Random graph directed Markov systems

Mario Roy; Mariusz Urbański

doi:10.3934/dcds.2011.30.261

Article Contents

2011, Volume 30, Issue 1: 261-298. Doi: 10.3934/dcds.2011.30.261

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Random graph directed Markov systems

Mario Roy^1, and
Mariusz Urbański^2,

1.
Glendon College, York University, 2275 Bayview Avenue, Toronto
2.
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received: December 31, 2009

Revised: September 30, 2010

Published: December 2010

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Abstract

We introduce and explore random conformal graph directed Mar-kov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise topological pressure. This enables us to define the expected topological pressure $\mathcal EP(t)$ and to prove a variant of Bowen's formula which identifies the Hausdorff dimension of almost every limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the unique zero of the expected pressure if the alphabet is finite or the system is regular. We introduce the class of essentially random systems and we show that in the realm of systems with finite alphabet their limit set fibers are never homeomorphic in a bi-Lipschitz fashion to the limit sets of deterministic systems; they thus make up a drastically new world. We also provide a large variety of examples, with exact computations of Hausdorff dimensions, and we study in detail the small random perturbations of an arbitrary elliptic function.

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Mathematics Subject Classification: Primary: 37H99, 37D35, 37F35; Secondary: 37F10.

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