Abstract
We prove that polygonal billiards with contracting reflection laws exhibit hyperbolic attractors with countably many ergodic SRB measures. These measures are robust under small perturbations of the reflection law, and the tables for which they exist form a generic set in the space of all convex polygons. Specific polygonal tables are studied in detail.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Afraĭ movich, V.S., Pesin, Ya.B.: Dimension of Lorenz type attractors. In: Mathematical physics reviews. vol. 6, 169–241, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 6 Harwood Academic Publ., Chur
Afraĭmovich V.S., Chernov N.I., Sataev E.A.: Statistical properties of 2-D generalized hyperbolic attractors. Chaos 5(1), 238–252 (1995)
Altmann, E.G., Del Magno, G., Hentschel, M.: Non-Hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics. Europhys. Lett. EPL 84, 6 (2008)
Arroyo A., Markarian R., Sanders D.P.: Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries. Nonlinearity 22(7), 1499–1522 (2009)
Arroyo A., Markarian R., Sanders D.P.: Structure and evolution of strange attractors in non-elastic triangular billiards. Chaos 22, 026107 (2012)
Belykh V.M.: Qualitative Methods of the Theory of Nonlinear Oscillations in Point Systems. Gorki University Press, New York (1980)
Bunimovich L.: Billiards and Other Hyperbolic Systems, in Encyclopedia of Mathematical Sciences, pp. 192–233. Springer-Verlag, New York (2000)
Chernov N., Zhang H.-K.: On statistical properties of hyperbolic systems with singularities. J. Stat. Phys. 136 (4), 615–642 (2009)
Chernov N., Markarian R.: Chaotic Billiards, Mathematical Surveys and Monographs. American Mathematical Society, Providence (2006)
Chernov N.I., Korepanov A., Simányi N.: Stable regimes for hard disks in a channel with twisting walls. Chaos 22, 026105 (2012)
Gutkin E.: Billiards in polygons: survey of recent results. J. Stat. Phys. 83(1–2), 7–26 (1996)
Del Magno G., Lopes Dias J., Duarte P., Gaivão J.P., Pinheiro D.: Chaos in the square billiard with a modified reflection law. Chaos 22, 026106 (2012)
Del Magno, G., Lopes Dias, J., Duarte, P., Gaivão, J.P.: Ergodicity of polygonal billiards with contracting reflection laws (2014, preprint)
Katok A.: The growth rate for the number of singular and periodic orbits for a polygonal billiard. Commun. Math. Phys. 111(1), 151–160 (1987)
Katok A. et al.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, Lecture Notes in Mathematics. Springer, Berlin (1986)
Lehmer D.H.: Questions, discussions, and notes: a note on trigonometric algebraic numbers. Amer. Math. Monthly 40(3), 165–166 (1933)
Lozi, R.: Un attracteur étrange du type attracteur de Hénon, J. Physc. 39(Coll. C5), 9–10 (1978)
Markarian R., Pujals E.J., Sambarino M.: Pinball billiards with dominated splitting. Ergodic Theory Dynam. Syst. 30(6), 1757–1786 (2010)
Pesin Ya.B.: Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergodic Theory Dynam. Syst. 12(1), 123–151 (1992)
Richter K.: Semiclassical Theory of Mesoscopic Quantum Systems, Springer Tracts in Modern Physics. Springer-Verlag, Berlin (1999)
Sataev, E.A.: Uspekhi Mat. Nauk 47(283), 147–202,240 (1992)
Sataev E.A.: Invariant measures for hyperbolic mappings with singularities. Transl. Russ. Math. Surveys 47(1), 191–251 (1992)
Schmeling J., Troubetzkoy S.: Dimension and invertibility of hyperbolic endomorphisms with singularities. Ergodic Theory Dynam. Syst. 18(5), 1257–1282 (1998)
Tabachnikov, S.: Billiards, Panor. Synth. No. 1, (1995)
Stewart I.: Galois Theory. Chapman & Hall/CRC, Boca Raton (2004)
Smillie J.: The Dynamics of Billiard Flows in Rational Polygons, Encyclopedia of Mathematical Sciences, pp. 360–382. Springer-Verlag, New York (2000)
Watkins W., Zeitlin J.: The minimal polynomial of cos(2π/n). Amer. Math. Monthly 100(5), 471–474 (1993)
Young L.-S.: Bowen–Ruelle measures for certain piecewise hyperbolic maps. Trans. Amer. Math. Soc. 287(1), 41–48 (1985)
Young L.S.: Statistical properties of dynamical systems. Ann. Math. 147(3), 585–650 (1998)
Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5–6), 733–754(2002)
Zhang H.-K.: Current in periodic Lorentz gases with twists. Commun. Math. Phys. 306(3), 747–776 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Lyubich
Rights and permissions
About this article
Cite this article
Del Magno, G., Lopes Dias, J., Duarte, P. et al. SRB Measures for Polygonal Billiards with Contracting Reflection Laws. Commun. Math. Phys. 329, 687–723 (2014). https://doi.org/10.1007/s00220-014-1960-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-1960-x