Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry

Abstract

In this paper we introduce measurable expanding random systems, develop the thermodynamical formalism and establish, in particular, exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen's formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we get a natural classifications of the systems into two classes: quasi-deterministic systems which share many properties of deterministic ones and essential random systems which are rather generic and never bilipschitz equivalent to deterministic systems. We show in the essential case that the Hausdorff measure vanishes which refutes a conjecture of Bogenschütz and Ochs. We finally give applications of our results to various specific conformal random systems and positively answer a question of Brück and Bürger concerning the Hausdorff dimension of random Julia sets.

Relative Equilibrium States and Dimensions of Fiberwise Invariant Measures for Random Distance Expanding Maps

“…5) with equilibrium states of Hölder continuous potentials and random distance expanding maps as D. Simmons & M. Urbański defined in [10]. We prove that the Gibbs states (known from [10] to be unique) of such potentials coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique.…”

“…The thermodynamic formalism of random distance expanding maps has been developed in [10]. It comprised and went beyond the previous work of Bogenschütz, Gundlach, Kifer, and others (see [1,3,4,6,7], and the references therein) on random symbolic dynamical systems and random infinitesimally expanding maps on smooth Riemannian manifolds.…”

“…It comprised and went beyond the previous work of Bogenschütz, Gundlach, Kifer, and others (see [1,3,4,6,7], and the references therein) on random symbolic dynamical systems and random infinitesimally expanding maps on smooth Riemannian manifolds. The work [10] thoroughly explored the concept of Gibbs states of appropriately defined Hölder continuous potentials. The work [12] substantially developed the theory of relative equilibrium states of holomorphic endomorphisms of the Riemann sphere and Hölder continuous potentials.…”

Regularity and irregularity of fiber dimensions of non-autonomous dynamical systems

“…Given Lemma 5.4, the expression (5.14) of the pressures along with the properties of the pressure functions (Proposition 5.8) the proof of the theorem is by now a standard application of the Frostman type lemma Theorem 7.6.1 in [PU]. For more details we refer the reader to Chapter 7 of [PU] or to the, parallel but technically more involved, proof of Theorem 5.2 in [MUS11].…”

Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

“…In the case of random dynamics, Bogenschütz and Gundlach [4] and also Kifer in [10] have considered the finite shift case; see also [14] for a general and systematic treatment of random expanding maps. Then Denker, Kifer and Stadlbauer in [7], Stadlbauer [21,22] and [17] dealt with the case of random countable topological Markov shifts.…”