1 Introduction
1.1 Background and motivation
The complexity of dynamical systems, a core focus in dynamical systems theory, is commonly assessed through mathematical indices such as Lyapunov exponents (measuring sensitivity to initial conditions), entropy (quantifying unpredictability), and Smale horseshoes (a topological indicator of chaos). In 1965, Adler, Konheim, and McAndrew [Reference Adler, Konheim and McAndrew1] introduced the concept of topological entropy using open covers. The widely accepted definition of topological entropy for systems on metric spaces, provided by Bowen [Reference Bowen2, Reference Bowen3], uses separating sets and spanning sets to describe the exponential growth rate of the number of separated orbit segments. The distribution properties of periodic orbits play an essential role in studying the complexity of dynamical systems. For Axiom A maps and flows, Bowen [Reference Bowen4, Reference Bowen5] proved the density of periodic orbits and the fact that the exponential, characterizing the growth rate of their number, is equal to the topological entropy. In the context of non-uniformly hyperbolic systems, Katok [Reference Katok7] established that for any given hyperbolic measure, its measure-theoretic entropy is bounded below by the exponential growth rate of the number of periodic points.
Non-autonomous and random dynamical systems often lack periodic orbits due to the absence of recurrence, which creates one of the most significant difficulties in establishing the conclusions of classical dynamical systems in these systems. In 2001, Klünger [Reference Klünger8] introduced the concept of random periodic points. Zhao and Zheng [Reference Zhao and Zheng9] proved the existence of random periodic solutions for a class of stochastic differential equations. Recently, Huang, Lian, and Lu [Reference Huang, Lian and Lu6] proved the existence and the density of random periodic orbits in Anosov systems driven by a quasi-periodic forcing. In particular, they pointed out that when the period is sufficiently large, the number of random periodic orbits will become uncountable. Therefore, its exponential growth rate cannot be related to the system’s topological entropy. It is worth noting that these random periodic points are Borel measurable mappings from the sample space (base space) to the phase space (fiber). The work [Reference Huang, Lian and Lu6] further provides an example where the system lacks continuous random periodic points and, hence, no invariant tori generated by them (see [Reference Huang, Lian and Lu6, Lemma 7.2]). However, for an affine Arnold’s cat map driven by a quasi-periodic forcing, they gave some conditions to assure the existence of an invariant torus, which sparked our interest in discovering the relationship between the exponential growth rate of the number of invariant tori and the topological entropy.
1.2 Setting and results
In this paper, we consider the skew-product system driven by an irrational rotation on
${\mathbb {T}}:=\mathbb {R}/ \mathbb {Z}$
.
Let
$R_{\alpha }: {\mathbb {T}}\to {\mathbb {T}}, \omega \mapsto \omega +\alpha \ \mod 1$
,
$\alpha \in \mathbb {R}\setminus \mathbb {Q}$
be the irrational rotation on
${\mathbb {T}}$
, and
$C({\mathbb {T}},{\mathbb {T}}^2)$
be the space of continuous mappings from
${\mathbb {T}}$
to
${\mathbb {T}}^2$
. We consider the system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
as follows:
$$ \begin{align} \begin{aligned} \varphi: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega, x)\mapsto (R_{\alpha}(\omega),f_{\omega}(x)), \end{aligned} \end{align} $$
where
$f: {\mathbb {T}}\to C({\mathbb {T}}^2, {\mathbb {T}}^2)$
and
$f_{\omega }:=f(\omega )$
for convenience.
In this paper, we consider the following type of invariant structures of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
.
Definition 1.1. (
$\varphi ^n$
-invariant torus)
For the system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
as in (1.1), if there are
$m, n\in {\mathbb {N}}$
and a continuous map
$g: \mathbb {R}\to {\mathbb {T}}^2$
such that for any
$\omega \in \mathbb {R}$
, one has:
-
(1)
$g(\omega )=g(\omega + m)$
; -
(2)
$ \varphi ^n(\omega \ \mod 1, g(\omega ) )= (\omega +n\alpha \ \mod 1, g(\omega +n\alpha ))$
; -
(3)
$\{g(\omega ),\ldots , g(\omega +m-1)\}$
are pairwise distinct,
the graph of a multi-valued map
$$ \begin{align*}g_{\mathcal{T}}: {\mathbb{T}}\to {\mathbb{T}}^2, \omega\mapsto \{g(\omega+i): i\in \{0,\ldots, m-1\}\}\end{align*} $$
is called a
${\varphi }^{{n}}$
-invariant torus of degree
${m}$
and denoted by
$$ \begin{align*}\mathcal{T}:=\{(\omega,g_{\mathcal{T}}(\omega)): \text{for all } \omega\in {\mathbb{T}}\}.\end{align*} $$
Throughout this paper, the constant
$m\in {\mathbb {N}}$
is called the degree of
$\mathcal {T}$
and denoted by
$\mathrm {deg}({\mathcal {T}})$
. We denote by
$\mathcal {G}(\varphi )$
the collection of invariant tori of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
, and
$\mathcal {G}(\varphi; n,m)$
the collection of all
$\varphi ^k(k\leq n)$
-invariant tori of degree m.
In this paper, we consider affine Anosov mappings with a quasi-periodic forcing:
$$ \begin{align} \begin{aligned} \varphi: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega, x)\mapsto (R_{\alpha}(\omega), Ax+ h(\omega)), \end{aligned} \end{align} $$
where
$h(\omega )= (\kern -2pt\begin {smallmatrix} h_1(\omega )\\ h_2(\omega )\end {smallmatrix}\kern -2pt) \in C({\mathbb {T}}, {\mathbb {T}}^2)$
, the matrix
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic, and GL
$(2, \mathbb {Z})$
is the general linear group of order 2 over
$\mathbb {Z}$
.
Note that there are two kinds of degrees in this paper, the first is the degree of the invariant torus we defined in Definition 1.1; and the second is the degree of a continuous mapping
$f:{\mathbb {T}}\to {\mathbb {T}}$
, which is denoted by
$\mathrm {deg}(f)$
. Additionally,
$I-A$
is invertible as
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic. Throughout this paper, we use
$\sharp $
to represent the cardinality of the corresponding set.
Theorem 1.2. For
$\varphi : {\mathbb {T}}\times {\mathbb {T}}^2 \to {\mathbb {T}}\times {\mathbb {T}}^2 $
as in (1.2), let
$m\in {\mathbb {N}}$
be the smallest positive integer satisfying
$$ \begin{align*}m\cdot (I-A)^{-1}\begin{pmatrix} \mathrm{deg}(h_1)\\ \mathrm{deg}(h_2)\end{pmatrix}\in \mathbb{Z}^2.\end{align*} $$
Then, one has:
-
(1)
$\mathcal {G}(\varphi )=\bigcup _{n\in {\mathbb {N}}}\mathcal {G}(\varphi ;n,m)$
; -
(2)
$\sharp \mathcal {G}(\varphi ;n,m)<+\infty $
for all
$n\in {\mathbb {N}}$
; -
(3)
$ \lim _{n\rightarrow +\infty }(1/n) \log \sharp \mathcal {G}(\varphi; n,m)=h_{\mathrm {top}}(\varphi ).$
To get the above theorem, we first show the existence of a
$\varphi $
-invariant torus in §2. Then, we prove Theorem 1.2 in §4 through some properties of the direct product system in §3.
2 Existence of a
$\varphi $
-invariant torus
In this section, we show that there is a
$\varphi $
-invariant torus of a particular degree.
Proposition 2.1. For system
$( {\mathbb {T}}\times {\mathbb {T}}^2, {\varphi })$
as in (1.2), let m be the smallest positive integer such that
$$ \begin{align} m\cdot (I-A)^{-1}\left(\begin{matrix}\mathrm{deg}({h}_1)\\\mathrm{deg}({h}_2)\end{matrix} \right)\in\mathbb{Z}^2. \end{align} $$
Then, there exists a
${\varphi }$
-invariant torus
$\mathcal {T}$
of
$\mathrm {deg}(\mathcal {T})=m$
.
