Estimation of topological entropy in random dynamical systems

Abstract

For $C^r(r>1)$ random dynamical systems F on a compact smooth Riemannian manifold M, the fiber topological entropy is bounded above by an integral formula. Particularly, for the $C^{\infty }$ random dynamical systems, the integral formula coincides with the fiber topological entropy.

Introduction

In 1965, Adler, Konheim and McAndrew [1] introduced a notion called topological entropy into the study of topological dynamical systems. Afterwards, it turned out to be a canonical conjugate invariant, which measures the exponential complexity of dynamical systems. As one of the most critical invariants to categorize different dynamical systems, calculating the topological entropy in different dynamical systems is of great interest to people. However, calculation through various definitions is difficult. In [18], [6], [17], Goodwyn, Dinaburg, and Goodman proved a fundamental relationship between topological entropy and measure-theoretic entropy introduced by Kolmogorov in [9], which is called variational principle. The variational principle gives an estimation of topological entropy by the optimal measure-theoretic entropy of invariant measures, but finding all invariant measures is hardly feasible. In this paper, we mainly concern the estimation of fiber topological entropy of differentiable random dynamical systems (RDS).

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a complete probability space, and θ be an invertible $\mathbb{P}$-invariant map on it, then $(\mathrm{\Omega },\mathcal{F},\mathbb{P},\theta )$ is called a metric dynamical system. Let M be a compact smooth Riemannian manifold without boundary, ${\mathrm{Diff}}^r(M)$ be the space of $C^r(r>1)$ diffeomorphisms on M. Let $F:\mathrm{\Omega }\to {\mathrm{Diff}}^r(M)$, $\omega \mapsto F_{\omega }$ be a Borel measurable map. We consider the RDS F generated by $\{F_{\omega },\omega \in \mathrm{\Omega }\}$ over the base space $(\mathrm{\Omega },\mathcal{F},\mathbb{P},\theta )$, and let

$$\mathrm{\Theta }:\mathrm{\Omega }\times M\to \mathrm{\Omega }\times M,(\omega ,x)\mapsto (\theta \omega ,F_{\omega }x)$$

be the corresponding skew product transformation.

For $n\in \mathbb{N}$ and a positive random variable $\epsilon =\epsilon (\omega )$ define a family of metrics $d_{\epsilon ,n}^{\omega }$ on $M_{\omega }$ by the formula

$$d_{\epsilon ,n}^{\omega }=\underset{0\le i<n}{\max }\{d(F_{\omega }^{i}x,F_{\omega }^{i}y)\epsilon {({\theta }^i\omega )}^{-1},x,y\in M_{\omega }\},$$

where $d(\cdot ,\cdot )$ is the Riemannian metric on M. Denote by $B_{(\omega ,x)}^n(\epsilon )$ the closed ball in $M_{\omega }$ centered at x of radius 1 under metric $d_{\epsilon ,n}^{\omega }$. We say $x,y\in M_{\omega }$ are $(\omega ,\epsilon ,n)$-close if $d_{\epsilon ,n}^{\omega }(x,y)\le 1$. A set $Q\subset M_{\omega }$ is called $(\omega ,\epsilon ,n)$-separated set if for $x,y\in Q$, $x\ne y$ implies $d_{\epsilon ,n}^{\omega }(x,y)>1$.

Due to compactness of $M_{\omega }$, there exists a smallest natural number $s(\omega ,\epsilon ,n)$ such that $\mathrm{Card}(Q)\le s(\omega ,\epsilon ,n)<\infty$ for every $(\omega ,\epsilon ,n)$-separated set Q. Furthermore, there exists a maximal $(\omega ,\epsilon ,n)$-separated set $Q(\omega ,\epsilon ,n)$ such that for any $x\in M_{\omega }\setminus Q$, $Q\cup \{x\}$ is not $(\omega ,\epsilon ,n)$-separated set anymore. Therefore, $M_{\omega }\subset {\cup }_{x\in Q}{B}_{(\omega ,x)}^n(\epsilon )$.

