A $C^{\infty}$ diffeomorphism with infinitely many intermingled basins

Abstract

Abstract. Let M be the four-dimensional compact manifold M = T 2 × S 2 and let k ≥ 2. We construct a C ∞ diffeomorphism F : M → M with precisely k intermingled minimal attractors A 1 , . . . , 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 non-empty 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. We say that attractors A 1 , . . . , A k for a dynamical system are intermingled if the basins of attraction are intermingled. Similarly, we can speak of countably many intermingled sets/attractors. IntroductionNumerical evidence for the existence of intermingled attractors was first presented in Alexander et al [2] for a certain class of non-invertible maps of the plane. A proof is presented in [1]. They did not verify that the basins occupy a set of full measure, but did show that the regular parts of the basin (those characterized by typical Lyapunov exponents) are intermingled for a set of parameters with positive measure.

Anatole Katok

“…This itself is surprising, not least because elliptic islands around generic elliptic fixed points are an obstruction to ergodicity. Some experts called this work ‘the main event of the year in ergodic theory’, and the method has over the decades yielded ever more astonishing examples of dynamical systems [24–26, 31, 52, 87–91, 115, 116, 153, 156, 157, 238–241, 250, 279, 280]†. It is an ingenious tour de force in elliptic dynamics, and its framework owes to the – theory of periodic approximations.…”

Measures maximizing the entropy for Kan endomorphisms

“…Given ǫ > 0, taking a forward iterate large enough, we can assume that γ j transversally intersects the rectangle B 0 (ǫ) defined by (23) in a set γ j µ 0 = γ j ∩B 0 (ǫ) such that ν 0 (π(γ j µ 0 )) > 0.…”

“…Since Kan's work, the intermingled phenomenon has been the subject of various studies. For instance, there are some new examples that have more than two physical measures, and their basins are intermingled (see [1,23,6,14]). Some accurate estimations for the Hausdorff dimension of the set that does not belong to the basins of physical measures were obtained by [17,20].…”

On the non-robustness of intermingled basins

“…The situation is different in the absence of domination as showed by Fayad [11]. Inspired in the Fayad example, Melbourne and Windsor [20] give a family of C ∞ -diffeomorphisms on T 2 × S 2 with arbitrary number of physical measures with intermingled basins.…”