The structure of the space of ergodic measures of transitive partially hyperbolic sets

Abstract

We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an "exposed" piece which is poorly homoclinically related with the rest and a "core" with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic measures.Measures of maximal entropy are discussed in more detail. We present examples where the measure of maximal entropy is nonhyperbolic. We also present examples where the measure of maximal entropy is unique and nonhyperbolic, however in this case the dynamics is nontransitive.

Mingled hyperbolicities: Ergodic properties and bifurcation phenomena (an approach using concavity)

“…In item a) in the above corollary, we call the measures µ and µ twin-measures. Similar phenomena are observed in [22,51].…”

Alsedà–Misiurewicz systems with place-dependent probabilities*

“…The general overview of the theory of groups acting on the circle and real line may be found in book and article by Navas [34,35]. Moreover, certain examples of skew-products with the Bernoulli shift in the base are investigated as models of partially hyperbolic dynamics (see [14][15][16]29]).…”