At first, we give a relationship between random periodic points and invariant tori. Recall the definition of random periodic points as in [Reference Huang, Lian and Lu6]. For a given system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
as in (1.1), the set
$\{\mathrm {graph}(g_i)| g_i\in L^{\infty }({\mathbb {T}},{\mathbb {T}}^2)\}_{0\leq i\leq n-1}$
is called a random periodic orbit of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
of period n if
$$ \begin{align*} \varphi(\mathrm{graph}(g_i))=\mathrm{graph}(g_{i+1\ \mod n})\quad\text{for all } 0\leq i\leq n-1,\end{align*} $$
and each
$g_i$
is called a random periodic point of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
of period n. Furthermore, if
$g_i$
is a continuous map, then
$g_i:{\mathbb {T}}\to {\mathbb {T}}^2$
is called a continuous random periodic point.
For system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
as in (1.1), we call a system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _m)$
for some
$m\in {\mathbb {N}}$
, which is
$$ \begin{align*} \begin{aligned} \varphi_{m}: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega', x)\mapsto (R_{\alpha/m}(\omega'), f_{m\omega'}(x)), \end{aligned} \end{align*} $$
the induced system of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
.
Lemma 2.2. For a system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
, as in (1.1), and
$m,n\in {\mathbb {N}}$
, there exists a
$\varphi ^n$
-invariant torus
$\mathcal {T}$
of degree m if and only if the following conditions hold:
-
(1) there is a continuous random periodic point
$\hat {g}\in C({\mathbb {T}},{\mathbb {T}}^2)$
of period n of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _m)$
; -
(2) and
$\hat {g}$
is not periodic of period
${1}/{N}$
for any
$N\in {\mathbb {N}}\cap [2, +\infty )$
in the case of
$m\geq 2$
.
Proof. Assume that there is a
$\varphi ^n$
-invariant torus of degree m, that is, there is a continuous map
$g:\mathbb {R}\to {\mathbb {T}}^2$
satisfying conditions (1)–(3) in Definition 1.1. Define a continuous map
$\hat {g}: \mathbb {R}\to {\mathbb {T}}^2$
such that
$\hat {g}({\omega }/{m})=g(\omega )$
for any
$\omega \in \mathbb {R}$
. It is straightforward that
$\hat {g}$
is a continuous periodic map of period
$1$
since
$g\in C(\mathbb {R},{\mathbb {T}}^2)$
is of period m. Moreover, if
$m\geq 2$
, by condition (3) in Definition 1.1,
$\hat {g}$
is not periodic of period
$1/N$
for any
${N\in {\mathbb {N}}\cap [2, +\infty )}$
. Due to condition (3) in Definition 1.1, one has
$$ \begin{align} g(\omega+n\alpha)=f_{(n-1)\omega}\circ \cdots\circ f_{\omega}(g(\omega)). \end{align} $$
Therefore, we have
$$ \begin{align*} \begin{aligned} \varphi_m^n\bigg(\frac{\omega}{m} \ \mod 1, \hat{g}\bigg(\frac{\omega}{m}\bigg)\bigg)&= \bigg(\frac{\omega}{m}+n\frac{\alpha}{m} \ \mod 1, f_{(n-1)m({\omega}/{m})}\circ\cdots\circ f_{m({\omega}/{m})}\bigg(\hat{g}\bigg(\frac{\omega}{m}\bigg)\bigg)\bigg)\\ &= \bigg(\frac{\omega}{m}+n\frac{\alpha}{m} \ \mod 1, f_{(n-1){\omega}}\circ\cdots \circ f_{\omega}({g}({\omega}))\bigg)\\ &\overset{({2.2})}{=}\bigg(\frac{\omega}{m}+n\frac{\alpha}{m} \ \mod 1, {g}(\omega+n\omega)\bigg)\\ &=\bigg(\frac{\omega}{m}+n\frac{\alpha}{m} \ \mod 1, \hat{g}\bigg(\frac{\omega}{m}+n\frac{\alpha}{m}\bigg)\bigg). \end{aligned} \end{align*} $$
Thus,
$\hat {g}:\mathbb {R}\to {\mathbb {T}}^2$
is a continuous map satisfying:
-
(I)
$\hat {g}(\omega ')=\hat {g}(\omega '+1)$
for all
$\omega '\in \mathbb {R}$
; -
(II)
$\varphi _{m}^n(\omega ' \ \mod 1, \hat {g}(\omega '))=(\omega '+n({\alpha }/{m}) \ \mod 1, \hat {g}(\omega '+n({\alpha }/{m})))$
; -
(III)
$\hat {g}$
is not periodic of period
$1/N$
for any
$N\in {\mathbb {N}}\cap [2, +\infty )$
in the case of
$m\geq 2$
.
Then, we have a continuous random periodic point
$\hat {g}\in C({\mathbb {T}},{\mathbb {T}}^2)$
of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _m)$
, the lift of which is
$\hat {g}\in C(\mathbb {R},{\mathbb {T}}^2)$
.
For given
$m, n\in {\mathbb {N}}$
, assume that there is a continuous random periodic point of period n of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _m)$
, which is not periodic of period
$1/N$
for any
$N\in {\mathbb {N}}\cap [2, +\infty )$
in the case of
$m\geq 2$
. That is, there is a
$\hat {g}\in C(\mathbb {R},{\mathbb {T}}^2)$
satisfying conditions (I)–(III) as above. Let
$g(\omega )=\hat {g}({\omega }/{m})$
for any
$\omega \in \mathbb {R}$
. It is straightforward that
$g\in C(\mathbb {R}, {\mathbb {T}}^2)$
satisfies conditions (1)–(3) in Definition 1.1. Then, we get a
$\varphi ^n$
-invariant torus
$\mathcal {T}$
.
2.1 Existence of a
${\varphi }$
-invariant torus in the case where h is
$C^2$
In this section, we study a special case of (1.2), namely the system
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi })$
, defined as follows:
$$ \begin{align} \begin{aligned} \tilde{\varphi}: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega, x)\mapsto (R_{\alpha}(\omega), Ax+ \tilde{h}(\omega)), \end{aligned} \end{align} $$
where
$\tilde {h}(\omega )= (\kern -2pt\begin {smallmatrix} \tilde {h}_1(\omega )\\ \tilde {h}_2(\omega )\end {smallmatrix}\kern -2pt) \in C^2({\mathbb {T}}, {\mathbb {T}}^2)$
, the matrix
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic.
The key distinction between systems (2.3) and (1.2) lies in the regularity of the pertubation term
$\tilde {h}$
: in (2.3),
$\tilde {h}$
is of class
$C^2$
, whereas in (1.2), it is only assumed to be continuous, that is,
$h\in C({\mathbb {T}},{\mathbb {T}}^2)$
. The following lemma proves Proposition 2.1 for the case when
$\tilde {h}$
is
$C^2$
. Here, the
$C^2$
regularity assumption on
$h_1$
and
$h_2$
ensures that the Fourier series converges absolutely and uniformly to a continuous function, as detailed in the proof of Claim 2.4.
Lemma 2.3. For a system
$( {\mathbb {T}}\times {\mathbb {T}}^2, \tilde {\varphi })$
as in (2.3), let m be the smallest positive integer such that
$$ \begin{align} m\cdot (I-A)^{-1}\left(\begin{matrix}\mathrm{deg}(\tilde{h}_1)\\\mathrm{deg}(\tilde{h}_2)\end{matrix} \right)\in\mathbb{Z}^2. \end{align} $$
Then, there exists a
$\tilde {\varphi }$
-invariant torus
$\mathcal {T}$
of
$\mathrm {deg}(\mathcal {T})=m$
.
Proof of Lemma 2.3
To find an invariant torus, we need to find
$m, n\in {\mathbb {N}}$
and a periodic continuous map
$g: \mathbb {R}\to {\mathbb {T}}^2$
satisfying conditions (1)–(3) in Definition 1.1. By Lemma 2.2, to get an invariant torus of
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi })$
, it is equivalent to find
$m,n\in {\mathbb {N}}$
and a continuous mapping
$\tilde {g}:\mathbb {R}\to {\mathbb {T}}^2$
satisfying:
-
(I)
$\tilde {g}(\omega ')=\tilde {g}(\omega '+1)$
for all
$\omega '\in \mathbb {R}$
; -
(II)
$\tilde {\varphi }_{m}^n(\omega ' \ \mod 1, \tilde {g}(\omega '))=(\omega '+n({\alpha }/{m}) \ \mod 1, \tilde {g}(\omega '+n({\alpha }/{m})))$
; -
(III)
$\tilde {g}$
is not periodic of period
$1/N$
for any
$N\in {\mathbb {N}}\cap [2, +\infty )$
in the case of
$m\geq 2$
.
