Thick attractors with intermingled basins

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop theory for it and use it in particular to provide examples of thick metric attractors with intermingled basins.

💡 Research Summary

The paper investigates metric attractors in the sense of Milnor and introduces the notion of “thick” attractors—closed invariant sets whose basin has positive but not full Lebesgue measure (0 < μ(A) < 1). While classical examples such as Kan’s map produce attractors whose basins have zero volume, the authors aim to construct attractors that occupy a substantial portion of phase space and whose basins are intermingled, i.e., each basin is dense in the other in a measure‑theoretic sense.

The central technical device is a random walk performed along the orbit of a deterministic dynamical system, where the step direction (±1) is chosen with a probability that depends continuously on the current position. Formally, given a compact manifold N, a diffeomorphism f : N→N and a continuous function p : N→