On Ilyashenko's Statistical Attractors

We define a minimal alpha-observability of Ilyashenko's statistical attractors. We prove 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 minimal alpha-observable statistical attractors. Among other examples, we analyze the Bowen homeomorphisms with non robust topological heteroclinic cycles. We prove the existence of three types of statistical behaviours for these examples.