Mingled hyperbolicities: Restricted variational principles

We quantify ergodically the lack of hyperbolicity in transitive diffeomorphisms with one-dimensional nonhyperbolic center. For that, we investigate step skew products whose fiber dynamics are circle diffeomorphisms. Such dynamics captures the key mechanisms of the dynamics of robustly transitive and nonhyperbolic maps with one-dimensional center. It also arises from the projective action of certain $2\times 2$ elliptic matrix cocycles.
A key feature of these systems is the coexistence of saddles of different types of hyperbolicity, described in terms of fiber-contracting and -expanding regions which are mingled by the dynamics. It gives also rise to nonhyperbolic ergodic measures characterized in terms of a zero Lyapunov exponent in the circle-fiber direction. Some of those measures have positive entropy.
We describe the topological entropy of each level set of points with fiber-Lyapunov exponent $\alpha$ in terms of a restricted variational principle. Here $\alpha$ takes negative and positive values and also $\alpha=0$. We will particularly focus on the latter case. For that we construct a nonhyperbolic ergodic measure of high entropy inspired by a periodic orbit approximation-technique of Gorodetski, Ilyashenko, Kleptsyn, and Nalski.
The talk is based on a joints works with K. Gelfert (UFRJ, Brazil) and M. Rams (IMPAN, Poland).