On finite-dimensional global attractors of homeomorphisms

Abstract

Let E be a normed linear space and suppose that A is the global attractor of either (i) a homeomorphism F : E → E or (ii) a semigroup S(·) on E that is injective on A. In both cases, A has trivial shape, and the dynamics on A can be described by a homeomorphism F : A → A (in the second case we set F = S(t) for some t > 0). If the topological dimension of A is finite, then we show that for any ϵ > 0 there is an embedding e: A → Rk, with k ∼ dim(A), and a (dynamical) homeomorphism f : Rk → Rk such that F is conjugate to f on A (that is, F|A = e−1 ◦ f ◦ e) and f has a global attractor Af with e(A) ⊆ Af ⊆ N(e(A), ϵ). In other words, we show that the dynamics on A are essentially finite dimensional. We characterise subsets of Rn that can be the global attractors of homeomorphisms as cellular sets, give elementary proofs of various topological results connected to Borsuk’s theory of shape and cellularity in Euclidean spaces, and prove a controlled homeomorphism extension theorem. We also show that we could achieve e(A) = Af under the assumption of a stronger controlled homeomorphism extension theorem.