Weak Horseshoe with bounded-gap-hitting times

TL;DR

This paper investigates weak horseshoe structures with bounded-gap-hitting times in dynamical systems, establishing their invariance under time changes and linking positive entropy to such structures in affine group homeomorphisms.

Contribution

It demonstrates that weak horseshoe with bounded-gap-hitting times is preserved under time rescaling and characterizes positive entropy in affine group systems through this property.

Findings

Weak horseshoe with bounded-gap-hitting times is invariant under time change.

Positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times in affine systems.

The results connect entropy and complex dynamical structures in specific flows.

Abstract

In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow $(M,\phi)$, it is shown that if the time one map $(M,\phi_1)$ has weak horseshoe with bounded-gap-hitting times, so is $(M,\phi_\tau)$ for all $\tau\neq 0$. In addition, we prove that for an affine homeomorphsim of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.