Towards computer-assisted proofs for stochastic dynamics in the quadratic family
Stefano Luzzatto (ICTP, Itália)

The quadratic family of one-dimensional maps contains an incredible richness of dynamical phenomena which depend in a very complex way on the parameter in the interval [1.4,2]. In the 1980s and 1990s some deep results were obtained by various people, which showed that the only two phenomena which occur for positive Lebesgue measure ets of parameters are: regular dynamics, where there exists a (unique) attracting periodic orbit, and stochastic dynamics, where there exists a (unique) invariant probability measure absolutely continuous with respect to Lebesgue. Moreover, the corresponding sets of parameters have very dierent topological structures: the set of regular parameters is open and dense in the parameter space whereas the set of stochastic parameters is nowhere dense (albeit of positive Lebesgue measure).
Even though regular and stochastic dynamics are completely dierent, it turns out to be extremely dicult to distinguish them in practice and to decide if any given parameter is regular or stochastic. It is also completely unknown what the actual measures of the two parameter sets, or if one is signicantly greater than the other, nor are there any heuristics to argue in favour of one or the other. I will give an overview of the results above, discuss the somewhat "philosophical” question about the dierence between regular and stochastic dynamics, and present some work in progress towards a computer-assisted approach to estimating the measure of regular and stochastic dynamics.