Continuum-wise expansive homeomorphisms on Peano continua

Abstract

On a Peano continuum, all local stable and unstable components of a continuum-wise expansive homeomorphism are non trivial. In particular, there is sensitive dependence on initial conditions. This generalizes results in \cite{h,l} about lack of Lyapunov stable points (weak sinks) and existence of non trivial stable and unstable components for expansive homeomorphisms on Peano continua. We also use this fact to generalize a result in \cite{ktt}: a Peano curve $X$ admitting a continuum-wise expansive homeomorphism is nowhere rim-countable. However, it is not resolved the question of whether such a dynamics could be possible in a locally planar Peano curve. Some other questions are posed.