A C^∞ diffeomorphism with infinitely many intermingled basins.

Ergodic Theory Dyn. Syst. 25 (2005) 1951-1959

Ian Melbourne and Alistair Windsor

Abstract

Let M be the four-dimensional compact manifold M=T²xS² and let k>1. We construct a C^∞ diffeomorphism F : M -> M with precisely k intermingled minimal attractors A₁,...,A_k. Moreover the union of the basins is a set of full Lebesgue measure. This means that Lebesgue almost every point in M lies in the basin of attraction of A_j for some j, but every open set in M has a positive measure intersection with each basin.

We also construct F : M -> M with a countable infinity of intermingled minimal attractors.