Generic surface homeomorphisms are almost continuum-wise expansive

Abstract

We show that for a compact surface without boundary $M$ the set of cw-expansive homeomorphisms is dense in the set of all the homeomorphisms of $M$ with respect to the $C^0$ topology. After this we show that for a generic homeomorphism $f$ of $M$ it holds that: for all $\epsilon>0$ there is a cw-expansive homeomorphism $g$ of $M$ which is $\epsilon$-close to $f$ and is semiconjugate to $f$; moreover, if $\pi\colon M\to M$ is this semiconjugacy then $\pi^{-1}(x)$ is connected, does not separate $M$ and has diameter less than $\epsilon$ for all $x\in M$.