Andrey Gogolev (Ohio State University, Estados Unidos)
Rigidity in dynamics and geometry
Abstract: When a weak form of equivalence of two dynamical systems, such as coincidence of some numerical invariants, implies a stronger equivalence we say that dynamics exhibits a rigidity phenomenon. Rigidity occurs in various dynamical contexts ranging from elliptic to hyperbolic. I will survey some classical examples of rigidity in the first part of the mini-course. In the second half I will focus on newly discovered rigidity phenomena in hyperbolic dynamics. I will also explain an interplay of dynamical rigidity and geometric rigidity. In particular, the effect of rigidity results for hyperbolic flows on marked length spectrum rigidity problems for negatively curved manifolds.
Anna Miriam Benini (Universita' di Parma, Itália)
A current-free introduction to complex Henon maps
Abstract: Complex (polynomial) Henon maps are polynomial automorphisms of $\mathbb{C}^2$ whose dynamics was first studied by Hubbard and Oversteegen, then developed by Bedford-Smillie and Fornaess-Sibony independently, and is currently studied by many authors including some of the original ones.
One of the reasons for their popularity is a result by Friedland and Milnor according to which all polynomial automorphisms with positive entropy are conjugate to a finite composition of such maps. Another one, is that they represent one of the easiest complex dynamical systems that one may study in $\mathbb{C}^2$; yet another motivation, is that under appropriate conditions one can recover dynamical features of the most beloved families of unicritical polynomials of the complex plane.
While much of the contemporary research uses advanced geometric tools like currents and algebraic geometry, we will give an introduction to the basic dynamical features which do not require such tools. Only some knowledge of dynamical systems and complex analysis is required.
Cecilia González Tokman (University of Queensland, Austrália)
Ergodic theory for random dynamical systems
Abstract: This minicourse starts with an introduction to random dynamical systems, including simple and elaborate examples. It covers recent developments in ergodic theory, yielding fundamental information about statistical properties as well as the long- and medium-term behavior of such systems.
Sylvain Crovisier (Université Paris-Saclay, França)
Ergodic theory of non-uniformly hyperbolic diffeomorphisms
Abstract: The dynamics of uniformly hyperbolic diffeomorphisms is well understood: it decomposes into a finite number of basic sets, most of their orbits equidistribute towards natural invariant measures (physical or maximizing the entropy), and their statistics are described by limit theorems. (See for instance Bowen’s monograph.)
In these lectures we develop a framework that allows us to extend this analysis to general diffeomorphisms. Our approach consists in focusing on an invariant subset of the dynamics (the non-uniformly hyperbolic locus). We introduce a property - the strong positive recurrence - that allows to recover the main statistical features of uniformly hyperbolic systems. This applies in particular to smooth surface diffeomorphisms with positive topological entropy. Part of the results that we will present have been obtained in collaboration with J. Buzzi and O. Sarig.