Homeomorphisms of zero-dimensional spaces
The author investigates the ``size'' of a class of self-homeomorphisms of the Cantor set of interest in topological dynamics. Specifically he shows that the set of all expansive homeomorphisms with the pseudo orbit tracing property (POTP) of the Cantor set C is dense in the set of all homeomorphism. (\(\phi\) is expansive if there is a \(c>0\) such that for every \(x,y\in C\) with \(x\neq y\) there is \(n\in Z\) for which \(d(\phi^ n(x),\phi^ n(y))>c\). \(\phi\) has POTP if for every \(\epsilon >0\) there is a \(\delta >0\) such that if we have a sequence \(\{x_ i\); \(i\in Z\}\) with \(d(\phi (x_ i),x_{i+1})<\delta\) for every \(i\in Z\), then we can find \(y\in C\) with \(d(\phi^ i(y),x_ i)<\epsilon\) for every \(i\in Z).\) It is also shown that on the space \(\{\) 0,1/2,1/3,...\(\}\) the sets of expansive and POTP homeomorphisms are both dense but are disjoint. Finally the author gives an example of a zero dimensional space with no expansive homeomorphisms. (The identity map always hat POTP on a zero dimensional space).
scientific article; zbMATH DE number 4073951