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Conditionally invariant measures for Anosov maps with small holes

Published online by Cambridge University Press:  01 October 1998

N. CHERNOV
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, Birmingham, AL 35294, USA
R. MARKARIAN
Affiliation:
Instituto de Matemática y Estadística ‘Prof. Ing. Rafael Laguardia’, Facultad de Ingeniería, Universidad de la República, C.C. 30, Montevideo, Uruguay
S. TROUBETZKOY
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, Birmingham, AL 35294, USA

Abstract

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ 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 never enter holes under the past iterations of the map is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

Type
Research Article
Copyright
© 1998 Cambridge University Press

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