Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin

Abstract

Abstract.We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive 3-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations. IntroductionThrough the numerical experiments and theoretical works of many researchers we have gained some understanding of a dynamical system which is "typical" in the "standard" setting; that is, the dynamical system / : M -► M is a smooth map and M is a smooth «-dimensional manifold. Special circumstances change the meaning of "typical" and add possibilities to the list of what we are likely to see, say, by requiring / to be in some particular subset of smooth maps on M. For example, if / is symplectic, then we see the KAM tori and other phenomena of Hamiltonian dynamics; or if / has symmetries, we are likely to see "symmetry-breaking" bifurcations. The setting we examine here is where / maps some fixed submanifold K c M of dimension k < m to itself ( / is invariant on K ). That is, we discuss some new phenomena which are "typical" in the sense that they occur for some large set of perturbations of / which continue to map K to itself. This is the natural setting for modeling physical systems with constraints or reflectional symmetries.We find some unexpected new limiting behaviors in the new setting. More precisely, the co-limit set of a point x g M is the set eo(x) = f)k (J¡>kfl{x) > the forward limit set of x under iteration by /. We say that a compact set A is an attractor if {x e M : eo(x) = A} has positive «-dimensional Lebesgue measure and the basin of attraction of A is 3$ (A) = {x e M : <y(x) c A} .In the standard setting the basins of attraction typically seem to be open sets, although the boundaries of these basins are often fractal sets [1]. Certainly, in the standard setting, it has long been known that the only structurally stable attractors are the Axiom A attractors, and these have open basins of attraction; see, for example, [2]. In the setting where we have a fixed invariant manifold K, we find that attractors may have basins with some radically new features. We find that some maps have several coexisting attractors whose basins are measure-theoretically dense in each other. That is, any open set which intersects one basin in a set of positive measure also intersects each of the other basins in a set of positive measure. We call such basins intermingled.

The control of chaos: theory and applications

“…Rigorous results on the dynamics of riddled basins for discrete maps were presented in Refs. [90,91]. The dynamics of riddled basins was subsequently investigated in [92] using a more realistic physical model.…”

Skew products of interval maps over subshifts

“…Such systems gained interest with an example by Kan [4] where they gave rise to intermingled basins. This example is over a full shift on two symbols and the end points of the interval are attracting on average.…”

Riddled basins in complex physical and biological systems

“…Hence we may regard the extinction of either species as an asymptotic state with a well-defined attractor in the phase space [25]. The observed extreme sensitivity on the initial condition has led to the hypothesis that the basins of these attractors are riddled [26]. In fact, the basins of attraction are intermingled, for each basin is riddled with holes belonging to the other basin.…”