Séminaires SymPA à venir

In the setting of smooth dynamical systems, the notion of non-uniform hyperbolicity was introduced by Y. Pesin, in order to show that certain maps, although not being hyperbolic, act as hyperbolic maps along almost every orbit. More precisely, Pesin theory deals with measure-preserving ergodic maps, satisfying that Lyapunov exponents do not vanish on a set of full measure.
In the setting of complex dynamics, similar ideas were developed by F. Przytycki, in order to show that certain features of hyperbolic maps on their Julia sets are accomplished by general rational maps almost everywhere on their Julia sets. Note that the maps considered are no longer globally invertible. The key ingredients are, on the one hand, the existence of an ergodic invariant measure supported on the Julia set, with positive Lyapunov exponent, and that singular values (i.e. points in which the function is not a local homeomorphism) are finite.
In this talk I will present F. Przytycki’s work on rational maps, and how one can extend it to transcendental maps (allowing infinitely many singular values). This is work in progress with Prof. Núria Fagella.