On finite-dimensional attractors of homeomorphisms

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.