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Growth rates for geometric complexities and counting functions in polygonal billiards

Published online by Cambridge University Press:  01 August 2009

EUGENE GUTKIN  and
MICHAŁ RAMS
EUGENE GUTKIN
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil UMK, Chopina 12/18, Torun 87-100, Poland (email: gutkin@impa.br, gutkin@mat.uni.torun.pl)
MICHAŁ RAMS
Affiliation:
IMPAN, S̀niadeckich 8, 00-956 Warszawa 10, Poland (email: rams@impan.gov.pl)
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Abstract

We introduce a new method for estimating the growth of various quantities arising in dynamical systems. Applying our method to polygonal billiards on surfaces of constant curvature, we obtain estimates almost everywhere on direction complexities and position complexities. As a byproduct, we recover the power bounds of Boshernitzan on the number of billiard orbits between almost all pairs of points in a planar polygon [M. Boshernitzan. Private letter to H. Masur (1986)].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1163 - 1183
Copyright
Copyright © Cambridge University Press 2008

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