Scaled pressure of continuous flows^*

Abstract

In this paper we introduce the notion of scaled pressure of continuous flows. It is proved that scaled pressure is preserved under certain reparametrizations of orbits and Lipschitz conjugations of fixed-point free continuous flows. Moreover, for expansive flows, scaled pressure can be described by period-orbits. In particular, a variational principle is obtained by comparing scaled pressures for continuous flows with their time one maps.

Metric mean dimension of flows

“…The following argument is also used by Li and Cheng [CL21] to study the scaled pressure of fixed-point free continuous flows.…”

“…Recently, the notion of mean dimension was introduced by Gutman and Jin [GJ20] when investigating embedding problems of flows. Moreover, Cheng and li successfully extended the notion of metric mean dimension to impulsive semi-flows and the notion of scaled pressure to fixed-point free continuous flows, see [CL21,CL22]. Following Lindenstrauss and Weiss's ideas, in this paper we introduced the notion of metric mean dimension of flows on the whole phase space to capture the complexity of infinite topological entropy of flows introduced by Bowen and Ruelle [BR75], which turns out that the classical Lindenstrauss-Weiss's inequality still holds.…”