Attractors in Euclidean spaces and shift maps on polyhedra (Q2705820)

Let \((X,d)\) be a separable metric space, let \(F:X\rightarrow X\) be a homeomorphism and let \(\Lambda\) be a compact subset of \(X\). The set \(\Lambda\) is called an attractor of \(F\) provided that there is an open neighbourhood \(U\) of \(\Lambda\) such that \(F(\text{Cl } U)\subset U\) and \(\Lambda=\bigcap_{n=0}^\infty F^n(U)\). If moreover \(\lim_{n\rightarrow \infty} d(F^n(x),\Lambda)=0\) for any \(x\in X\) then \(\Lambda\) is called a global attractor of \(F\). NEWLINENEWLINENEWLINEAssume additionally that \(X\) is compact and let \(f:X\rightarrow X\) be continuous. The inverse limit space for \(f\) is defined by \((X,f)=\{(x_i)_{i=1}^\infty: x_i\in X\) and \(f(x_{i+1})=x_i\) for any \(i\geq 1\}\). Then \(\widetilde{f}:(X,f)\rightarrow (X,f)\) given by \(\widetilde{f}(x_1,x_2,\ldots)=(f(x_1),x_1,\ldots)\) is a homeomorphism which is called the shift map of \(f\). NEWLINENEWLINENEWLINEThe main results of the paper are the following: NEWLINENEWLINENEWLINE(i) If \(X\) is a compact absolute neighbourhood retract (in particular, a compact polyhedron) in \({\mathbb R}^n\) and \(f:X\rightarrow X\) is continuous, then there is a homeomorphism \(F:{\mathbb R}^{2n}\rightarrow {\mathbb R}^{2n}\) such that \((X,f)\) is (homeomorphic to a set) contained in \({\mathbb R}^{2n}\), \(F\) is an extension of the shift map \(\widetilde{f}:(X,f)\rightarrow (X,f)\), and \((X,f)\) is an attractor of \(F\). Moreover, if \(X\) is an absolute retract (in particular, a contractible polyhedron), then \(F\) can be chosen so that \((X,f)\) is a global attractor of \(F\). NEWLINENEWLINENEWLINE(ii) If \(f:S\rightarrow S\) is a continuous map of the unit circle \(S\) then there is a homeomorphism \(F:{\mathbb R}^3\rightarrow {\mathbb R}^3\) such that \((S,f)\) is contained in \({\mathbb R}^3\), \(F\) is an extension of the shift map \(\widetilde{f}\), and \((S,f)\) is an attractor of \(F\). NEWLINENEWLINENEWLINE(iii) Let \(\Lambda\) be compact and let \(f:\Lambda\rightarrow \Lambda\) be a homeomorphism. Then the following statements are equivalent: (a) There are a topological manifold (with or without boundary) \(M\) containing \(\Lambda\) and a homeomorphism \(F:M\rightarrow M\) such that \(\Lambda\) is an attractor of \(F\) and \(F\) is an extension of \(f\); (b) there are a finite dimensional compact absolute neighbourhood retract (or, equivalently, a compact polyhedron) \(P\) and a continuous map \(g:P\rightarrow P\) such that \(f\) is topologically conjugate to the shift map \(\widetilde{g}:(P,g)\rightarrow (P,g)\), that is, there is a homeomorphism \(h:\Lambda\rightarrow (P,g)\) such that \(\widetilde{g}=hfh^{-1}\). NEWLINENEWLINENEWLINEFrom Theorem (iii), which answers a question by \textit{B. Günther} and \textit{J. Segal} [Proc. Am. Math. Soc. 119, No. 1, 321-329 (1993; Zbl 0822.54014)], it follows that a compact subset \(\Lambda\) of a topological manifold is an attractor of some homeomorphism if and only if there is a map \(g:P\rightarrow P\) of a finite dimensional compact absolute neighbourhood retract \(P\) such that \(\Lambda\) is homeomorphic to \((P,g)\).