On Ilyashenko's Statistical Attractors

TL;DR

This paper introduces a new concept of minimal alpha-observability for Ilyashenko's statistical attractors, demonstrating how the space decomposes into basins of these attractors and analyzing specific examples like Bowen homeomorphisms.

Contribution

It defines minimal alpha-observability for statistical attractors and proves a space decomposition into basins of finitely many such attractors, including analysis of Bowen homeomorphisms.

Findings

Space decomposes into disjoint basins of minimal alpha-observable attractors.

Existence of three types of statistical behaviors in Bowen homeomorphisms.

Examples include non-robust topological heteroclinic cycles.

Abstract

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.