H\"older continuity of measures for heavy tail potentials

TL;DR

This paper establishes a H"older continuity property for measures associated with certain heavy tail potentials in dynamical systems, revealing new phase transition phenomena in intermittent flows and maps.

Contribution

It introduces an EKP inequality for potentials depending on roof functions, linking measure regularity to pressure and $L^q$ spaces, and uncovers novel phase transitions.

Findings

Proves a H"older continuity property in the weak* norm for measures.

Identifies a new type of phase transition in intermittent flows.

Connects measure regularity to the $L^q$-space of potentials.

Abstract

For a class of potentials $\psi$ satisfying a condition depending on the roof function of a suspension (semi)flow, we show an EKP inequality, which can be interpreted as a H\"older continuity property in the weak${^*}$ norm of measures, with respect to the pressure of those measures, where the H\"older exponent depends on the $L^q$-space that $\psi$ belongs to. This also captures a new type of phase transition for intermittent (semi)flows (and maps).