Packing entropy for fixed-point free flows

Abstract

Let $(X,\phi)$ be a compact flow without fixed points. We define the packing topological entropy $h_{\mathrm{top}}^P(\phi,K)$ on subsets of $X$ through considering all the possible reparametrizations of time. For fixed-point free flows, we prove the following result: for any non-empty compact subset $K$ of $X$, $$h_{\mathrm{top}}^P(\phi,K)=\sup\{\overline{h}_{\mu}(\phi):\mu(K)=1,\mu\text{ is a Borel probability measure on} X\},$$ where $\overline{h}_{\mu}(\phi)$ denotes the upper local entropy for a Borel probability measure $\mu$ on $X$.