报告题目：Equilibrium States beyond uniform hyperbolicity and IFS examples

报告人：Prof. Krerley, Oliveira(Universidade Federal de Alagoas)

报告时间：11月17日下午14:00-15:00

报告地点：数学楼2楼学术报告厅

报告摘要：

In this tak we discuss two recent results on uniqueness of equilibrium states for some Iterate Function Systems (IFS). 
 
In the first setting, we prove existence of relative  maximal entropy measures for certain random dynamical systems that are skew products of the type $F(x,y)=(/theta(x), f_x(y))$,  where $/theta$ is an invertible map preserving an ergodic measure   $/mathbb{P}$ on a Polish space and $f_x$ is a local diffeomorphism of a compact Riemannian manifold exhibiting some non-uniform expansion. As a consequence of our proofs, we obtain an integral formula for  the relative topological entropy as the integral the of logarithm of the topological degree of $f_x$ with respect to $/mathbb{P}$.   When $F$ is topologically exact and the supremum of the topological degree of $f_x$ is finite, the maximizing measure is unique and positive on open sets.
 
The second setting we discuss uniqueness of equilibrium states for Partially Hyperbolic Horseshoes studied in a previous article with R. Leplaideur and I. Rios and in a previous article by Diaz, Horita, Sambarino and Rios. These families of  horseshoes have interesting features, as dense sets of segments in its central direction on its non-wandering set.  They have heteroclinical cycles and are extensions of the golden shift. From one hand, they have phase transitions for smooth potentials. From the other hand, one expect uniqueness for potentials that are close to zero.  We make use of the semiconjugacy with the shift to characterize  the set of Holder potentials with unique equilibrium state