Generic homeomorphisms have no smallest attractors

Abstract

We show that if M is a compact manifold then there is a residual subset £% of the set of homeomorphisms on M with the property that if f £31 then / has no smallest attractor (that is, an attractor with the property that none of its proper subsets is also an attractor). Part of the motivation for this result comes from portions of a recent paper by Lewowicz and Tolosa that deal with properties of smallest attractors of generic homeomorphisms. In this paper we consider the set of homeomorphisms of a compact manifold M with metric d ; this set of homeomorphisms is denoted as Hom(Af) and is given the topology of uniform convergence, which makes Hom(Ai) a Baire space. An attractor for / e Hom(Af ) is a compact, nonempty subset A of M with the following property^ there is an open neighborhood U of A satisfying the two conditions (i) f(U) C U and (ii) n>0/(i/) = A. We will refer to such an open set U as an attractor block that determines A (this is slightly different from the standard definition of an attractor block). Note that M itself is an attractor for any / £ Hom(Af) . The attractor A is a smallest attractor for / if no proper subset of A is also an attractor for / ; in other words, a smallest attractor is one that is minimal in the class of all attractors of / when they are partially ordered by inclusion. The main result of this note is that there is a residual subset & of Hom(Af) with the property that if g e 31 then g has no smallest attractor.