Skew product Smale endomorphisms over countable shifts of finite type

Abstract

We introduce and study skew product Smale endomorphisms over finitely irreducible shifts with countable alphabets. This case is different from the one with finite alphabets and we develop new methods. In the conformal context we prove that almost all conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional and their dimension is equal to the ratio of (global) entropy and Lyapunov exponent. We show that the exact dimensionality of conditional measures on fibers implies global exact dimensionality of the original measure. We then study equilibrium states for skew products over expanding Markov–Rényi transformations and settle the question of exact dimensionality of such measures. We apply our results to skew products over the continued fraction transformation. This allows us to extend and improve the Doeblin–Lenstra conjecture on Diophantine approximation coefficients to a larger class of measures and irrational numbers.