On Ilyashenko’s statistical attractors

Abstract

We define a minimal α-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 α-observable statistical attractors. Among other examples, we analyse the Bowen homeomorphisms with non-robust topological heteroclinic cycles. We prove the existence of three types of statistical behaviours for these examples.