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

Abstract

Abstract. In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved 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 exhibiting here is induced by the positive entropy but not the randomness of the system.

SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

“…Thus, with a positive entropy, Lian and Young's result implies that the hyperbolic system has a horseshoe. The existence of SRB measures also yields that the partially hyperbolic attractor is chaotic and contains a full weak horseshoe following from the recent result by Huang and Lu [16] since the entropy is positive. By a full horseshoe of two symbols we mean that there exist subsets U 1 , U 2 of Hilbert space H such that the following properties hold (1) U 1 and U 2 are non-empty bounded closed subsets of H and d(U 1 , U 2 ) > 0.…”

“…Remark: With a positive entropy, Lian and Young's result implies that the hyperbolic system has a horseshoe. The existence of SRB measures also yields that the partially hyperbolic attractor is chaotic and contains a full weak horseshoe following from the recent result by Huang and Lu [16] since the entropy is positive.…”

Weak Horseshoe with bounded-gap-hitting times

“…The notion of weak horseshoe first appeared in [7] (see also [8,9]). We introduce several kinds of weak horseshoes in the following.…”

“…For a flow (M, φ), it is well known that h top (φ t ) = |t|h top (φ 1 ) for any t ∈ R (see [1]). In [8], Huang and Ye showed that a TDS (X, T ) has positive topological entropy if and only if (X, T ) has a weak horseshoe with positive density hitting times (see also [7,9]). Thus, it is clear that for a flow (M, φ), if the time one map (M, φ 1 ) has a weak horseshoe with positive density hitting times then (M, φ τ ) has the same property for all τ = 0.…”