Topological and Metric Pressure for Singular Flows

2025/10/31 by Wen Xiao, Zhao, Meijie, Wen, Xiao
Mathematics · Physics and Astronomy · #37B05 #37C10 #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2510.27216

openalex publication_date 2025/10/31 · openalex created_date 2025/11/05 · openalex updated_date 2026/07/28

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(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.

Citations

• A rescaled expansiveness for flows
• Morse-Smale systems and horseshoes for three dimensional singular flows
• Repellers for real analytic maps

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