Weak Horseshoe with bounded-gap-hitting times

Abstract

In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow $(M,\varphi )$, it is shown that if the time one map $(M,{\varphi }_1)$ has weak horseshoe with bounded-gap-hitting times, so is $(M,{\varphi }_{\tau })$ for all $\tau \ne 0$. 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.