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 an homeomorphism C-0-clo se to an expansive map of M. In general, it is not true that g is also expa nsive, 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 f uture and the past, we obtain an equivalence relation similar to The quotie nt space M/ similar to is a compact, metric space and g induces an expansiv e homeomorphism (g) over bar on that quotient. If M is a surface, we show t hat for any (x) over bar epsilon M/ similar to the connected component of t he local stable (unstable) set containing (x) over bar is nontrivial and ar e-wise connected. AMS classification scheme numbers: 58F30, 58F15, 54H20, 5 4F15.