Bowen entropy of generic point for fixed-point free flows

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.