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/