Packing entropy for fixed-point free flows

TL;DR

This paper introduces a new notion of packing topological entropy for fixed-point free flows and establishes a variational principle linking it to the supremum of upper local entropies over measures supported on subsets.

Contribution

It defines packing topological entropy considering reparametrizations of time and proves a variational principle for fixed-point free flows.

Findings

Packing entropy equals the supremum of upper local entropies over measures supported on the set.

The result applies to any non-empty compact subset of the flow space.

Provides a new tool for analyzing complexity in fixed-point free dynamical systems.

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 $X$.