Bowen entropy of generic point for fixed-point free flows

TL;DR

This paper extends Bowen's classical entropy result to fixed-point free flows by establishing a Bowen entropy formula for generic points and connecting it to local entropies of measures.

Contribution

It proves a Bowen entropy formula for flows without fixed points, extending Bowen's 1973 result to a broader class of dynamical systems.

Findings

Established Brin-Katok's entropy formula for flows without fixed points.

Proved Bowen entropy equals measure-theoretic entropy for generic points.

Demonstrated Bowen entropy can be computed via local measure entropies.

Abstract

Let $(X, \phi)$ be a compact metric flow without fixed points. We will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of the flows. In this paper, by establishing the Brin-Katok's entropy formula for flows without fixed points in non-ergodic case, we prove the following result: for an ergodic $\phi$-invariant measure $\mu$, $$ h_{top}^{B}(\phi, G_{\mu}(\phi))=h_{\mu}(\phi_{1}),$$ where $G_{\mu}(\phi)$ is the set of generic points for $\mu$ and $h_{top}^{B}(\phi, G_{\mu}(\phi))$ is the Bowen entropy on $G_{\mu}(\phi)$. This extends the classical result of Bowen in 1973 to fixed-point free flows. Moreover, we show that the Bowen entropy can be determined via the local entropies of measures.