Additionally, due to condition (II) above, we have
$$ \begin{align} \tilde{g}\bigg(\omega' + n\frac{\alpha}{m}\bigg)= A^n \tilde{g}(\omega')+ A^{n-1}\tilde{h}(m\omega' \ \mod 1)+\cdots + \tilde{h}(m\omega'+(n-1)\alpha \ \mod 1). \end{align} $$
First, we show when (2.4) holds, we may have solutions. By comparing the degrees of both sides of (2.5), we have
$$ \begin{align*} \begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\\ \mathrm{deg}(\tilde{g}_2)\end{pmatrix}=A^n\begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\\ \mathrm{deg}(\tilde{g}_2)\end{pmatrix}+ mA^{n-1}\begin{pmatrix} \mathrm{ deg}(\tilde{h}_1)\\ \mathrm{deg}(\tilde{h}_2)\end{pmatrix}+\cdots+ m \begin{pmatrix} \mathrm{deg}(\tilde{h}_1)\\ \mathrm{deg}(\tilde{h}_2)\end{pmatrix}. \end{align*} $$
Due to the hyperbolicity of
$A\in \mathrm {GL}(2, \mathbb {Z})$
, we have
$\det (I-A)\neq 0$
. Therefore,
$$ \begin{align} \begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\\ \mathrm{deg}(\tilde{g}_2)\end{pmatrix}=m \cdot (I-A)^{-1}\begin{pmatrix} \mathrm{deg}(\tilde{h}_1)\\ \mathrm{ deg}(\tilde{h}_2)\end{pmatrix},\end{align} $$
as
$(I-A^n)=(I-A)(I+A+\cdots + A^{n-1})$
. Thus, for a given system
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi })$
, we can only have desired
$\tilde {g}\in C(\mathbb {R},{\mathbb {T}}^2)$
when (2.4) is satisfied.
Second, we show the existence of a continuous random fixed point of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2, \tilde {\varphi }_{m})$
. Let m be the smallest positive integer satisfying (2.4), we get
$\mathrm {deg}(\tilde {g}_1), \mathrm {deg}(\tilde {g}_2)\in \mathbb {Z}$
by (2.6). Let
$$ \begin{align} \begin{cases} \tilde{g}_1(\omega)=\mathrm{deg}(\tilde{g}_1)\omega+\tilde{r}_1(\omega) &\text{with }\mathrm{deg}(\tilde{r}_1)=0;\\ \tilde{g}_2(\omega)=\mathrm{deg}(\tilde{g}_2)\omega+\tilde{r}_2(\omega) &\text{with }\mathrm{deg}(\tilde{r}_2)=0;\\ \tilde{h}_1(\omega)=\mathrm{deg}(\tilde{h}_1)\omega+s_1(\omega) &\text{with }\mathrm{deg}(s_1)=0;\\ \tilde{h}_2(\omega)=\mathrm{deg}(\tilde{h}_2)\omega+s_2(\omega) &\text{with }\mathrm{deg}(s_2)=0. \end{cases} \end{align} $$
According to (2.5) for the case where
$n=1$
, we have
$$ \begin{align*} \begin{aligned} &\begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\omega+\mathrm{deg}(\tilde{g}_1)\dfrac{\alpha}{m}+\tilde{r}_1\bigg(\omega+\dfrac{\alpha}{m}\bigg)\\[8pt] \mathrm{deg}(\tilde{g}_2)\omega+\mathrm{deg}(\tilde{g}_2)\dfrac{\alpha}{m}+\tilde{r}_2\bigg(\omega+\dfrac{\alpha}{m}\bigg) \end{pmatrix}\\&\quad=A\begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\omega+\tilde{r}_1(\omega)\\ \mathrm{deg}(\tilde{g}_2)\omega+\tilde{r}_2(\omega)\end{pmatrix}+\begin{pmatrix}\mathrm{ deg}(\tilde{h}_1)m\omega+s_1(m\omega)\\ \mathrm{deg}(\tilde{h}_2)m\omega+s_2(m\omega)\end{pmatrix}\\&\quad\overset{({2.6})}{=}A\begin{pmatrix} \tilde{r}_1(\omega)\\ \tilde{r}_2(\omega)\end{pmatrix}+(A+I-A)\begin{pmatrix} \mathrm{deg}(\tilde{g}_1)\omega\\ \mathrm{ deg}(\tilde{g}_2)\omega\end{pmatrix}+\begin{pmatrix} s_1(m\omega)\\ s_2(m\omega)\end{pmatrix}. \end{aligned} \end{align*} $$
Thus, one has
$$ \begin{align} \begin{aligned} \begin{pmatrix}\tilde{r}_1\bigg(\omega+\dfrac{\alpha}{m}\bigg)\\[8pt] \tilde{r}_2\bigg(\omega+\dfrac{\alpha}{m}\bigg)\end{pmatrix}&=A\begin{pmatrix} \tilde{r}_1(\omega)\\ \tilde{r}_2(\omega)\end{pmatrix}+\begin{pmatrix} s_1(m\omega)-\mathrm{deg}(\tilde{g}_1)\dfrac{\alpha}{m}\\\\[-10pt] s_2(m\omega)-\mathrm{deg}(\tilde{g}_2)\dfrac{\alpha}{m}\end{pmatrix}. \end{aligned} \end{align} $$
Assume the Fourier series of
$\tilde {r}_1, \tilde {r}_2$
,
$s_1$
, and
$s_2$
is as follows:
$$ \begin{align} \begin{cases} \tilde{r}_1(\omega)=\sum\limits_{k\in \mathbb{Z}}a_k^1e^{2\pi i k\omega}\ \text{almost everywhere (a.e.)}&\text{with }a_k^1=\overline{a_{-k}^1};\\ \tilde{r}_2(\omega)=\sum\limits_{k\in \mathbb{Z}}a_k^2e^{2\pi i k\omega}\ \text{a.e.}&\text{with }a_k^2=\overline{a_{-k}^2};\\ s_1(\omega)=\sum\limits_{k\in \mathbb{Z}}b_k^1e^{2\pi i k\omega}\ \text{a.e.}&\text{with }b_k^1=\overline{b_{-k}^1};\\ s_2(\omega)=\sum\limits_{k\in \mathbb{Z}}b_k^2e^{2\pi i k\omega}\ \text{a.e.}&\text{with }b_k^2=\overline{b_{-k}^2}. \end{cases} \end{align} $$
According to (2.8) and the comparison of the coefficients of
$e^{2\pi i k\omega }$
for
$k\in \mathbb {Z}$
, we have the following three cases. When
$k=0$
, one has
$$ \begin{align} \begin{pmatrix}a_0^1\\ a_0^2\end{pmatrix}=A\begin{pmatrix} a_0^1\\ a_0^2 \end{pmatrix}+\begin{pmatrix} b_0^1-\mathrm{deg}(\tilde{g}_1)\dfrac{\alpha}{m}\\\\[-10pt] b_0^2-\mathrm{ deg}(\tilde{g}_2)\dfrac{\alpha}{m}\end{pmatrix}. \end{align} $$
When
$k\not = 0$
and
$k/m\in \mathbb {Z}$
, one has
$$ \begin{align*} e^{{2\pi ik\alpha}/{m}}\begin{pmatrix}a_k^1\\ a_k^2\end{pmatrix}=A\begin{pmatrix} a_k^1\\ a_k^2 \end{pmatrix}+\begin{pmatrix} b_{{k}/m}^1\\ b_{{k}/m}^2\end{pmatrix}. \end{align*} $$
Due to the hyperbolicity of
$A\in \mathrm {GL}(2, \mathbb {Z})$
, one has
$$ \begin{align*} \det(e^{2\pi i k\alpha/m}I- A) \not= 0,\end{align*} $$
and then
$e^{2\pi i k\alpha /m}I- A$
is invertible. Therefore, we have
$$ \begin{align} \begin{pmatrix}a_k^1\\ a_k^2\end{pmatrix}=(e^{{2\pi ik\alpha}/{m}}I-A)^{-1}\begin{pmatrix} b_{{k}/m}^1\\ b_{{k}/m}^2\end{pmatrix}. \end{align} $$
When
$k\not = 0$
and
$k/m\notin \mathbb {Z}$
, by comparing the coefficients, one has
$$ \begin{align} \begin{pmatrix}a_k^1\\ a_k^2\end{pmatrix}=\begin{pmatrix} 0\\ 0 \end{pmatrix}. \end{align} $$
Thus, according to (2.10), (2.11), and (2.12), one has
$$ \begin{align} \begin{pmatrix}a_k^1\\ a_k^2\end{pmatrix}= \begin{cases} (I-A)^{-1}\begin{pmatrix} b_0^1-\mathrm{deg}(\tilde{g}_1)\frac{\alpha}{m}\\ b_0^2-\mathrm{deg}(\tilde{g}_2)\frac{\alpha}{m}\end{pmatrix} &\text{when }k=0;\\[12pt](e^{2\pi i k\alpha/m}I- A)^{-1}\begin{pmatrix} b_{{k}/m}^1\\ b_{{k}/m}^2\end{pmatrix} &\text{when }k/m\in \mathbb{Z}\setminus\{0\};\\[12pt]\begin{pmatrix} 0\\ 0 \end{pmatrix} &\text{otherwise.} \end{cases} \end{align} $$
Next, we make the following claim.
