Multidimensional bony attractors

Abstract

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., "big." It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.

Attractors. Then and now

“…First of all, there are local attractors which are very natural for the modern theory of dynamical systems (see, for example, [116]) and which attract only the trajectories starting from some (small) neighbourhood of the attractor. A more advanced version is a Milnor attractor, where the attraction property holds for the initial data up to a zero-measure set: see [176], see also [48], [88], [100], [108], and [202] for more delicate versions of attractors.…”