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Topological entropy of generalized polygon exchanges
Authors:
Eugene Gutkin and Nicolai Haydn
Journal:
Bull. Amer. Math. Soc. 32 (1995), 50-56
MSC:
Primary 58F11; Secondary 54H20, 58F99
MathSciNet review:
1273398
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Abstract: We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons and rational polytops have zero topological entropy.
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E. Gutkin and N. Haydn, Generalized polygon exchanges: 1. Topological entropy, preprint, Univ. of Southern California, 1994.
- [1]
- V. I. Arnold and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, New York, 1968. MR 0232910 (38:1233)
- [2]
- C. Boldrighini, M. Keane, and F. Marchetti, Billiards in polygons, Ann. Probab. 6 (1978), 532-540. MR 0644840 (58:31007b)
- [3]
- I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic theory, Springer-Verlag, Berlin, 1982. MR 832433 (87f:28019)
- [4]
- G. Galperin, T. Kruger, and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, preprint, Univ. Bielefeld, 1993; Comm. Math. Phys. (to appear). MR 1328732 (96d:58106)
- [5]
- M. Gromov, Entropy, homology and semialgebraic geometry (after Y. Yomdin), Seminaire Bourbaki 1985-86, expose 663, Astérisque (1987), 145-146. MR 880035 (89f:58082)
- [6]
- E. Gutkin, Billiard flows on almost integrable polyhedral surfaces, Ergodic Theory Dynamical Systems 4 (1984), 569-584. MR 779714 (86m:58123)
- [7]
- E. Gutkin and N. Haydn, Generalized polygon exchanges with zero topological entropy, Proceedings of workshop on dynamical systems and related topics, Univ. of Maryland (1993).
- [8]
- E. Gutkin and N. Simanyi, Dual polygonal billiards and necklace dynamics, Comm. Math. Phys. 143 (1992), 431-449. MR 1145593 (92k:58139)
- [9]
- A. Katok, Entropy and closed geodesies, Ergodic Theory Dynamical Systems 2 (1982), 339-365. MR 721728 (85b:53047)
- [10]
- -, The growth rate for the number of singular and periodic orbits for a polygonal billiard, Comm. Math. Phys. 111 (1987), 151-160. MR 896765 (88g:58162)
- [11]
- A. Katok and J.-M. Strelcyn, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Math., vol. 1222, Springer, New York, 1986. MR 872698 (88k:58075)
- [12]
- A. G. Kushnirenko, An estimate from above for the entropy of a classical dynamical system, Soviet Math. Dokl. 161 (1965), 360-362.
- [13]
- R. Mañe, Ergodic theory and differentiable dynamics, Springer-Verlag, New York, 1987. MR 889254 (88c:58040)
- [14]
- A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), 567-573. MR 554385 (81e:58044)
- [15]
- G. Margulis, Applications of ergodic theory to the investigations of manifolds of negative curvature, Functional Anal. Appl. 3 (1969), 335-336. MR 0257933 (41:2582)
- [16]
- S. Newhouse, Entropy and volume, Ergodic Theory Dynamical Systems 8 (1988), 283-299. MR 967642 (90g:58068)
- [17]
- F. Przytycki, An upper estimation for the topological entropy of diffeomorphisms, Invent. Math. 59 (1980), 205-213. MR 579699 (82a:58032)
- [18]
- P. Sarnak, Entropy estimates for geodesic flows, Ergodic Theory Dynamical Systems 2 (1982), 513-524. MR 721737 (85f:58094)
- [19]
- P. Walters, An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)
- [20]
- Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), 285-317. MR 889979 (90g:58008)
- [21]
- E. Gutkin and N. Haydn, Generalized polygon exchanges: 1. Topological entropy, preprint, Univ. of Southern California, 1994.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1995-00555-0
PII:
S 0273-0979(1995)00555-0
Keywords:
Polygon exchange,
entropy,
billiard dynamics
Article copyright:
© Copyright 1995 American Mathematical Society
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