Topological and Metric Pressure for Singular Flows

TL;DR

This paper advances the thermodynamic formalism for flows with singularities by introducing three rescaled Bowen balls that incorporate flow speed via the vector field magnitude and time reparametrizations, effectively neutralizing singular behavior. It proves the equivalence of metric and topological pressures defined via these three balls, and establishes a Katok-type formula $P_\mu^*(X,f)=h_\mu(\phi_1)+\int f\,d\mu$ under mild integrability and singularity assumptions, along with a partial variational principle linking rescaled metric and topological pressures. The work combines flowbox techniques, bounded variation results, time discretization, and classical entropy tools (Mañé, SMB) to connect local geometric control with global thermodynamic quantities. The results extend the pressure framework to singular flows, providing robust tools for equilibrium states and multifractal analysis in this broader setting.

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 $μ(\mathrm{Sing}(X))=0$, we prove Katok's formula of pressure. We establish a partial variational principle that relates the rescaled metric pressure and the rescaled topological pressure.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2