Rescaled topological entropy

TL;DR

This paper introduces the rescaled topological entropy, a new invariant for smooth vector fields on closed manifolds, which bounds classical entropy measures, detects positivity in certain cases, and controls periodic orbit growth.

Contribution

It defines the rescaled topological entropy and proves its key properties, including bounds, invariance, and applications to periodic orbit growth for expansive flows.

Findings

Rescaled topological entropy bounds classical topological and metric entropy.

It coincides with topological entropy for nonsingular vector fields.

It is positive for certain surface vector fields, unlike classical topological entropy.

Abstract

We prove that to any smooth vector field of a closed manifold it can be assigned a nonnegative number called {\em rescaled topological entropy} satisfying the following properties: it is an upper bound for both the topological entropy and the rescaled metric entropy \cite{ww}; coincides with the topological entropy for nonsingular vector fields; is positive for certain surface vector fields (in contrast to the topological entropy); is invariant under rescaled topological conjugacy; and serves as an upper bound for the growth rate of periodic orbits for rescaling expansive flows with dynamically isolated singular set. Therefore, the rescaled topological entropy bounds such growth rates for $C^r$-generic rescaling (or $k^*$) expansive vector fields on closed manifolds.