Smale endomorphisms over graph-directed Markov systems

Eugen Mihailescu, Mariusz Urbanski

Published 2019-07-30Version 1

In this paper we study Smale skew product endomorphisms (introduced in [21]) 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. In particular, this applies to natural extensions of graph-directed Markov systems, and to skew products over parabolic systems. This comprises large classes of examples. In the end, we apply these results to obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to induced maps of natural extensions $\mathcal T_\beta$ of $\beta$-maps, for arbitrary $\beta > 1$.