On finite-dimensional attractors of homeomorphisms

TL;DR

This paper demonstrates that finite-dimensional attractors of homeomorphisms can be embedded into finite-dimensional Euclidean spaces with conjugate dynamics, showing their essentially finite-dimensional nature.

Contribution

It establishes that finite-dimensional attractors of homeomorphisms are topologically equivalent to attractors in Euclidean spaces, providing new embedding and conjugacy results.

Findings

Attractors have trivial shape and finite topological dimension.

Existence of embeddings into Euclidean space with conjugate dynamics.

Characterization of attractors as cellular sets in Euclidean spaces.

Abstract

Let $E$ be a linear space and suppose that $A$ is the global attractor of either (i) a homeomorphism $F:E\rightarrow E$ or (ii) a semigroup $S(\cdot)$ 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 $F:A\rightarrow A$ (in the second case we set $F=S(t)$ for some $t>0$). If the topological dimension of $A$ is finite we show that for any $\epsilon>0$ there is an embedding $e:A\rightarrow{\mathbb R}^k$, with $k\sim{\rm dim}(A)$, and a (dynamical) homeomorphism $f:\R^k\rightarrow\R^k$ such that $F$ is conjugate to $f$ on $A$ (i.e.\ $F|_A=e^{-1}\circ f\circ e$) and $f$ has an attractor $A_f$ with $e(A)\subset A_f\subset N(e(A),\epsilon)$. In other words, we show that the dynamics on $A$ is essentially finite-dimensional. We characterise subsets of ${\mathbb R}^n$ that can be the 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.