30º SIES: 20 de agosto de 2026 (quinta-feira)

We construct an example of a smooth (C-infinity) circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point. By relaxing the smooth condition at a single point, we also construct an example where the basin of the physical measure has full measure. A key technical step is a realization method of independent interest, which gives a canonical way to construct a full branch map given its induced map.

Given an ergodic measure, Lyapunov exponents can be viewed as functions of the linear cocycle. A natural question is to understand under which conditions this map is continuous. In the setting of continuous cocycles taking values in SL(2,R), Bochi and Mañé (2002) obtained a complete characterization of this behavior. When considering finer topologies, subsequent results by Backes, Brown, and Butler provide sufficient conditions for continuity in the context of Hölder continuous cocycles.
In this talk, we discuss examples that do not satisfy these conditions and exhibit discontinuity of the Lyapunov exponents. More precisely, we show that the cocycles under consideration can be approximated by cocycles with zero Lyapunov exponents. In particular, these examples generalize the example constructed by Bocker and Viana (2010). This is joint work with Edhin Mamani.

We give a complete classification of $\mathrm{cw}_F$-hyperbolic surface homeomorphisms. More precisely, we prove that a surface homeomorphism is $\mathrm{cw}_F$-hyperbolic if and only if it is pseudo-Anosov and all of its singularities are one-prongs, which we call spines. We then classify these systems up to topological conjugacy: every such map is conjugate either to an Anosov automorphism of the torus $\mathbb{T}^2$ or to the standard quotient of such an automorphism on the sphere $\mathbb{S}^2$. As a consequence, this type of dynamics is obstructed on orientable surfaces of genus greater than one, as well as on the Klein bottle and the projective plane. This is joint work with Carvalho, Sarmiento, and Oprocha.

In recent years, two tools have stood out in the study of the statistical properties of chaotic systems: anisotropic Banach spaces and standard pairs. The former provides a method for applying transfer operators to systems with some hyperbolicity, overcoming the challenge of different behaviors in the stable and unstable directions. Standard pairs, on the other hand, are measures supported on unstable curves that are frequently combined with coupling techniques and other probabilistic ideas to prove the exponential decay of correlations and other mixing properties. In this talk, I will explain how to use standard pairs to construct anisotropic norms for a class of piecewise hyperbolic systems.