The nonexistence of expansive homeomorphisms of $1$-dimensional compact ANRs

Abstract

It is well known that if $X$ is an arc or a circle, then there is no expansive homeomorphism on $X$ (see [2] and [3]). In this note, we show that if $X$ is a Peano continuum which has a neighborhood $M$ such that ${\text {cl}}\left ( M \right )$ is a $1$-dimensional AR, then there is no expansive homeomorphism on $X$ . In particular, no $1$-dimensional compact ANR admits an expansive homeomorphism.