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.

We study certain one-parameter families of partially hyperbolic maps Ft : Σ 2 ×R → Σ 2 ×R of skew-product type generating so-called porcupinelike horseshoes. Such sets are topologically transitive and semiconjugate to the shift map in two symbols. They exhibit a very rich fiber structure characterized by the fact that the set Σ 2 is the disjoint union of two dense and uncountable subsets with opposite behavior: corresponding spines (preimage of a sequence by the semiconjugation) are nontrivial and trivial, respectively, that is, the semiconjugation is noninjective and injective, respectively. We will study the bifurcation process of creation and annihilation of nontrivial spines as the parameter t evolves. In particular, we focus on the Hausdorff dimension of these subsets of Σ 2. This study illustrates the richness of the process.

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