Generation of spines in porcupine-like horseshoes

Abstract

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.