Claim 2.4. For given
$h_1, h_2\in C^2({\mathbb {T}},{\mathbb {T}})$
,
$\tilde {r}_{1}, \tilde {r}_2$
are continuous periodic functions from
$\mathbb {R}$
to
$\mathbb {R}$
of period 1.
The proof of Claim 2.4 can be found in the proof of [Reference Huang, Lian and Lu6, Lemma 7.3]. For convenience, we give the proof explicitly here. View
$\tilde {h}_1$
and
$\tilde {h}_2$
as periodic functions from
$\mathbb {R}$
to
$\mathbb {R}$
with period
$1$
. Due the compactness and the
$C^2$
smoothness
$\tilde {h}_{\tau }$
,
$\tau =1,2$
, one has the uniform continuity of the first and second derivative of
$\tilde {h}_{\tau }$
,
$\tau =1,2$
. Then, one has
$$ \begin{align*} b_{k}^{\tau}&=\int_{0}^{1}\tilde{h}_{\tau}(\omega)e^{-2\pi i k\omega}\,d\omega\\&=\frac14\bigg(\int_{0}^1 \tilde{h}_{\tau}(\omega)e^{-2\pi i k\omega}d\omega-2\int_{0}^1 \tilde{h}_{\tau}(\omega)e^{-2\pi i k\omega+\pi i}\,d\omega\\&\quad+\int_{0}^1 \tilde{h}_{\tau}(\omega)e^{-2\pi i k\omega+2\pi i}\,d\omega\bigg)\\&=\frac14\bigg(\int_{0}^1 \tilde{h}_{\tau}(\omega)e^{-2\pi i k\omega}\,d\omega-2\int_{0}^1 \tilde{h}_{\tau}\bigg(\omega+\frac{1}{2k}\bigg)e^{-2\pi i k\omega}\,d\omega\\&\quad+\int_{0}^1 \tilde{h}_{\tau}\bigg(\omega+\frac1k\bigg)e^{-2\pi i k\omega}\,d\omega\bigg)\\&=\int_{0}^1 \frac14 \bigg(\tilde{h}_{\tau}(\omega)-2\tilde{h}_{\tau}\bigg(\omega+\dfrac{1}{2k}\bigg)+\tilde{h}_{\tau}\bigg(\omega+\frac1k\bigg)\bigg)e^{-2\pi i k\omega}\,d\omega\\&=\int_{0}^1\bigg(\frac{1}{16k^2}\tilde{h}"_{\tau}(\omega)+o\bigg(\dfrac{1}{k^2}\bigg)\bigg)e^{-2\pi i k\omega}\,d\omega \sim {O}\bigg(\dfrac{1}{k^2}\bigg). \end{align*} $$
Note that norms of the entries of
$(e^{2\pi i k\alpha /m}I- A)^{-1}$
are uniformly bounded for
$k/m \in \mathbb {Z}\setminus \{0\}$
. Then, the Fourier series of
$\tilde {r}_1$
and
$\tilde {r}_2$
, viewed as 1-periodic functions from
$\mathbb {R}$
to
$\mathbb {R}$
satisfying condition (2.13), converge uniformly. Consequently, their limits are continuous functions. We complete the proof of Claim 2.4.
By Claim 2.4,
$\tilde {r}_1, \tilde {r}_2$
can be viewed as continuous functions from
${\mathbb {T}}$
to
${\mathbb {T}}$
. Then, for the smallest positive integer m satisfying (2.6), we have a desired
$\tilde {g}\in C(\mathbb {R},{\mathbb {T}}^2)$
. Moreover, in the case where
$m\geq 2$
, for any
$m'\in {\mathbb {N}}\cap (0,m)$
, there is no continuous random periodic point of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2, \tilde {\varphi }_{m'})$
. By Lemma 2.2, we have shown the existence of a
$\tilde {\varphi }$
-invariant torus
$\mathcal {T}$
of degree m, that is, Lemma 2.3.
2.2 Proof of Proposition 2.1
To get Proposition 2.1, we consider the perturbed system
$({\mathbb {T}}\times {\mathbb {T}}^2, \tilde {\varphi }_r)$
:
$$ \begin{align} \begin{aligned} \tilde{\varphi}_{r}: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega, x)\mapsto (R_{\alpha}(\omega), Ax+ \tilde{h}(\omega)+r(\omega)), \end{aligned} \end{align} $$
where
$\alpha \in \mathbb {R}\setminus \mathbb {Q}$
,
$A\in GL_{2}(\mathbb {Z})$
is hyperbolic,
$\tilde {h}=(\kern -2pt\begin {smallmatrix} \tilde {h}_1 \\ \tilde {h}_2\end {smallmatrix}\kern -2pt)\in C^2({\mathbb {T}},{\mathbb {T}}^2)$
, and
$r\in C({\mathbb {T}} ,{\mathbb {T}}^2)$
.
Lemma 2.5. Let
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi }_r)$
be given as in (2.14). If
$\|r\|_{C^0}<1$
, there is an
${\eta \in C({\mathbb {T}},{\mathbb {T}}^2)}$
with
$\mathrm { deg}(\eta )=(\kern -2pt\begin {smallmatrix}0\\ 0\end {smallmatrix}\kern -2pt)$
such that
$$ \begin{align} \eta\circ R_{\alpha}(\omega)=A\eta(\omega)+r(\omega)\quad \text{for all } \omega\in {\mathbb{T}}. \end{align} $$
Proof. By lifting to the universal covering, finding the solution
$\eta $
of (2.15) is equivalent to solving
$$ \begin{align} \tilde{r}(\omega)=\tilde{\eta}(\omega+\alpha)-A\tilde{\eta}(\omega), \end{align} $$
where
$\tilde {r}:\mathbb {R}\to \mathbb {R}^2$
is the lift of r, and
$\tilde {\eta }:\mathbb {R}\to \mathbb {R}^2$
is the lift of
$\eta : {\mathbb {T}}\to {\mathbb {T}}^2$
.
