Generic surface homeomorphisms are almost continuum-wise expansive
Alfonso Artigue
TL;DR
This work addresses whether continuum-wise expansivity (cw-expansivity) is typical among surface homeomorphisms. It develops a decompositional framework and a cw-transversality mechanism to control intersections of stable and unstable continua, enabling perturbations that promote cw-expansive behavior. The authors prove that cw-expansive maps are dense on compact surfaces and that a generic map is almost cw-expansive, admitting a cw-expansive model via a monotone semiconjugacy with fibers of arbitrarily small diameter. The approach combines perturbations of diffeomorphisms to one-dimensional stable/unstable structures with quotient dynamics from derived-from-Anosov/pseudo-Anosov constructions, yielding a robust generic picture for surface dynamics with cw-expansive structure.
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 $ε>0$ there is a cw-expansive homeomorphism $g$ of $M$ which is $ε$-close to $f$ and is semiconjugate to $f$; moreover, if $π\colon M\to M$ is this semiconjugacy then $π^{-1}(x)$ is connected, does not separate $M$ and has diameter less than $ε$ for all $x\in M$.