Stable and unstable sets of C 0 perturbations of expansive homeomorphisms of surfaces.

Let M be a compact metric space and g : M → M be a homeomorphism C 0 -close to an expansive map of M. In general, it is not true that g is also expansive, but it still has some properties resembling the expansivity. In fact, if we identify pairs of points whose g-orbits stay nearby, both for the future and the past, we obtain an equivalence relation ∼. The quotient space M/ ∼ is a compact metric space and g induces an expansive homeomorphism ge on that quotient. If M is a surface, we show that for any xe ∈ M/ ∼, the connected component of the local stable (unstable) set containing xe is nontrivial and arcwise connected.