Since A is hyperbolic, we have a splitting
$\mathbb {R}^2=E^{u}\oplus E^s$
, where
$E^u$
/
$E^{s}$
is the unstable/stable subspace, and denote by
$\pi ^{u}$
and
$\pi ^s$
the corresponding projections. For
$\tau =u,s$
, we have
$$ \begin{align*}\begin{aligned} \pi^{\tau}\tilde{r}(\omega)&=\pi^{\tau}\tilde{\eta}(\omega+\alpha)-\pi^\tau A\tilde{\eta}(\omega)=\pi^{\tau}\tilde{\eta}(\omega+\alpha)-A|_{E^\tau}\pi^\tau\tilde{\eta}(\omega), \end{aligned} \end{align*} $$
where
$A|_{E^{\tau }}$
,
$\tau =u,s$
represents the operator restricted on the corresponding subspace. For any
$k\in {\mathbb {N}}$
, we have
$$ \begin{align*} (A|_{E^s})^{k-1}\pi^s\tilde{r}(\omega-k\alpha)=(A|_{E^s})^{k-1}\pi^s\tilde{\eta}(\omega-(k-1)\alpha)-(A|_{E^s})^k\pi^s\tilde{\eta}(\omega-k\alpha) \end{align*} $$
and
$$ \begin{align*} -(A|_{E^u})^{-(k+1)}\pi^u\tilde{r}(\omega+k\alpha)=(A|_{E^u})^{-k}\pi^u\tilde{\eta}(\omega+k\alpha)-(A|_{E^u})^{-k-1}\pi^u\tilde{\eta}(\omega+(k+1)\alpha). \end{align*} $$
Then, we have
$$ \begin{align} \begin{aligned} \pi^{s}\tilde{\eta}(\omega)&=\sum_{k=1}^{+\infty}(A|_{E^s})^{k-1}\pi^s\tilde{r}(\omega-k\alpha),\\ \pi^u\tilde{\eta}(\omega)&=-\sum_{k=0}^{+\infty}(A|_{E^u})^{-k-1}\pi^u\tilde{r}(\omega+k\alpha). \end{aligned}\end{align} $$
Let
$\|r\|_{C^0}<1$
. One has
$\mathrm {deg}(r)=(\kern -2pt\begin {smallmatrix} \mathrm {deg}(r_1)\\ \mathrm { deg}(r_2)\end {smallmatrix}\kern -2pt)=(\kern -2pt\begin {smallmatrix}0\\0\end {smallmatrix}\kern -2pt)$
, where
$r=(\kern -2pt\begin {smallmatrix}r_1\\r_2\end {smallmatrix}\kern -2pt)\in C({\mathbb {T}},{\mathbb {T}}^2)$
. Hence,
$\tilde {r}:\mathbb {R}\to \mathbb {R}^2$
is a periodic continuous mapping of period 1, and thus bounded. Then, we have the convergence of
$\pi ^u\tilde {\eta }$
and
$\pi ^s\tilde {\eta }$
in (2.17).
According to (2.15) and
$\mathrm {deg}(r)=(\kern -2pt\begin {smallmatrix}0\\ 0\end {smallmatrix}\kern -2pt)$
, we have
$$ \begin{align*} \begin{pmatrix}0\\0\end{pmatrix}=\mathrm{deg}(r)=(I-A)\mathrm{deg}(\eta). \end{align*} $$
Since
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic, one has
$\det (I-A)\not =0$
. Then, the solution
${\eta \in C({\mathbb {T}},{\mathbb {T}}^2)}$
to (2.15) should be of degree
$\mathrm { deg}(\eta )=(\kern -2pt\begin {smallmatrix}0\\ 0\end {smallmatrix}\kern -2pt)$
. Therefore, we need to prove that
$\tilde {\eta }:=\pi ^u\tilde {\eta }+\pi ^s\tilde {\eta }: \mathbb {R}\to \mathbb {R}^2$
is a periodic continuous mapping of period
$1$
, which will be carried out by contradiction.
Since
$\tilde {r}$
is periodic of period
$1$
, for any
$i\in \mathbb {Z}, \omega \in \mathbb {R}$
, one has
$$ \begin{align*} \tilde{\eta}(\omega+\alpha)-A\tilde{\eta}(\omega)=\tilde{r}(\omega)=\tilde{r}(\omega+i)=\tilde{\eta}(\omega+i+\alpha)-A\tilde{\eta}(\omega+i). \end{align*} $$
Then, for any
$i\in \mathbb {Z}$
, one has
$$ \begin{align*} \tilde{\eta}(\omega+i+\alpha)-\tilde{\eta}(\omega+\alpha)=A(\tilde{\eta}(\omega+i)-\tilde{\eta}(\omega)), \end{align*} $$
and thus for any
$i, n\in {\mathbb {N}}$
, one has
$$ \begin{align} \begin{aligned} \tilde{\eta}(\omega+i+n\alpha)-\tilde{\eta}(\omega+n\alpha)&=A^n(\tilde{\eta}(\omega+i)-\tilde{\eta}(\omega)),\\ \tilde{\eta}(\omega+i-n\alpha)-\tilde{\eta}(\omega-n\alpha)&=A^{-n}(\tilde{\eta}(\omega+i)-\tilde{\eta}(\omega)). \end{aligned} \end{align} $$
Assume there is an
$\omega _0\in \mathbb {R}$
such that
$$ \begin{align*} \tilde{\eta}(\omega_0+1)-\tilde{\eta}(\omega_0)=\begin{pmatrix}x_0\\y_0\end{pmatrix}\not=\begin{pmatrix} 0\\ 0\end{pmatrix}.\end{align*} $$
According to (2.18), one has
$$ \begin{align*} \begin{aligned} \tilde{\eta}(\omega_0+1+n\alpha)-\tilde{\eta}(\omega_0+n\alpha)&=A^n\begin{pmatrix}x_0\\ y_0\end{pmatrix},\\ \tilde{\eta}(\omega_0+1-n\alpha)-\tilde{\eta}(\omega_0-n\alpha)&=A^{-n}\begin{pmatrix} x_0\\ y_0\end{pmatrix}. \end{aligned} \end{align*} $$
Since
$A: \mathbb {R}^2\to \mathbb {R}^2$
is a given hyperbolic operator and
$\tilde {\eta }$
is bounded due to (2.17), the equations stated above cannot hold simultaneously, which is a contradiction.
In summary, we have shown that
$\tilde {\eta }\in C(\mathbb {R},\mathbb {R}^2)$
is a bounded periodic mapping of period 1. Then, we get an
$\eta : {\mathbb {T}}\to {\mathbb {T}}^2$
, the lift of which is
$\tilde {\eta }$
. Thus, we get the desired
$\eta \in C( {\mathbb {T}}, {\mathbb {T}}^2)$
satisfying
$$ \begin{align*}\eta\circ R_{\alpha}(\omega)=A\eta(\omega)+r(\omega)\quad\text{and}\quad\mathrm{deg}(\eta)=\begin{pmatrix} 0\\ 0\end{pmatrix}\!,\end{align*} $$
which completes the proof of Lemma 2.5.
Now, we show Proposition 2.1 by combining Lemmas 2.3 and 2.5.
Proof of Proposition 2.1
As
$C^2({\mathbb {T}},{\mathbb {T}}^2)$
is
$C^{0}$
-dense in
$C({\mathbb {T}},{\mathbb {T}}^2)$
, for any
$h\in C({\mathbb {T}},{\mathbb {T}}^2)$
, there is an
$\tilde {h}=(\kern -2pt\begin {smallmatrix} \tilde {h}_1\\ \tilde {h}_2\end {smallmatrix}\kern -2pt)\in C^2({\mathbb {T}},{\mathbb {T}}^2)$
such that
$\|h-\tilde {h}\|_{C^0}<1$
. Let
$r=h-\tilde {h}\in C({\mathbb {T}},{\mathbb {T}}^2)$
. The system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
in (1.2) is the same as the system
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi }_r)$
in (2.14), and
$\mathrm {deg}(r)=(\kern -2pt\begin {smallmatrix} \mathrm {deg}(h_1-\tilde {h}_1)\\ \mathrm { deg}(h_2-\tilde {h}_2)\end {smallmatrix}\kern -2pt)=(\kern -2pt\begin {smallmatrix}0\\ 0\end {smallmatrix}\kern -2pt)$
.
