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

Abstract

We show that the Gibbs states (known from [10] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps, we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.

Critically finite random maps of an interval

“…The potentials −t log |T ω | we consider in the current paper are however unbounded because of the presence of critical points. Nevertheless (see Theorem 5.6 in [32]) if a random system is good enough (in particular the fiberwise potentials must be continuous), then both the random topological pressure and the expected pressure (defined in an analogous way as in the current paper) coincide. So, our expected pressure can be viewed as an extension of topological pressure to the case of unbounded potentials although only for the special random critically finite dynamical systems we consider and for potentials being equal to −t log |T ω |.…”

“…We also recommend the little, well written, book of Hans Crauel [13]. Additional papers on random dynamics, somewhat related to our current work include [2], [4]- [9], [14], [16], [17]- [21], [22], [24], [3], [30] and [31], [32]. This list is not by any means complete.…”

“…Secondly, there is however a concept of topological pressure for random dynamical systems, see [4], comp. [32] for a brief account, where as in our setting the base Ω is an arbitrary probability space and the fibers are compact. But this definition (and the variational principle which is then proved) require the potential restricted to each (compact) fiber to be continuous, thus bounded.…”

“…The measures h λ ν λ , λ ∈ Λ, need not be random probability measures but, due to (32), their fiber total values h θ(ω),λ ν λ θ(ω) (J (k) (T )) are uniformly bounded away from zero and ∞. Since M m (J (k) (T )) is compact, it follows that {h λ ν λ : λ ∈ Λ} is a tight family.…”

Thermodynamic Formalism for Random Weighted Covering Systems

“…Mayer, Skorulski, and Urbański developed distance expanding random mappings in [33], which generalize the works of [25] and [7]. Simmons and Urbański established a variational principle and the existence of a unique relative equilibrium state for these maps in [41].…”