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.
Categories: math.DS