According to Lemma 2.3, there is an
$m\in {\mathbb {N}}$
and a continuous mapping
$\hat {g}: \mathbb {R}\to {\mathbb {T}}^2$
satisfying conditions (1)–(3) of Definition 1.1 for the system
$({\mathbb {T}}\times {\mathbb {T}}^2,\tilde {\varphi })$
, where m is the smallest positive integer satisfying
$$ \begin{align*}m\cdot (I-A)^{-1}\begin{pmatrix}\mathrm{deg}(\tilde{h}_1)\\ \mathrm{deg}(\tilde{h}_2) \end{pmatrix}=m\cdot (I-A)^{-1}\begin{pmatrix}\mathrm{deg}({h}_1)\\ \mathrm{deg}({h}_2) \end{pmatrix}\in\mathbb{Z}^2.\end{align*} $$
By Lemma 2.5, there is an
$\eta \in C({\mathbb {T}},{\mathbb {T}}^2)$
with
$\mathrm {deg}(\eta )=(\kern -2pt\begin {smallmatrix}0\\0\end {smallmatrix}\kern -2pt)$
. Then, the lifts of
$r, \eta \in C({\mathbb {T}},{\mathbb {T}}^2)$
can be seen as continuous maps from
$\mathbb {R}$
to
${\mathbb {T}}^2$
. That is to say, there are
$\tilde {r}, \tilde {\eta }\in C(\mathbb {R},{\mathbb {T}}^2)$
of period
$1$
satisfying
$$ \begin{align*}\tilde{\eta}({\omega}+\alpha)=A\tilde{\eta}(\omega)+\tilde{r}(\omega)\quad \text{for all } \omega\in \mathbb{R}.\end{align*} $$
Consider the system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
as in (1.2). Conditions (1) and (3) of Definition 1.1 hold for the map
$\hat {g}+\tilde {\eta }\in C(\mathbb {R},{\mathbb {T}}^2)$
. Now, we prove condition (2) of Definition 1.1 still holds for
$\hat {g}+\tilde {\eta }$
. For any
$\omega \in \mathbb {R}$
, one has
$$ \begin{align*} \hat{g}(\omega+\alpha)=A\hat{g}(\omega)+\tilde{h}(\omega \ \mod 1).\end{align*} $$
Then, for any
$\omega \in \mathbb {R}$
, one has
$$ \begin{align*} \begin{aligned} &\varphi(\omega \ \mod 1, \hat{g}(\omega)+\tilde{\eta}(\omega))=\tilde{\varphi}^n_r(\omega \ \mod 1, \hat{g}(\omega)+\tilde{\eta}(\omega))\\ &\quad= (\omega+\alpha \ \mod 1, A\hat{g}(\omega)+A\tilde{\eta}(\omega)+ \tilde{h}(\omega \ \mod 1)+\tilde{r}(\omega \ \mod 1))\\ &\quad=(\omega+\alpha \ \mod 1, \hat{g}(\omega+\alpha)+ \tilde{\eta}(\omega+\alpha)). \end{aligned} \end{align*} $$
Therefore, we have shown the existence of a
$\varphi $
-invariant torus of degree m.
3 Direct product systems
In this section, we consider the following type of systems. We call the system
$$ \begin{align} \begin{aligned} \varphi_0: {\mathbb{T}}\times {\mathbb{T}}^2 &\to {\mathbb{T}}\times {\mathbb{T}}^2, \\ (\omega,x )&\mapsto (R_{\alpha}(\omega), Ax) \end{aligned} \end{align} $$
the direct product system of (1.2).
First, we show that if
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic, then the set of periodic points of
$A: {\mathbb {T}}^2\to {\mathbb {T}}^2$
is one-to-one corresponding to the set of invariant tori of the direct product system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
.
Lemma 3.1. Let
$\mathcal {G}(\varphi _0)$
be the collection of all invariant tori of the direct product system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
as in Definition 1.1, and
$P(A):=\{x\in {\mathbb {T}}^2: A^nx=x \text { for some }n\in {\mathbb {N}}\}$
be the set of all periodic points of A. If
$A\in \mathrm {GL}(2, \mathbb {Z})$
is hyperbolic, then
$$ \begin{align*} \mathcal{G}(\varphi_0)= \{\mathrm{graph}(g)&: \text{there exists } x\in P(A) \text{ such that }g(\omega)= x\ \text{ for all } \omega\in{\mathbb{T}}\}\\&\quad=\{{\mathbb{T}}\times\{x\}: x\in P(A)\}.\end{align*} $$
Proof. The second equality is trivial, so we only prove the first. It is straightforward that
$$ \begin{align*} \{\mathrm{graph}(g): \text{there exists } x\in P(A) \text{ such that }g(\omega)= x\ \text{for all } \omega\in{\mathbb{T}}\} \subseteq \mathcal{G}(\varphi_0).\end{align*} $$
We only need to prove
$$ \begin{align*}\mathcal{G}(\varphi_0)\subseteq \{\mathrm{graph}(g): \text{there exists } x\in P(A),\text{ such that }g(\omega)= x \text{ for all } \omega\in{\mathbb{T}}\},\end{align*} $$
which is by contradiction.
For each
$\varphi _0^{n_0}$
-invariant torus
$\mathcal {T}_{0}:=\{(\omega , g_{\mathcal {T}_{0}}(\omega )): \text { for all } \omega \in {\mathbb {T}}\}$
of degree
$m_0$
, where
$m_0, n_0\in {\mathbb {N}}$
, denote the collection of all images of the multi-valued map
$g_{\mathcal {T}_0}: {\mathbb {T}}\to {\mathbb {T}}^2$
by
$$ \begin{align*}\mathrm{Im}(g_{\mathcal{T}_0}):= \{g_{\mathcal{T}_0}(\omega)\in {\mathbb{T}}^2: \omega\in {\mathbb{T}}\}.\end{align*} $$
Due to Definition 1.1,
$\mathcal {T}_0$
is a
$\varphi _{0}^{n_0}$
-invariant torus of degree
$m_0$
, where
$m_0,n_0\in {\mathbb {N}}$
, there is a continuous map
$g: \mathbb {R}\to {\mathbb {T}}^2$
, such that for any
$\omega \in \mathbb {R}$
, one has:
-
(1)
$g(\omega )=g(\omega +m_0)$
; -
(2)
$\varphi _0^{n_0}(\omega \ \mod 1, g(\omega ))=(\omega +n_0\alpha \ \mod 1, g(\omega +n_0\alpha ))$
; -
(3)
$\{g(\omega ),\ldots , g(\omega +m_0-1)\}$
are pairwise distinct.
Therefore,
$\mathrm {Im}(g_{\mathcal {T}_0})$
is either path connected, which wraps around
${\mathbb {T}}^2$
with
$m_0\geq 1$
times, or
$\mathrm {Im}(g_{\mathcal {T}_0})$
is a singleton, which is equivalent to
$$ \begin{align*}\mathcal{G}(\varphi_0)\subseteq \{\mathrm{graph}(g): \text{there exists } x\in P(A) \text{ such that }g(\omega)= x \text{ for all } \omega\in{\mathbb{T}}\}.\end{align*} $$
Now, we show that the first case will not happen. For given
$m_0, n_0\in \mathbb {Z}_{+}$
and a periodic continuous map
$g: \mathbb {R}\to {\mathbb {T}}^2$
satisfying conditions (1)–(3) as above, by Lemma 2.2, there is a continuous random periodic point of period
$n_0$
of the induced direct product system
$$ \begin{align*} \begin{aligned} \varphi_{m_0,0}: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega', x)\mapsto (R_{\alpha/m_0}(\omega'), Ax). \end{aligned} \end{align*} $$
That is,
$\tilde {g}: \mathbb {R}\to {\mathbb {T}}^2$
is a periodic continuous map with:
-
(I)
$\tilde {g}(\omega ')=\tilde {g}(\omega '+k)$
for all
$\omega '\in \mathbb {R}$
; -
(II)
$\varphi _{m_0, 0}^{n_0}(\omega ' \ \mod 1, \tilde {g}(\omega '))=(\omega '+n_0({\alpha }/{m}) \ \mod 1, \tilde {g}(\omega '+n_0({\alpha }/{m})))$
.
It is clear that
$\mathrm {Im}(\tilde {g})\setminus \{\tilde {g}(0)\}$
is path connected as
${\mathbb {T}}$
is path connected and
$\tilde {g}$
is continuous, where
$\tilde {g}$
is a continuous random periodic point of the induced direct product system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{m_0,0})$
as above. Due to the fact that
$A: {\mathbb {T}}^2\to {\mathbb {T}}^2$
is a hyperbolic linear automorphism, the global stable and unstable manifold
$W^{\tau }(0), \tau =u,s$
of
$0$
are orthogonal and dense in
${\mathbb {T}}^2$
. Suppose that
$\sharp \mathrm {Im}(\tilde {g}) \geq 2$
, where
$\sharp $
denotes the cardinality of the corresponding subset. Then, either
$(\mathrm {Im}(\tilde {g})\setminus \{\tilde {g}(0)\}) \cap W^s(0)\not = \emptyset $
or
$(\mathrm {Im}(\tilde {g})\setminus \{\tilde {g}(0)\}) \cap W^u(0)\not = \emptyset $
.
