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On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces

Published online by Cambridge University Press:  18 January 2012

JEAN-PIERRE CONZE  and
EUGENE GUTKIN
JEAN-PIERRE CONZE
Affiliation:
IRMAR, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France (email: conze@univ-rennes1.fr)
EUGENE GUTKIN
Affiliation:
Nicolaus Copernicus University, Chopina 18/12, Torun 87-100, Poland IM PAN, Sniadeckich 8, 00-956 Warszawa, Poland (email: gutkin@mat.umk.pl, gutkin@impan.pl)
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Abstract

We study billiard dynamics on non-compact polygonal surfaces with a free, cocompact action of ℤ or ℤ2. In the ℤ-periodic case, we establish criteria for conservativity. In the ℤ2-periodic case, we study a particular family of such surfaces, the rectangular Lorenz gas. Assuming that the obstacles are sufficiently small, we obtain the ergodic decomposition of directional billiards for a finite but asymptotically dense set of directions. This is based on our study of ergodicity for ℤd-valued cocycles over irrational rotations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 491 - 515
Copyright
Copyright © Cambridge University Press 2012

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