Estimation of topological entropy in random dynamical systems
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 be a complete probability space, and θ be an invertible -invariant map on it, then is called a metric dynamical system. Let M be a compact smooth Riemannian manifold without boundary, be the space of diffeomorphisms on M. Let , be a Borel measurable map. We consider the RDS F generated by over the base space , and let be the corresponding skew product transformation.
For and a positive random variable define a family of metrics on by the formula where is the Riemannian metric on M. Denote by the closed ball in centered at x of radius 1 under metric . We say are -close if . A set is called -separated set if for , implies .
Due to compactness of , there exists a smallest natural number such that for every -separated set Q. Furthermore, there exists a maximal -separated set such that for any , is not -separated set anymore. Therefore, .
The fiber topological entropy is defined as
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 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 satisfies , we have the following estimate: Furthermore, if satisfies , then
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 RDSs, the fiber topological entropy plus an 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.
Section snippets
Preliminaries
In this section, we will demonstrate the Multiplicative Ergodic Theorem and Lyapunov charts for random dynamical systems for completeness of this paper.
Let F be a RDS over , μ be a Θ-invariant probability of F with marginal on Ω, i.e. it can be disintegrated into . The disintegration , for is called the sample measures of μ and is invariant in the sense of , which means for each Borel measurable subset , we have . In the following, we
The upper bound of topological entropy of RDS
In this section, we consider the first part of Theorem 1. We rephrase the first part of Theorem 1 in the following lemma for convenience. Lemma 5 Let F be a Borel measurable mapping from Ω to satisfies , we have the following estimate:
The lower bound of topological entropy of RDS
In this section, our goal is to estimate the upper bound of the integral formula for RDSs. There are two steps of the estimation, locally and globally. In other words, we estimate the integral formula on a Bowen ball, then on the whole space, which gives a lower bound of the fiber topological entropy for RDSs.
Let M be a compact smooth manifold with Riemannian metric d. The random dynamical system be Borel measurable, where . And assume that where
Acknowledgement
The author would like to thank Prof. Wen Huang, Prof. Zeng Lian, and Dr. Rui Gao for their insights and opinions. Many thanks to the referee for the careful reading of the manuscript and valuable comments and suggestions. The author is partially supported by NNSF of China (11971455) and The Fundamental Research Funds for the Central Universities (WK3470000014).
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