Institute for Mathematical Physics Conditionally Invariant Measures for Anosov Maps with Small Holes Conditionally Invariant Measures for Anosov Maps with Small Holes

We study Anosov diieomorphisms on surfaces in which some smalìholes' are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories stay away from holes in the past is a Cantor-like union of unstable bers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable bers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.