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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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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. 13 (1976), 486–488.CrossRefGoogle Scholar
[3]Brémont, J.. Ergodic non-abelian smooth extensions of an irrational rotation. J. Lond. Math. Soc. 81 (2010), 457–476.CrossRefGoogle Scholar
[4]Chevallier, N. and Conze, J.-P.. Examples of Recurrent or Transient Stationary Walks in ℝd Over a Rotation of 𝕋2 (Contemporary Mathematics, 485). American Mathematical Society, Providence, RI, 2009, pp. 71–84.Google Scholar
[5]Cirilo, P., Lima, Y. and Pujals, E.. On rational ergodicity of certain cylinder flows. arXiv:1108.3519 (2011).Google Scholar
[6]Conze, J.-P.. Transformations cylindriques et mesures finies invariantes. Ann. Sci. Univ. Clermont Math. 17 (1979), 25–31.Google Scholar
[7]Conze, J.-P.. Recurrence, Ergodicity and Invariant Measures for Cocycles Over a Rotation (Contemporary Mathematics, 485). AMS, Providence, RI, 2009, pp. 45–70.Google Scholar
[8]Conze, J.-P. and Fraczek, K.. Cocycles over interval exchange transformations and multivalued Hamiltonian flows. Adv. Math. 226 (2011), 4373–4428.CrossRefGoogle Scholar
[9]Conze, J.-P. and Gutkin, E.. On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces. arXiv:1008.0136, August 2010.Google Scholar
[10]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[11]Delecroix, V.. Divergent trajectories in the periodic wind-tree model. arXiv:1107.2418, July 2011.Google Scholar
[12]Denjoy, A.. Les trajectoires à la surface du tore. C. R. Acad. Sci. Paris 223 (1946), 5–8.Google Scholar
[13]Ehrenfest, P. and Ehrenfest, T.. Encyclopedia Article (1912); English Translation ‘The Conceptual Foundations of the Statistical Approach in Mechanics’. Cornell University Press, Ithaca, NY, 1959.Google Scholar
[14]Fraczek, K.. On ergodicity of some cylinder flows. Fund. Math. 163 (2000), 117–130.CrossRefGoogle Scholar
[15]Fraczek, K. and Ulcigrai, C.. Non-ergodic Z-periodic billiards and infinite translation surfaces. arXiv:1109.4584 (2011).Google Scholar
[16]Greschonig, G.. Recurrence in unipotent groups and ergodic nonabelian group extensions. Israel J. Math. 147 (2005), 245–267.CrossRefGoogle Scholar
[17]Gutkin, E.. Billiard flows on almost integrable polyhedral surfaces. Ergod. Th. & Dynam. Sys. 4 (1984), 560–584.CrossRefGoogle Scholar
[18]Gutkin, E.. Billiards in polygons: survey of recent results. J. Stat. Phys. 83 (1996), 7–26.CrossRefGoogle Scholar
[19]Gutkin, E.. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8 (2003), 1–13.CrossRefGoogle Scholar
[20]Gutkin, E.. Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Regul. Chaotic Dyn. 15 (2010), 482–503.CrossRefGoogle Scholar
[21]Gutkin, E. and Judge, C.. The geometry and arithmetic of translation surfaces with applications to polygonal billiards. Math. Res. Lett. 3 (1996), 391–403.CrossRefGoogle Scholar
[22]Gutkin, E. and Judge, C.. Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. 103 (2000), 191–213.CrossRefGoogle Scholar
[23]Gutkin, E. and Katok, A.. Caustics for inner and outer billiards. Comm. Math. Phys. 173 (1995), 101–133.CrossRefGoogle Scholar
[24]Gutkin, E. and Rams, M.. Growth rates for geometric complexities and counting functions in polygonal billiards. Ergod. Th. & Dynam. Sys. 29 (2009), 1163–1183.CrossRefGoogle Scholar
[25]Hardy, J. and Weber, J.. Diffusion in a periodic wind-tree model. J. Math. Phys. 21 (1980), 1802–1808.CrossRefGoogle Scholar
[26]Hooper, W. P. and Weiss, B.. Generalized staircases. Preprint.Google Scholar
[27]Hooper, W. P., Hubert, P. and Weiss, B.. Dynamics on the infinite staircase. Preprint.Google Scholar
[28]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[29]Kerckhoff, S., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124 (1986), 293–311.CrossRefGoogle Scholar
[30]Kesten, H.. Sums of stationary sequences cannot grow slower than linearly. Proc. Amer. Math. Soc. 49 (1975), 205–211.CrossRefGoogle Scholar
[31]Khinchin, A. Ya.. Continued Fractions. Dover Publications, Mineola, NY, 1997.Google Scholar
[32]Oren, I.. Ergodicity of cylinder flows arising from irregularities of distribution. Israel J. Math. 44 (1983), 127–138.CrossRefGoogle Scholar
[33]Oxtoby, J. C.. Measure and Category. A Survey of the Analogies between Topological and Measure Spaces. Springer, New York-Berlin, 1980.Google Scholar
[34]Schmidt, K.. Lectures on Cocycles of Ergodic Transformations Groups (Lecture Notes in Mathematics, 1). MacMillan Co., India, 1977.Google Scholar
[35]Schmidt, K.. On joint recurrence. C. R. Acad. Sci. Paris 327 (1998), 837–842.CrossRefGoogle Scholar
[36]Schmithüsen, G.. An algorithm for finding the Veech group of an origami. Experiment. Math. 13 (2004), 459–472.CrossRefGoogle Scholar
[37]Vorobets, Ya.. Ergodicity of billiards in polygons. Sb. Math. 188 (1997), 389–434.CrossRefGoogle Scholar
[38]Zemlyakov, A. and Katok, A.. Topological transitivity of billiards in polygons. Mat. Zametki 18 (1975), 291–300.Google Scholar