A C∞ diffeomorphism with infinitely many intermingled basins
Let M be the four-dimensional compact manifold $M = T2 × S2 and let $k\ge2$. We construct a $C^\infty$ diffeomorphism $F:M\to M$ with precisely k intermingled minimal attractors $A_1,\dotsc, A_k$. 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 Aj for some j, but every non-empty open set in M has a positive measure intersection with each basin. We also construct $F:M\to M$ with a countable infinity of intermingled minimal attractors. © 2005 Cambridge University Press.
Ergodic Theory and Dynamical Systems
Melbourne, I., & Windsor, A. (2005). A C∞ diffeomorphism with infinitely many intermingled basins. Ergodic Theory and Dynamical Systems, 25 (6), 1951-1959. https://doi.org/10.1017/S0143385705000325