Entropy, Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

Abstract

In this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.

Full-horseshoes for the Galerkin truncations of 2D Navier-Stokes equation with degenerate stochastic forcing

“…Let's end of this section by a result about conditional measure-theoretic entropy and relative Pinsker factor. The reader can refer to the proof for [19,Lemma 4.1] and [20,Lemma 3.3].…”

“…The purpose of this paper is to characterize turbulence of the GSNS with full-horseshoes. Full-horseshoe was firstly introduced by the first author and Lu in [19] to investigate the complex behaviors of infinite-dimensional random dynamical systems. It imitates the process of coin toss and is a weaker chaotic structure than Smale horseshoe.…”

Random semi-horseshoe for partially hyperbolic systems driven by an external force

“…In this paper, inspired by [15,18,20], we prove the following statement: If a system driven by an external force j has an equicontinuous uniformly expanding subbundle, then it has a random semi-horseshoe, which means the system has a subsystem randomly semi-conjugating to the full shift of two symbols.…”

“…One step towards this question is [20], in which Huang and Lu considered an injective infinite-dimensional continuous random dynamical system j on a Polish space X over an ergodic Polish system q W  , , ,  (…”

“…If (K, j) has positive topological entropy, then it has a weak horseshoe of two symbols. In [20], a random dynamical system (K, f) is said to have a weak horseshoe of two symbols if one can find two subsets U 0 , U 1 ⊂ X such that the followings hold:…”