The fiber topological entropy is defined as

$$h_{top}(F)=h_{top}^{(r)}(\mathrm{\Theta })=\underset{\epsilon \to 0}{\lim }\underset{n\to \infty }{\lim \sup }\frac{1}{n}\int \log s(\omega ,ϵ,n)d\mathbb{P}.$$

In deterministic dynamical systems, the estimation of topological entropy has a long history. In [3], [19], [20], and [22], the topological entropy for tori is the sum of logarithm of eigenvalues with modulo greater than 1. Afterwards, Bowen gave the same formula for entropy of an endomorphism of Lie group in [5]. Sacksteder and Shub made an attempt in [21] to estimate the topological entropy for differentiable map on a compact manifold. Then Przytycki proved that the topological entropy of a differentiable map on a compact manifold is bounded above by the (exponential) growth rate of an integral formula in [23]. Newhouse extended the result to non-invertible maps in [16]. In [25], Yomdin proved an inequality of volume growth and entropy, also pointed out that the topological entropy is equal to the volume growth for $C^{\infty }$ maps. In [10], Kozlovski combined the techniques from Przytycki, Newhouse, and Yomdin to estimate the topological entropy by the growth rate of the integral formula Przytycki used. In the case of RDSs, there are several proceeding works. In [7], Kifer and Yomdin extended the work of volume growth and entropy for a sequence of independent random smooth maps of a manifold. In [27], Zhu proved the growth in volume, fundamental groups, homological groups are bounded by topological entropy for smooth RDSs. In this paper, we have the following theorem, which estimates fiber topological entropy of RDSs by an integral formula.

Theorem 1

Let F be a Borel measurable mapping from Ω to ${\mathrm{Diff}}^2(M)$ satisfies ${\int }_{\mathit{\Omega }}{\log }^{+}{\Vert D{F}_{\omega }\Vert }_{C^2}d\mathbb{P}<+\infty$, we have the following estimate:

$$h_{top}(F)\le \int \underset{n\to \infty }{\lim \sup }\frac{1}{n}\log (\int \Vert {(D_x{F}_{\omega }^n)}^{\wedge }\Vert dx)d\mathbb{P}(\omega ).$$

Furthermore, if $F:\mathit{\Omega }\to {\mathrm{Diff}}^{\infty }(M)$ satisfies ${\int }_{\mathit{\Omega }}{\log }^{+}{\Vert D{F}_{\omega }\Vert }_{C^{\infty }}d\mathbb{P}<+\infty$, then

$$h_{top}(F)=\int \underset{n\to \infty }{\lim \sup }\frac{1}{n}\log (\int \Vert {(D_x{F}_{\omega }^n)}^{\wedge }\Vert dx)d\mathbb{P}.$$

Note that ${(D_x{F}_{\omega }^n)}^{\wedge }$ is a mapping between exterior algebras of the tangent space $T_x{M}_{\omega }$ and $T_{F_{\omega }^{n}x}{M}_{{\theta }^n\omega }$ induced by $D_x{F}_{\omega }^n$. For $k\in {\mathbb{Z}}^{+}$ and $C^k$ diffeomorphisms, ${\Vert D{F}_{\omega }\Vert }_{C^k}={\sup }_{s=1,\cdots ,k}\Vert D^s{F}_{\omega }\Vert$, and ${\Vert D{F}_{\omega }\Vert }_{C^{\infty }}={\sup }_{s\in {\mathbb{Z}}^{+}}\Vert D^s{F}_{\omega }\Vert$, where $\Vert D^s{F}_{\omega }\Vert :={\sup }_{x\in M}\Vert D_x^s{F}_{\omega }\Vert$.

The structure of this paper is as follows. Firstly, several preliminaries of RDSs. In Section 3, we illuminate that the fiber topological entropy is bounded above by the growth rate of the integral formula, by using the Multiplicative Ergodic Theorem, Lyapunov charts and variational principle. In Section 4, firstly, for the $C^k$ RDSs, the fiber topological entropy plus an $\frac{\ell }{k}$ correction is bounded below by the growth rate of the integral formula. The main technique is a modification of Yomdin's work to the RDSs. Then the second part of Theorem 1 holds by letting k tend to infinity and combining the result in Section 3.