The proofs for these two cases are similar, so we only give the proof of the first one. Let
$\omega '\in \mathbb {R}$
such that
$g(\omega ')\in (\mathrm {Im}(\tilde {g})\setminus \{\tilde {g}(0)\}) \cap W^s(0)$
. As
$\tilde {g}(\omega ')\in W^s(0)$
, we have
$$ \begin{align*} \tilde{g}\bigg(\omega'+ n \frac{\alpha}{m_0}\bigg)=A^n \tilde{g}(\omega')\to \tilde{g}(0) \quad\text{as } n\to \infty.\end{align*} $$
Due to the fact that
$\alpha \in \mathbb {R}\setminus \mathbb {Q}$
, there is a subsequence
$\{n_i\}_{i\in {\mathbb {N}}}\subseteq {\mathbb {N}}$
such that
$$ \begin{align*}\tilde{g}\bigg(\omega'+n_i\frac{\alpha}{m_0}\bigg)\to \tilde{g}(\omega') \quad\text{as } i\to \infty.\end{align*} $$
Recall the fact
$\tilde {g}(\omega ')\not = \tilde {g}(0)$
. We have a contradiction.
In summary, for the direct product system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
, one has that all invariant tori are the graphs of periodic points of the hyperbolic matrix
$A\in \mathrm {GL}(2, \mathbb {Z})$
.
Second, we show there is a one-to-one correspondence of random periodic points between system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
and its direct product system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
.
Lemma 3.2. For any
$n\in {\mathbb {N}}$
, let
$\mathcal {A}_0^{n}=\{\mathfrak {g}_i\}$
and
$\mathcal {A}^{n}=\{{g}_i\}$
be the set of all random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
of period n, respectively. Then, for any
$g\in \mathcal {A}^n$
and
$n\in {\mathbb {N}}$
, one has
$$ \begin{align*} g+\mathcal{A}^n_{0}:=\{g+\mathfrak{g_i}: \mathfrak{g}_i\in \mathcal{A}^n_0\}=\mathcal{A}^n.\end{align*} $$
Proof. Fix
$n\in {\mathbb {N}}$
, let
$g_1, g_2$
be any two random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
of period n, then
$g_1-g_2$
is a random periodic point of period n of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
. This is a straightforward calculation. For any
$\omega \in {\mathbb {T}}$
,
$i=1,2$
, we have
$$ \begin{align*} \varphi^n(\omega, g_i(\omega))&=(R^n_{\alpha}(\omega), g_i(\omega+n\alpha))\\ &= \bigg(R^n_{\alpha}(\omega), A^ng_i(\omega)+ \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg).\end{align*} $$
Then,
$$ \begin{align*} \varphi^n_0(\omega, (g_1-g_2)(\omega))&=(R^n_{\alpha}(\omega), A^ng_1(\omega) -A^ng_2(\omega))\\ &=\bigg(R^n_{\alpha}(\omega), A^ng_1(\omega)+ \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\\ &\quad -A^ng_2(\omega) - \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg)\\ &= (R^n_{\alpha}(\omega), g_1\circ R^n_{\alpha}(\omega)-g_2\circ R^n_{\alpha}(\omega))\\ &= (R^n_{\alpha}(\omega), (g_1-g_2)\circ R^n_{\alpha}(\omega)). \end{align*} $$
Thus,
$$ \begin{align} \text{ for all } g\in \mathcal{A}^n,\quad \mathcal{A}^n\subseteq g+\mathcal{A}^n_0.\end{align} $$
Similarly, for any
$g\in \mathcal {A}^n$
,
$\mathfrak {g}\in \mathcal {A}^n_0$
, and
$\omega \in {\mathbb {T}}$
, one has
$$ \begin{align*} \varphi^n(\omega,(g+\mathfrak{g})(\omega))&= \bigg(\omega+n\alpha, A^ng(\omega)+A^n\mathfrak{g}(\omega)+ \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg) \\ &= (\omega+n\alpha, g\circ R^n_{\alpha}(\omega)+\mathfrak{g}\circ R^n_{\alpha}(\omega)). \end{align*} $$
Thus,
$$ \begin{align*} g+\mathfrak{g}\in \mathcal{A}^n,\end{align*} $$
which shows that
$$ \begin{align} \text{ for all } g\in \mathcal{A}^n,\quad g+\mathcal{A}^n_0\subseteq \mathcal{A}^n.\end{align} $$
Therefore, by combining (3.2) and (3.3), the lemma is proved.
Remark 3.3. If there is a continuous random fixed point
$g\in C({\mathbb {T}},{\mathbb {T}}^2)$
of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
, then due to Lemmas 3.1 and 3.2, we have that for any
$n\in {\mathbb {N}}$
,
$\mathcal {A}^n\subseteq C({\mathbb {T}},{\mathbb {T}}^2),$
which means all random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
are continuous.
Third, we show the topological entropies of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
and
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _0)$
are equal if there is a continuous random periodic point of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
.
Lemma 3.4. Let
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
be as in (1.2) and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _0)$
be its direct product system. If there is a continuous random periodic point g of period
$n\in {\mathbb {N}}$
of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
, then
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi ^n)$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi ^n_0)$
are topologically conjugate and thus
$h_{\mathrm {top}}(\varphi )=h_{\mathrm {top}}(\varphi _0)$
.
Proof of Lemma 3.4
By assumption, there is a continuous random periodic point g of period
$n\in {\mathbb {N}}$
of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi )$
, that is,
$g\in C({\mathbb {T}},{\mathbb {T}}^2)$
and
$$ \begin{align*}\varphi^n(\omega, g(\omega))= (R^n_{\alpha}(\omega), g\circ R^n_{\alpha}(\omega))=\bigg(R^n_{\alpha}(\omega), A^ng(\omega)+ \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg).\end{align*} $$
Denote
$$ \begin{align*} \begin{aligned} T_{g}: &\,{\mathbb{T}}\times {\mathbb{T}}^2\to {\mathbb{T}}\times {\mathbb{T}}^2 \\ &\,(\omega, x) \mapsto (\omega, x+g(\omega)), \end{aligned} \end{align*} $$
and
$$ \begin{align*} \begin{aligned} \tilde{T}_{g}: &\,{\mathbb{T}}\times {\mathbb{T}}^2\to {\mathbb{T}}\times {\mathbb{T}}^2 \\ &\,(\omega, x) \mapsto (\omega, x-g(\omega)). \end{aligned} \end{align*} $$
It is clear that
$T_g\circ \tilde {T}_g=\tilde {T}_g\circ T_g=\mathrm {id}$
, and
$T_g$
,
$\tilde {T}_g$
are continuous as g is continuous. Thus,
$T_{g}$
is a homeomorphism.
For any
$(\omega , x)\in {\mathbb {T}}\times {\mathbb {T}}^2$
, we have
$$ \begin{align*} \varphi^n \circ T_g(\omega,x)&=\varphi^n(\omega, x+g(\omega))\\ &=\bigg(R^n_{\alpha}(\omega), A^n(x+g(\omega))+ \sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg)\\ &=\bigg(R^n_{\alpha}(\omega), A^n(x)+A^ng(\omega)+\sum_{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha}(\omega)\bigg), \end{align*} $$
and
$$ \begin{align*} T_g\circ \varphi^n_0(\omega,x)=T_g(R^n_{\alpha}(\omega), A^nx) =(R^n_{\alpha}(\omega), A^nx+g\circ R^n_{\alpha}(\omega)). \end{align*} $$
Since,
$g\circ R^n_{\alpha }(\omega )=A^ng(\omega )+\sum _{k=0}^{n-1}A^{n-1-k}h\circ R^k_{\alpha }(\omega )$
, we have
$\varphi \circ T_g= T_g\circ \varphi _0$
, that is,
$$ \begin{align*}\begin{array}{c@{}c@{}c} \mathbb T\times \mathbb T^2 & \stackrel{\varphi_0}{\longrightarrow} & \mathbb T\times \mathbb T^2 \\ \Big\downarrow \rlap{$T_g$} & & \Big\downarrow\vcenter{ \rlap{$T_g$}}\\ \mathbb T\times \mathbb T^2 & \stackrel{\varphi}{\longrightarrow} & \mathbb T\times \mathbb T^2 \end{array}\end{align*} $$
Therefore,
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi ^n)$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi ^n_0)$
are topological conjugate. Thus,
$$ \begin{align*}h_{\mathrm{top}}(\varphi^n_0)=h_{\mathrm{top}}(\varphi^n)\end{align*} $$
as
$h_{\mathrm {top}}(\varphi _0)<+\infty $
and the topological entropy is a topological invariant. Then, one has
$$ \begin{align*}h_{\mathrm{top}}(\varphi_0)=h_{\mathrm{top}}(\varphi),\end{align*} $$
which completes the proof of Lemma 3.4.
