09:30 to 10:15 
Sylvain Crovisier (Université ParisSud, France) 
Lecture II  Homoclinic classes and equilibrium measures The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this minicourse we will be interested by the nonuniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.
We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, BuzziCrovisierSarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.
Reference: arXiv:1811.02240.



10:45 to 11:30 
Stefano Luzzatto (ICTP, Italy) 
Lecture II  Hyperbolicity and Physical Measures A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.



11:45 to 12:30 
Dalia Terhesiu (Leiden University, Netherlands) 
Rates of mixing for the periodic Z^d billiard map (infinite horizon) Optimal rates of mixing (of arbitrarily high order) for the periodic Z^d billiard map with finite horizon have been obtained recently by F. Pene. First order mixing for the infinite horizon case is also known. Rates of mixing (higher order mixing) in the infinite horizon case poses new challenges. In joint work with F. Pene, we obtain second order mixing.



14:30 to 14:50 
Peyman Eslami (University of Roma Tor Vergata , Italy) 
Inducing schemes for piecewise expanding maps of R^n For piecewise expanding maps of Rn Rn I will show how to construct an inducing scheme where the base map is GibbsMarkov and the return times have exponential tails. The existence of such a structure has many consequences in regards to the statistical properties of systems with discontinuities and nonuniform expansion.



14:50 to 15:10 
Gabriella Keszthelyi (Alfréd Rényi Institute of Mathematics, Hungary) 
Dynamical properties of biparametric skew tent maps (joint work with Zoltan Buczolich) We consider skew tent maps Tα,β(x) with Tα,β(x) = βαx for 0 ≤ x ≤ αand Tα,β(x) = β1−α(1 − x) for α < x ≤ 1. With this choice of parametersTα,β maps [0, 1] into [0, 1] and (α, β) is the vertex of Tα,β. The dynamics ofTα,β for (α, β) ∈ [0, 1]2
is interesting when (α, β) ∈ U = {(α, β) : 0.5 < β ≤1, 1 − β < α < β}. Denote by h(α, β) the topological entropy of Tα,β. It is wellknown that h(α, β) is strictly monotone increasing along vertical line segments in U. It is natural to ask what happens if we move in the horizontal direction, that is for fixed β we consider h(α) = h(α, β), for (α, β) ∈ U. Turned out that h(α) is strictly monotone increasing on (1 − β, β). To deal with this question one needs to consider equitopological entropy curves in the square. We denote by M = K(α, β) the kneading sequence of Tα,β and by Λ = Λα,β its Lyapunov exponent. For a given kneading squence M we consider isentropes (or equitopological entropy, or equikneading curves), (α, ΨM(α)) such that K(α, ΨM(α)) = M. On these curves the topological
entropy h(α, ΨM(α)) is constant. We show that Ψ0 M(α) exists and the Lyapunov exponent Λα,β can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function ΘM a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.



15:10 to 15:30 
Luciana Silva Salgado (Federal University of Rio de Janeiro, Brazil) 
Weak Hyperbolicity for Singular Flows Dominated splittings perhaps is the first notion of weak hyperbolicity that have emerged after the uniform hyperbolic theory.
In the case of singular flows, the study of Lorenz attractors have inspired the definition of the so called Singular Hyperbolicity. Nowadays, there are several other notions of weak hyperbolicity for
flows beyond the classical singular hyperbolicity one.
In this talk, I am interested in weak and uniform hyperbolic structures in a broad sense and its characterizations, mainly by using the notion of infinitesimal Lyapunov functions.
References:
[1] L. S. Salgado. Singular hyperbolicity and sectional Lyapunov exponents of various orders. Proc. of the Amer. Math. Soc., Vol. 147, doi.org/10.1090/proc/14254. 2019.
[2] L. S. Salgado. Star flows: a characterization via Lyapunov functions. preprint ArXiv1704.01987. 2017.
[3] V. Araujo, L. S. Salgado. Infinitesimal Lyapunov functions for singular flows. Math. Z., 275, 34, 863–897. 2013.
[4] A. Arbieto, L. S. Salgado. On critical orbits and sectional hyperbolicity of the nonwandering set for flows. J. of diff. equations 250 (2011) 2927–2939.



16:00 to 17:30 
Marcelo Viana (IMPA, Brazil) 
Lyapunov exponents, from the 1960's to the 2020's The ergodic theory of Lyapunov exponents, initiated by the work of Furstenberg and Kesten at the dawn of the 1960s, has been a remarkably active field of mathematics over all these decades and, through the works of Oseledets, Pesin and many, many others, found extremely important applications in the realm of smooth dynamics. The speaker will survey some of the classical results and more recently developments, with a view to the future of the field.


