We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of contains a gap and hence gives rise to a first order phase transition. A major part of the proofs relies on the analysis of an associated iterated function system that is genuinely non-contracting.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-2, author = {Lorenzo J. D\'\i az and Katrin Gelfert}, title = {Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions}, journal = {Fundamenta Mathematicae}, volume = {219}, year = {2012}, pages = {55-100}, zbl = {1273.37027}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-2} }
Lorenzo J. Díaz; Katrin Gelfert. Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions. Fundamenta Mathematicae, Tome 219 (2012) pp. 55-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-2/