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

A. Ehsani, A. Fakhari, F. H. Ghane, J. Nazarian

TL;DR

This work develops a rigorous thermodynamic framework for topologically mixing locally conformal expanding semigroup actions on compact manifolds by constructing a countable Markov partition with finite images and finite cycle properties, and by formulating inducing schemes of hyperbolic type. It then models the dynamics via induced towers and a skew product $F_A$, introducing liftable measures through a random Markov-chain construction and establishing a comprehensive thermodynamic formalism for these liftable states. By extending Sarig’s thermodynamics to two-sided countable shifts and linking the original semigroup action to the induced system, the authors prove existence and uniqueness results for equilibrium measures under summability conditions and derive Abramov/Kac-type relations between base and induced dynamics. The results provide a symbolic and probabilistic bridge between semigroup actions and countable-state thermodynamic formalism, with potential applications to ergodic properties, Gibbs states, and equilibrium states in non-uniformly hyperbolic settings.

Abstract

For topologically mixing locally conformal semigroup actions generated by a finite collection of $C^{1+α}$ 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.

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

TL;DR

This work develops a rigorous thermodynamic framework for topologically mixing locally conformal expanding semigroup actions on compact manifolds by constructing a countable Markov partition with finite images and finite cycle properties, and by formulating inducing schemes of hyperbolic type. It then models the dynamics via induced towers and a skew product , introducing liftable measures through a random Markov-chain construction and establishing a comprehensive thermodynamic formalism for these liftable states. By extending Sarig’s thermodynamics to two-sided countable shifts and linking the original semigroup action to the induced system, the authors prove existence and uniqueness results for equilibrium measures under summability conditions and derive Abramov/Kac-type relations between base and induced dynamics. The results provide a symbolic and probabilistic bridge between semigroup actions and countable-state thermodynamic formalism, with potential applications to ergodic properties, Gibbs states, and equilibrium states in non-uniformly hyperbolic settings.

Abstract

For topologically mixing locally conformal semigroup actions generated by a finite collection of 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.

Paper Structure

This paper contains 15 sections, 25 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Markov partition for Locally conformal expanding semigroup actions
  4. Inducing scheme of hyperbolic type for semigroups
  5. Inducing scheme for the induced skew product
  6. Equilibrium measures and Potentials
  7. Countable Markov partitions and ergodicity of induced map
  8. Countable Markov partitions associated to locally conformal actions
  9. Bounded distortion property and ergodicity of induced maps
  10. Markov towers and Thermodynamics of two-sided countable shifts
  11. Markov towers for locally conformal semigroup actions
  12. Thermodynamics of two-sided countable shifts
  13. Thermodynamic formalism of the induced skew product
  14. Random Markov chains and liftable measures
  15. Thermodynamic formalism with respect to liftable measures

Key Result

Lemma 3.1

Take a dynamical $n$-ball of the form (01). Then $f_w^{n}(B(x,n,w,\varepsilon))=B(f_w^{n}(x),\varepsilon)$, for any admissible $n$-word of $x$.

Figures (2)

  • Figure 1: Graph of $f_0$ and $f_1$
  • Figure 2: A countable Markov partition with FIP and FCP

Theorems & Definitions (47)

  • Definition 2.1: Locally conformal semigroup
  • Definition 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 37 more