Thermodynamics via inducing

TL;DR

This paper develops thermodynamic formalism for inducing schemes of hyperbolic type and their induced and tower maps, analyzing equilibrium measures and their ergodic properties, including Bernoulli property, decay of correlations, CLT, and pressure analyticity.

Contribution

It extends thermodynamic formalism to hyperbolic inducing schemes and relates equilibrium measures across different associated systems.

Findings

Established thermodynamic formalism for inducing schemes and associated maps.

Analyzed ergodic properties of equilibrium measures including Bernoulli property and CLT.

Proved analyticity of the pressure function for these systems.

Abstract

We consider continuous maps $f:X\to X$ on compact metric spaces admitting inducing schemes of hyperbolic type introduced in [15] as well as the induced maps $\tilde{f}:\tilde{X}\to\tilde{X}$ and the associated tower maps $\hat{f}:\hat{X} \to \hat {X}$. For a certain class of potential functions $\varphi$ on $X$, which includes all H\"older continuous functions, we establish thermodynamic formalism for each of the above three systems and we describe some relations between the corresponding equilibrium measures. Furthermore we study ergodic properties of these equilibrium measures including the Bernoulli property, decay of correlations, and the Central Limit Theorem (CLT). Finally, we prove analyticity of the pressure function for the three systems.