  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.
