Rigidity of pressures of H\"older potentials and the fitting of analytic functions via them

TL;DR

This paper investigates the higher order differentials of pressure functions of H"older potentials, revealing rigidity properties that limit their ability to approximate certain analytic functions, and explores conditions under which these functions can be realized by locally constant potentials.

Contribution

It establishes new rigidity constraints on pressure functions of H"older potentials and demonstrates when analytic germs can be realized by pressures of locally constant potentials.

Findings

Rigidity relationships between differentials of pressure functions.

Obstructions to fitting convex analytic functions globally.

Conditions for realizing analytic germs with locally constant potentials.

Abstract

The first part of this work is devoted to the study of higher differentials of pressure functions of H\"older potentials on shift spaces of finite type. By describing the differentials of pressure functions via the Central Limit Theorem for the associated random processes, we discover some rigid relationships between differentials of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of H\"older potentials globally, which answers a question of Kucherenko-Quas. In the second part of the work we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as number of the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.