# Skew product Smale endomorphisms over countable shifts of finite type

## TL;DR

This paper develops a thermodynamic formalism for skew product Smale endomorphisms over countable shifts, proving dimension formulas and exact dimensionality of measures, with applications to Diophantine approximation and beta-maps.

## Contribution

It introduces a new thermodynamic formalism for countable Markov shifts and skew products, establishing dimension formulas and exact dimensionality results for a broad class of measures.

## Key findings

- Almost all equilibrium measures are dimensionally exact.
- Dimension equals entropy divided by Lyapunov exponent.
- Established formulas for Hausdorff dimension of fibers.

## Abstract

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi(EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the renowned Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we prove exact dimensionality and find a computable formula for the dimension of equilibrium measures, for induced maps of natural extensions $\mathcal T_\beta$ of beta-maps, for arbitrary \beta > 1.