Georgia Institute of Technology College of Sciences

We introduce the notion of TDFOE studying the thermodynamic formalism on topological Markov shifts where the potential is random, depending on a Gibbs random walk on a compact metric space.

We introduce the associated Pressure Out of Equilibrium, and discuss its properties (e.g. variational principle), and applications (Azuma inequality for potentials expanding on average w.r.t. to a Gibbs measure).

We discuss the associated Semi-Ruelle operator which generates the POE, and its properties (e.g. existence of conformal measures, spectral radius). We provide an application by studying Effectively Expanding on Average diffeomorphisms on closed manifolds (this is a $C^1$-open condition, extends previously studied conditions).

We prove quasi-compactness (and later a spectral gap) for the Averaged Semi-Ruelle operator on a Sobolev space.

Notably, the diffeos are allowed to be highly dissipative. Finally, we mention a few applications such as: Dimension bounds on the stationary measure, and CLT and LLT for the (not necessarily invariant) volume measure.