Random Graph Directed Markov Systems

We introduce and explore random conformal graph directed Markov 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 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 : EP (t) ≤ 0}, 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.