# Smale endomorphisms over graph-directed Markov systems

## TL;DR

This paper investigates Smale skew product endomorphisms over graph-directed Markov systems, establishing their measure dimensionality properties and deriving a general formula for the Hausdorff dimension of equilibrium measures, with applications in ergodic number theory.

## Contribution

It extends the theory of Smale endomorphisms to countable graph-directed systems and provides new results on measure dimensionality and dimension formulas for natural extensions.

## Key findings

- Proves exact dimensionality of conditional measures in fibers.
- Establishes global exact dimensionality of equilibrium measures.
- Derives a general Hausdorff dimension formula for natural extensions of β-maps.

## Abstract

We study Smale skew product endomorphisms (introduced in [27]) now over countable graph directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply for instance to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We give also applications in ergodic number theory, for example to the continued fraction expansion, and the backward fractions expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions $\mathcal T_\beta$ of $\beta$-maps $T_\beta$, for arbitrary $\beta > 1$.