Nonhyperbolic step skew-products: Ergodic approximation
L.J. Díaz
Departamento de Matemática PUC-Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro 22451-900, BrazilK. Gelfert
Instituto de Matemática Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária – Ilha do Fundão, Rio de Janeiro 21945-909, BrazilM. Rams
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
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Abstract
We study transitive step skew-product maps modeled over a complete shift of k, , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents.
We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves.
Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.
Cite this article
L.J. Díaz, K. Gelfert, M. Rams, Nonhyperbolic step skew-products: Ergodic approximation. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 6, pp. 1561–1598
DOI 10.1016/J.ANIHPC.2016.10.009