NONHYPERBOLIC ERGODIC MEASURES

Abstract

We discuss some methods for constructing nonhyperbolic ergodic measures and their applications in the setting of nonhyperbolic skew-products, homoclinic classes, and robustly transitive diffeomorphisms. to Wellington de Melo in memoriam 1 The transitive and nonhyperbolic setting Irrational rotations of the circle T 1 and Anosov maps of the two-torus T 2 are emblematic examples of transitive systems (existence of a dense orbit). Small perturbations of Anosov systems are also transitive. This property fails however for irrational rotations. Anosov diffeomorphisms are also paradigmatic examples of hyperbolic maps and, by definition, hyperbolicity persists by small perturbations. Our focus are systems which are robustly transitive. In dimension three or higher, there are important examples of those systems that fail to be hyperbolic. They are one of the main foci of this paper. A second focus is on nonhyperbolic elementary pieces of dynamics. We discuss how their lack of hyperbolicity is reflected at the ergodic level by the existence of nonhyperbolic ergodic measures. We also study how this influences the structure of the space of measures. In this discussion, we see how this sort of dynamics gives rise to robust cycles and blenders. This research has been partially supported by CNE-Faperj and CNPq-grants (Brazil). The author warmly thanks J. Bochi, Ch. Bonatti, S. Crovisier, K. Gelfert, A. Gorodetski, Y. Ilyashenko, D. Kwietniak, J. Palis, M. Rams, A. Tahzibi, C. Vásques, and J. Yang for their useful comments and conversations. MSC2010: primary 37D25; secondary 37D30, 28D20, 28D99.