $z_{n_{\mathrm{1}}+\cdots +n_q}-z_{\mathrm{0}}-(n_{\mathrm{1}}+\cdots +n_q)\omega >q.$ Hence,

$x_{k_q}-x_{\mathrm{0}}-k_q\omega >q,$(7)

in which $k_q=n_{\mathrm{1}}+\cdots +n_q,q\ge \mathrm{1}$. We proceed to show that $(k_q{)}_{q\ge \mathrm{1}}$ is unbounded.

Indeed, assuming to the contrary that we can find $\alpha \in \mathbb{N}$ such that $|k_q|\le \alpha ,q\ge \mathrm{1}$. We choose $A=\mathrm{m}\mathrm{a}{\mathrm{x}}_{-\alpha \le i\le \alpha }\{x_i-x_{\mathrm{0}}-i\omega \}$. Since $(k_q{)}_{q\ge \mathrm{1}}$ is a sequence consisting of integers, there is at least one $\beta \in \{-\alpha ,\cdots ,\alpha \}$ that appears infinitely in $(k_q{)}_{q\ge \mathrm{1}}$. Hence there exists$q>A$ satisfying $x_{\beta }-x_{\mathrm{0}}-\beta \omega >q.$ But this is a contradiction to the choice of A.

Since $\{k_q\}$ is unbounded, there must be a subsequence, not relabeled, such that $k_q\to +\mathrm{\infty },q\to +\mathrm{\infty }$ or $k_q\to -\mathrm{\infty },q\to +\mathrm{\infty }$.

If $\underset{q\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{k}_q=+\mathrm{\infty }$, then $k_q>\mathrm{0}$ for $q$ large enough. Hence by (7),

$\frac{x_{k_q}-x_{\mathrm{0}}}{k_q}>\omega +\frac{q}{k_q}\ge \omega +\frac{\mathrm{1}}{M},\mathrm{f}\mathrm{o}\mathrm{r}q\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h},$(8)

where $M=\mathrm{m}\mathrm{a}{\mathrm{x}}_{\mathrm{1}\le i\le j}\{m({\mathit{y}}^i)\}>\mathrm{0}$.

For each $n$ large enough, there exists a large $q$ such that $k_q\le n<k_{q+\mathrm{1}}$ since $\underset{q\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{k}_q=+\mathrm{\infty }$. It is clear that $\mathrm{0}\le n-k_q\le M$. Note that

$\frac{x_n-x_{\mathrm{0}}}{n}=\frac{x_n-x_{k_q}}{n}+\frac{x_{k_q}-x_{\mathrm{0}}}{k_q}\cdot \frac{k_q}{n}.$

Since $\mathit{x}$ is a solution with bounded action, there exists $L>\mathrm{0}$ such that $|x_{i+\mathrm{1}}-x_i|\le L$ for all $i\in \mathbb{Z}.$ As a result, we obtain

$\frac{x_n-x_{\mathrm{0}}}{n}\ge -\frac{ML}{n}+\frac{x_{k_q}-x_{\mathrm{0}}}{k_q}\cdot \frac{n-M}{n}.$

Then $\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}{x}_n/n>\omega$ holds automatically, which is a contradiction to the assumption that $\underset{n\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}{x}_n/n\le \omega$. Similarly, $\underset{q\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{k}_q=-\mathrm{\infty }$ will lead to a contradiction that $\underset{n\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}{x}_n/n<\omega$.

Hence there exists$\underline{\mathit{y}}=({\underline{y}}_i)\in \mathrm{\Gamma }$ such that ${\underline{y}}_i-{\underline{y}}_{\mathrm{0}}-i\omega \le \mathrm{1}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}$. Analogously, there exists $\overline{\mathit{y}}=({\overline{y}}_i)\in \mathrm{\Gamma }$ such that ${\overline{y}}_i-{\overline{y}}_{\mathrm{0}}-i\omega \ge -\mathrm{1}$ for all $i\in \mathbf{\mathbb{Z}}$.

