Nonadditive families of potentials: physical equivalence and some regularity relations

TL;DR

This paper demonstrates the physical equivalence of additive and asymptotically additive potential families in suspension flows, extending thermodynamic formalism and multifractal analysis, with applications to hyperbolic and expansive flows.

Contribution

It establishes the physical equivalence of additive and asymptotically additive potentials and extends nonadditive thermodynamic formalism and multifractal analysis.

Findings

Equivalence holds for hyperbolic flows and some expansive flows.

Extension of thermodynamic formalism to nonadditive potentials.

Results on regularity and Livšic-like theorems for nonadditive families.

Abstract

We show that additive and asymptotically additive families of continuous functions with respect to suspension flows are physically equivalent. In particular, the equivalence result holds for hyperbolic flows and some classes of expansive flows in general. Moreover, we show how this equivalence result can be used to extend the nonadditive thermodynamic formalism and multifractal analysis. In the second part of this work, we obtain a Liv\v{s}ic-like result for nonadditive families of potentials and also address the H\"older and Bowen regularity problem for the physical equivalence relations with respect to hyperbolic symbolic flows.