Chain recurrence rates and topological entropy

Abstract

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the wellknown notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.

The sequence asymptotic average shadowing property and transitivity

“…It is easy to see that f × f is a surjection from X × X to itself. By Corollary 12 of [11], f is chain mixing if and only if f × f is chain transitive, thus the proof is evident from Proposition 3.6 and Theorem 4.1.…”

Average Chain Transitivity and the Almost Average Shadowing Property

“…By the above implications we have that the tent map on unit interval, the doubling map on circle, irrational rotations on circle, identity map on a connected metric space are average chain mixing and the adding machine on Cantor space (see [11]) is average chain transitive.…”

“…The map f is said to be chain transitive if for any δ > 0 and any pair x, y ∈ X, there is a δ-chain of f from x to y and it is totally chain transitive if each f k , k ∈ N, is chain transitive. The map f is said to be chain mixing if for any δ > 0 and any pair x, y ∈ X, there exists N ∈ N such that for any n N , there is a δ-chain of f from x to y of length n [11].…”

“…We recall that if (X, f ) is a compact dynamical system, then f is totally chain transitive if and only if f is chain mixing [11]. Theorem 4.11.…”

On shadowing: Ordinary and ergodic

“…By [15], f is chain transitive and satisfy the shadowing property. By Corollary 4.7, f has the ergodic shadowing property.…”