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Smale endomorphisms over graph-directed Markov systems

Part of: Dynamical systems with hyperbolic behavior Probability theory on algebraic and topological structures Smooth dynamical systems: general theory Ergodic theory Measures, integration, derivative, holomorphy Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  08 June 2020

EUGEN MIHAILESCU  and
MARIUSZ URBAŃSKI
EUGEN MIHAILESCU
Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO 014700, Bucharest, Romania (e-mail: Eugen.Mihailescu@imar.ro)
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX76203-1430, USA (e-mail: urbanski@unt.edu)
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Abstract

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) 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 also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction 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 ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

Keywords

Hausdorff dimension of measuresthermodynamic formalism of Smale endomorphismsbeta-mapsinverse limitscontinued fractions

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37A35: Entropy and other invariants, isomorphism, classification 37C45: Dimension theory of dynamical systems 37A45: Relations with number theory and harmonic analysis 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 46G10: Vector-valued measures and integration 60B05: Probability measures on topological spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2508 - 2541
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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