Locally conformal expanding Actions: Markov Partition and Thermodynamic of the induced skew product

TL;DR

This paper develops a Markov partition and thermodynamic formalism for locally conformal semigroup actions generated by conformal local diffeomorphisms, analyzing their ergodic properties and tower structures.

Contribution

It introduces a countable Markov partition with finite images and cycle properties for such actions, and establishes a thermodynamic formalism for the associated skew product.

Findings

Existence of countable Markov partitions for the actions.

Construction of inducing schemes and tower models.

Proof of thermodynamic formalism and ergodic properties.

Abstract

For topologically mixing locally conformal semigroup actions generated by a finite collection of $C^{1+\alpha}$ conformal local diffeomorphisms, we provide a countable Markov partition satisfying the finite images and the finite cycle properties. We show that they admit inducing schemes and describe the tower constructions associated with them. An important feature of these towers is that their induced maps are equivalent to a subshift of countable type. Through the investigating the ergodic properties of induced map, we prove the existence of liftable measures and establish a thermodynamic formalism of the induced skew product with respect to them.