On Bowen's entropy inequality and almost specification for flows

TL;DR

This paper extends Bowen's entropy inequality to flows, introduces an almost specification property for flows, and demonstrates that flows with this property are saturated, providing new insights into the entropy of irregular points.

Contribution

It defines the topological entropy for noncompact sets in flows, establishes Bowen's inequality for flows, and introduces the almost specification property for flows with related entropy results.

Findings

Topological entropy of noncompact sets for flows matches that of the time-1 map.

Bowen's inequality holds for flows, bounding metric entropy of invariant measures.

Flows with almost specification are saturated, extending previous results.

Abstract

We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is ergodic. We propose a definition of almost specification property for flows and prove that a continuous flow has the almost specification property if the time-1 map satisfies this property. Using Bowen's inequality for flows, we show that every continuous flow with the almost specification property is saturated, extending a result of Meson and Vericat in [22]. Under the same hypotheses, we extend a result of Thompson on the entropy of irregular points in [30].