Remark 3.5. There is no need to prove the topological conjugacy between
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi ^n)$
and
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi ^n_0)$
for some
$n\in {\mathbb {N}}$
to get
$h_{\mathrm { top}}(\varphi )=h_{\mathrm {top}}(\varphi _{0})$
. However, with the help of continuous random periodic points, we can get the topological conjugacy between these two systems straightforwardly, which may not hold without them.
4 Proof of Theorem 1.2
The proof is done by combining Proposition 2.1, Lemmas 3.1, 3.2, and 3.4.
Due to Proposition 2.1, there is an
$m\in {\mathbb {N}}$
and a
$\varphi $
-invariant torus of degree m, where m is the smallest positive integer satisfying
$m\cdot (I-A)^{-1}(\kern -2pt\begin {smallmatrix} \mathrm {deg}(h_1)\\ \mathrm {deg}(h_2)\end {smallmatrix}\kern -2pt)\in \mathbb {Z}^2$
. Moreover, due to Lemma 2.2, there is a continuous random fixed point
$g\in C({\mathbb {T}},{\mathbb {T}}^2)$
of the induced system
$$ \begin{align*} \begin{aligned} \varphi_{m}: &\,{\mathbb{T}}\times {\mathbb{T}}^2 \to {\mathbb{T}}\times {\mathbb{T}}^2\\ &\,(\omega', x)\mapsto (R_{\alpha/m}(\omega'), Ax+ h(m\omega' \ \mod 1)). \end{aligned} \end{align*} $$
Here, we introduce a finite-to-one map
$$ \begin{align*} \begin{aligned} \mathcal{K}_{m}: &\,{\mathbb{T}}\times {\mathbb{T}}^2\to {\mathbb{T}}\times {\mathbb{T}}^2 \\ &\,(\omega, x) \mapsto (m\omega \ \mod 1, x). \end{aligned} \end{align*} $$
Then, we have the following commuting diagram, that is,
$$ \begin{align*}\begin{array}{c@{}c@{}c} \mathbb T\times \mathbb T^2 & \stackrel{\varphi_m}{\longrightarrow} & \mathbb T\times \mathbb T^2 \\ \Big\downarrow \rlap{$\mathcal{K}_{m}$} & & \Big\downarrow\vcenter{ \rlap{$\mathcal{K}_{m}$}}\\ \mathbb T\times \mathbb T^2 & \stackrel{\varphi}{\longrightarrow} & \mathbb T\times \mathbb T^2 \end{array}\ \end{align*} $$
Thus, by [Reference Bowen3, Theorem 17], we have
$$ \begin{align} h_{\mathrm{top}}(\varphi_m)=h_{\mathrm{top}}(\varphi). \end{align} $$
Since
$g\in C({\mathbb {T}},{\mathbb {T}}^2)$
is a continuous fixed point of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _m)$
, by (4.1) and Lemma 3.4, we have
$$ \begin{align} h_{\mathrm{top}}(\varphi)=h_{\mathrm{top}}(\varphi_m)=h_{\mathrm{top}}(\varphi_{m,0}), \end{align} $$
where
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{m,0})$
is the direct product system of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{m})$
. Moreover, let
$$ \begin{align*}T_{A}: {\mathbb{T}}^2\to {\mathbb{T}}^2,\quad x\mapsto Ax,\end{align*} $$
where
$A\in \mathrm {GL}(2, \mathbb {Z})$
is a hyperbolic matrix. By combining with the fact that
${h_{\mathrm {top}}(R_{\alpha /m})=0}$
, we have
$$ \begin{align} h_{\mathrm{top}}(\varphi_{m,0})= h_{\mathrm{top}}(R_{\alpha/m})+ h_{\mathrm{top}}(T_{A})=\lim_{n\to \infty}\frac1n \log \sharp P(A; n), \end{align} $$
where
$P(A;n)$
is the collection of all periodic points of A with periods less than or equal to n.
Now, we consider the relationship between the number of the periodic points of the hyperbolic matrix
$A\in \mathrm {GL}(2, \mathbb {Z})$
and the invariant tori of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi )$
. Again, due to Lemma 2.3, there is a
$\varphi $
-invariant torus of degree m, which can induce a continuous random fixed point
$g\in C({\mathbb {T}},{\mathbb {T}}^2)$
of the induced system
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _m)$
. Then, due to Lemmas 3.1, 3.2, and Remark 3.3, we have
$$ \begin{align} \sharp P(A; n)=\sharp \mathcal{G}(\varphi_{m,0}; n) =\sharp \bigcup_{k\leq n}\mathcal{A}_0^k=\sharp \bigcup_{k\leq n}\mathcal{A}^k=\sharp \mathcal{G}(\varphi; n, m), \end{align} $$
where
$\mathcal {G}(\varphi _{m,0}; n)$
is the collection of all
$ \varphi _{m,0}^k$
-invariant tori with
$k\leq n$
,
$\mathcal {A}_0^k$
,
$\mathcal {A}^k$
are the sets of all random periodic points with period k of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{m,0})$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{m})$
as in Lemma 3.2, and
$\mathcal {G}(\varphi; n, m)$
is the collection of
$\varphi ^k(k\leq n)$
-invariant tori of degree m.
Finally, we show the uniqueness of
$m\in {\mathbb {N}}$
. Since m is the smallest positive integer satisfying (2.4), due to the discussion in Lemma 2.3, there are continuous random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{lm})$
for all
$l\in {\mathbb {N}}$
. It is straightforward that all random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _m)$
are random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{lm})$
for any
$l\in {\mathbb {N}}$
, that is,
$$ \begin{align} \mathcal{A}^n_{m}\subseteq \mathcal{A}^n_{lm}\quad\text{for all } n,l\in {\mathbb{N}}, \end{align} $$
where
$\mathcal {A}_m^n$
,
$\mathcal {A}_{lm}^n$
are the sets of all random periodic points with period n of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{m})$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{lm})$
. Due to Lemmas 3.1, 3.2, and the fact the direct product systems of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{lm})\text { for all } l\in {\mathbb {N}}$
are the same, we have
$$ \begin{align} \sharp P(A; n)=\sharp \mathcal{G}(\varphi_{lm,0}; n) =\sharp \bigcup_{k\leq n}\mathcal{A}_{lm,0}^k=\sharp \bigcup_{k\leq n}\mathcal{A}_{lm}^k, \end{align} $$
where
$\mathcal {A}_{lm, 0}^k$
and
$\mathcal {A}_{lm}^k$
are the sets of all random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2,\varphi _{lm,0})$
and
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{lm})$
, respectively. By combining (4.5) and (4.6), one has
$$ \begin{align*} \mathcal{A}^n_{m}= \mathcal{A}^n_{lm}\quad\text{for all } n,l\in {\mathbb{N}}. \end{align*} $$
That is to say, all continuous random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{lm})$
for
$l\in {\mathbb {N}}$
are continuous random periodic points of
$({\mathbb {T}}\times {\mathbb {T}}^2, \varphi _{m})$
. Due to condition (3) of Definition 1.1, we only have
$\varphi ^n$
-invariant tori of degree m.
Combining the above argument with (4.2), (4.3), and (4.4), we complete the proof of Theorem 1.2.
Acknowledgement
The authors would like to thank the referees for their careful reading and valuable suggestions. We are also grateful to Wen Huang, Yi Shi, and Hui Xu for their discussions and feedback. This paper was partially supported by National Key R&D Program of China (Nos. 2024YFA1013602, 2024YFA1013600), and partially supported by National Natural Science Foundation of China (Nos. 12090012, 12471188, 12426201, 123B2006, 12090010).
