97-266 N. Chernov and R. Markarian

Anosov maps with rectangular holes. Nonergodic cases (69K, LATeX) May 13, 97

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Abstract. We study Anosov diffeomorphisms on manifolds in which some `holes' are cut. The points that are mapped into our holes will disappear and never return. We study the case where the holes are rectangles of a Markov partition. Such maps with holes generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of our map is a Cantor-like set we call a {\it repeller}. In our previous paper, we assumed that the map restricted to the remaining rectangles of the Markov partition is topologically mixing. Under this assumption we constructed invariant and conditionally invariant measures on the sets of nonwandering points. Here we relax the mixing assumption and extend our results to nonmixing and nonergodic cases.

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