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Topological entropy of polygon exchange transformations and polygonal billiards

Published online by Cambridge University Press:  02 April 2001

EUGENE GUTKIN and
NICOLAI HAYDN
EUGENE GUTKIN
Affiliation:
Mathematics Department, USC, Los Angeles, 90089-1113, USA (e-mail: egutkin@mtha.usc.edu) (e-mail: nhaydn@mtha.usc.edu)
NICOLAI HAYDN
Affiliation:
Mathematics Department, USC, Los Angeles, 90089-1113, USA (e-mail: egutkin@mtha.usc.edu) (e-mail: nhaydn@mtha.usc.edu)
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Abstract

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We study the topological entropy of a class of transformations with mildsingularities: the generalized polygon exchanges.This class contains, in particular, polygonal billiards. Our main result is ageometric estimate, from above, on the topologicalentropy of generalized polygon exchanges. One of the applications of ourestimate is that the topological entropy of polygonalbilliards is zero. This implies the subexponential growth of variousgeometric quantities associated with a polygon.Other applications are to the piecewise isometries in two dimensions, and tobilliards in rational polyhedra.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 17 , Issue 4 , August 1997, pp. 849 - 867
Copyright
© 1997 Cambridge University Press