On Ilyashenko’s Statistical Attractors

We revisit the notion of statistical attractor defined by Ilyashenko. We argue to show that they are optimally defined to exist and to describe the asymptotical statistics of Lebesgue-almost all the orbits. We contribute to the theory, defining a minimality concept of α-observable statistical attractors and proving that the space is always full Lebesgue decomposable into pairwise disjoint sets that are Lebesgue-bounded away from zero and included in the basins of a finite family of minimally observable statistical attractors. We illustrate the abstract theory including, among other examples, the Bowen homeomorphisms with non robust topological heteroclinic cycles. We prove the existence of three types of statistical behaviors for these examples.