EXPANSIVE HOMEOMORPHISMS WITH THE SHADOWING PROPERTY ON ZERO DIMENSIONAL SPACES

Let X = {a} ${\cup}$ {$a_{i}$ ${$\mid$}$i $\in$ N} be a subspace of Euclidean space $E^2$ such that $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a and $a_{i}\;{\neq}\;a_{j}$ for $i{\neq}j$. Then it is well known that the space X has no expansive homeomorphisms with the shadowing property. In this paper we show that the set of all expansive homeomorphisms with the shadowing property on the space Y is dense in the space H(Y) of all homeomorphisms on Y, where Y = {a, b} ${\cup}$ {$a_{i}{$\mid$}i{\in}Z$} is a subspace of $E^2$ such that $lim_{i}$-$\infty$ $a_{i}$ = b and $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a with the following properties; $a_{i}{\neq}a_{j}$ for $i{\neq}j$ and $a{\neq}b$.