Let $\underset{\_}{\mathit{x}}\in X$ and $\overline{\mathit{x}}\in X$ be a lift of $\underline{\mathit{y}}$ and $\overline{\mathit{y}}$, respectively. Since $\underline{\mathit{y}}$ and $\overline{\mathit{y}}$ are in $\mathrm{\Gamma }$, $\underline{\mathit{x}}$ and $\overline{\mathit{x}}$ are solutions of (1). Moreover,

${\underline{x}}_i-{\underline{x}}_{\mathrm{0}}-i\omega \le \mathrm{1}\mathrm{a}\mathrm{n}\mathrm{d}{\overline{x}}_i-{\overline{x}}_{\mathrm{0}}-i\omega \ge -\mathrm{1},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$

We choose $N=\lfloor |{\underline{x}}_{\mathrm{0}}+\mathrm{1}|\rfloor +\lfloor |\mathrm{1}-{\overline{x}}_{\mathrm{0}}|\rfloor +\mathrm{1}.$ Then

${\underline{x}}_i\le i\omega +N\mathrm{a}\mathrm{n}\mathrm{d}{\overline{x}}_i\ge i\omega -N,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i\in \mathbb{Z}.$

It follows immediately from Lemma 5 that there is a Birkhoff solution with rotation number $\omega$.

We are now in a position to prove Lemma 7.

Lemma 7   Equation (1) has a Birkhoff solution with rotation number $\omega$ if and only if (1) has a solution with bounded action and rotation number $\omega$.

Proof   Assume (1) has a Birkhoff solution with rotation number $\omega$. Then it is a solution with bounded action as we mentioned in Section 1.

For another direction, since $\mathit{x}$ is a solution of (1) with bounded action and rotation number $\omega$, we see $I_{\mathit{x}}$ defined in Lemma 6 is the singleton $\{\omega \}$. We obtain by Lemma 6 that there must be a Birkhoff solution with rotation number $\omega$.

Combining Lemma 4 and Lemma 7 can acquire Theorem 2.

3 Proof of Theorem 1

Let $F_{\mathrm{\Delta }}$ and $f_{\mathrm{\Delta }}$ be homeomorphisms defined in Introduction. The aim of this section is to show the proof of Theorem 1.

Proof  of Theorem 1:

Since $A$ is a compact forward-invariant set for $f_{\mathrm{\Delta }}$, one has

$f_{\mathrm{\Delta }}^{n+\mathrm{1}}(A)\subseteq f_{\mathrm{\Delta }}^n(A),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{N}.$(9)

Then $\{f_{\mathrm{\Delta }}^n(A)\}$ has finite intersection property and hence $B:={\displaystyle \underset{n=\mathrm{1}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^n(A)$$\subseteq A$ is a nonempty compact set. On the one hand, we have

$f_{\mathrm{\Delta }}(B)={\displaystyle \underset{n=\mathrm{1}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^{n+\mathrm{1}}(A)={\displaystyle \underset{n=\mathrm{2}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^n(A)\supseteq {\displaystyle \underset{n=\mathrm{1}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^n(A)=B.$

On the other hand, we see by (9) that

$f_{\mathrm{\Delta }}(B)={\displaystyle \underset{n=\mathrm{1}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^{n+\mathrm{1}}(A)\subseteq {\displaystyle \underset{n=\mathrm{1}}{\overset{+\mathrm{\infty }}{\cap }}}{f}_{\mathrm{\Delta }}^n(A)=B.$

From above, we derive that $B$ is a compact set invariant for $f_{\mathrm{\Delta }}$. Then for $\mathrm{1}\le i\le k+l-\mathrm{1}$,there exists $a_i<b_i$ such that $p_i(B)\subseteq [a_i,b_i]$ is compact, where$p_i:S^{\mathrm{1}}\times {\mathrm{\Pi }}_{n=\mathrm{1}}^{k+l-\mathrm{1}}[a_n,b_n]\to [a_i,b_i]$ denotes the projection. Therefore by definition, $F_{\mathrm{\Delta }}$ has an orbit $\mathit{x}=(x_n)$ satisfying

$\mathrm{m}\mathrm{a}{\mathrm{x}}_{\mathrm{1}\le i\le k+l-\mathrm{1}}\{a_i\}\le x_n\le \mathrm{m}\mathrm{i}{\mathrm{n}}_{\mathrm{1}\le i\le k+l-\mathrm{1}}\{b_i\},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}n\in \mathbb{Z}.$

In other words, $\mathit{x}$ is a solution of (1) with bounded action. Further by Theorem 2, (1) has a Birkhoff solution, implying $f_{\mathrm{\Delta }}$ has a Birkhoff orbit.

