Topological and Metric Pressure for Singular Flows

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 $\log \mathrm{\parallel }X(x)\mathrm{\parallel }$ is integrable and that $\mu (\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(X))=0$, we prove Katok’s formula of pressure. We establish a partial variational principle that relates the rescaled metric pressure and the rescaled topological pressure.