Topological and Metric Pressure for Singular Flows

TL;DR

This paper introduces rescaled topological and metric pressures for singular flows, accounting for flow velocity and reparametrization, and proves key equivalences and formulas under certain conditions.

Contribution

It defines new pressure notions for singular flows using rescaled Bowen balls and proves their equivalence and a variational principle.

Findings

Rescaled pressures effectively handle singularities.

Equivalence of pressures via different Bowen balls.

Proof of Katok's formula for pressure under specific conditions.

Abstract

In this paper, we introduce the notions of rescaled metric pressure and rescaled topological pressure for flows by considering three types of rescaled Bowen balls, which take the flow velocity and time reparametrization into account. This approach effectively eliminates the influence of singularities. It is demonstrated that defining both metric pressure and topological pressure via several distinct Bowen balls is equivalent. Furthermore, under the assumptions that $\log \|X(x)\|$ is integrable and that $\mu(\mathrm{Sing}(X))=0$, we prove Katok's formula of pressure. We establish a partial variational principle that relates the rescaled metric pressure and the rescaled topological pressure.