4 Examples

We shall explain by examples the relation between variational monotone recurrence relations and monotone recurrence relations of type $(k,l)$.

On the one hand, we consider the potential function of the classical Frenkel-Kontorova model^[2,18]:

$h({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}})=\frac{\mathrm{1}}{\mathrm{2}}({\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{)}^{\mathrm{2}}+\frac{K}{\mathrm{2}\mathrm{\pi }}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{2}\mathrm{\pi }{\xi }_{\mathrm{1}},\mathrm{i}\mathrm{n}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}K\in \mathbb{R}.$

Then $h$ is a $C^{\mathrm{2}}$ function satisfying (C1)-(C3). Let $W(\mathit{x})=\sum _{i\in \mathbb{Z}}h(x_i,x_{i+\mathrm{1}})$ and

$\begin{array}{l}\mathrm{\Delta }(x_{n-\mathrm{1}},x_n,x_{n+\mathrm{1}})=-{\partial }_{n}W(\mathit{x})=x_{n-\mathrm{1}}-\mathrm{2}{x}_n+x_{n+\mathrm{1}}+K\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}\mathrm{\pi }{x}_n\\ =\mathrm{0},n\in \mathbb{Z}.\end{array}$(10)

It turns out that $\mathrm{\Delta }$ is continuous with (H1)-(H3). Hence, variational monotone recurrence relation (10) is a monotone recurrence relation of type $(\mathrm{1,1})$.

On the other hand, we consider the dissipative system:

$\lambda x_{n-\mathrm{1}}-(\mathrm{1}+\lambda )x_n+x_{n+\mathrm{1}}+K\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}\mathrm{\pi }{x}_n=\mathrm{0},$(11)

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\lambda \in (\mathrm{0,1})\mathrm{a}\mathrm{n}\mathrm{d}n\in \mathbb{Z}.$ It is easily seen that (11) suits the definition of monotone recurrence relations of type $(\mathrm{1,1})$. We claim that (11) could not be generated by any $C^{\mathrm{2}}$ function $h$ satisfying (C1)-(C3). In fact, if there exists $\tilde{h}$, such that (11) is generated by $\tilde{h}$. Let

$m_n={\displaystyle \sum _{j=n-\mathrm{1}}^n}\tilde{h}(x_j,x_{j+\mathrm{1}}),n\in \mathbb{Z}.$

We have

$m_n=-\lambda x_{n-\mathrm{1}}{x}_n+\frac{\mathrm{1}+\lambda }{\mathrm{2}}{x}_n^{\mathrm{2}}-x_{n+\mathrm{1}}{x}_n+\frac{K}{\mathrm{2}\mathrm{\pi }}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{2}\mathrm{\pi }{x}_n+f(x_{n-\mathrm{1}},x_{n+\mathrm{1}}),$(12)

where $f$ is a function of $x_{n-\mathrm{1}}$ and $x_{n+\mathrm{1}}$.

Coefficients of the cross term $a(\lambda )x_{n-\mathrm{1}}{x}_n$ of $\tilde{h}(x_{n-\mathrm{1}},x_n)$ and the cross term $\tilde{a}(\lambda )x_n{x}_{n+\mathrm{1}}$ of $\tilde{h}(x_n,x_{n+\mathrm{1}})$ are equal, that is, $a(\lambda )=\tilde{a}(\lambda ).$ We obtain by (12) that $-\lambda =a(\lambda )=\tilde{a}(\lambda )=-\mathrm{1},$ which yields $\lambda =\mathrm{1}$, a contradiction to $\lambda \in (\mathrm{0,1})$.

References