New Horizons in Mathematical Physics
Ergodicity in Umbrella Billiards
Download PDF (7141.9 KB) PP. 56 - 67 Pub. Date: September 12, 2017
Author(s)
- Maria F. Correia
CIMA-UE, Dept of Mathematics, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal. - Christopher Cox*
Dept of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130-4899 - Hong-Kun Zhang
Dept of Mathematics and Statistics, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003.
Abstract
Keywords
References
[1] E. G. Altmann, T. Friedrich, A. E. Motter, H. Kantz, and A. Richter, "Prevalence of marginally unstable periodic orbits in chaotic billiards", Phys Rev E 77, 016205 (2008).
[2] D. Armstead, B. Hunt and E. Ott, "Power-law decay and self-similar distribution in stadium-type billiards", Physica D 193, 96–127 (2004).
[3] M. V. Berry, "Regularity and chaos in classical mechanics, illustrated by three deformations of a circular'billiard'", Eur. J. Phys., 2, 91–102 (1981).
[4] P. Bálint, M. Halász, J. Hernández-Tahuilán, and D. Sanders, "Chaos and stability in a two-parameter family of convex billiard tables", Nonlinearity 24, 1499–1521 (2011).
[5] M. Brack and R. K. Bhaduri, "Semiclassical Physics", Addison-Wesley, Reading, Massachusetts, (1997).
[6] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelcyn, "Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them", Meccanica 15, 9–30 (1980).
[7] G. Benettin and J. M. Strelcyn, "Numerical experiments on the free motion of a point mass moving in a plane convex region: stochastic transition and entropy", Phys. Rev. A 17, 773–785 (1978).
[8] O. Bohigas, D. Boosé, R. Egydio de Carvalho, and V. Marvulle, "Quantum tunneling and chaotic dynamics", Nucl. Phys. A 560, 197–210 (1993).
[9] L. A. Bunimovich, "On ergodic properties of certain billiards", Funct. Anal. Appl. 8, 254–255 (1974).
[10] L. A. Bunimovich, "On absolutely focusing mirror", Ergodic theory and related topics, III (Güstrow, 1990)", Lecturer Notes in Math 1514, Springer-Verlag, 62–82 (1992).
[11] L. Bunimovich and A. Grigo, "Focusing components in typical chaotic billiards should be absolutely focusing", Commun. Math. Phys. 293, 127–143 (2010).
[12] L. A. Bunimovich, H. K. Zhang, and P. Zhang, "On another edge of defocusing: hyperbolicity of asymmetric lemon billiards", arXiv preprint arXiv:1412.0173 (2014).
[13] J. Chen, L. Mohr, H. K. Zhang, and P. Zhang, "Ergodicity of the generalized lemon billiards", Chaos 23, 0431371–12 (2013).
[14] N. Chernov and H.-K. Zhang, "Billiards with polynomial mixing rates", Nonlinearity 18, 1527–1553 (2005).
[15] N. Chernov and R. Markarian, "Chaotic billiards", Mathematical Surveys and Monographs 127, American Mathematical Society, Providence, RI, (2006).
[16] M. F. Correia and H. K. Zhang, "Stability and ergodicity of moon billiards", Chaos 25, 083110 (2015).
[17] M.J. Dias Carneiro, S. Oliffson Kamphorst and S. Pintode Carvalho,"Elliptic islands in strictly convex billiards", Erg. Th. Dynam. Syst. 23, 799–812 (2003).
[18] C. Dettmann, V. Fain, "Linear and nonlinear stability of periodic orbits in annular billiards," arXiv:1609.03652.
[19] V. Donnay, "Using integrability to produce chaos: billiards with positive entropy", Commun. Math. Phys. 141, 225–257 (1991).
[20] H. R. Dullin, P. H. Richter, and A. Wittek, "A two-parameter study of the extent of chaos in a billiard system", Chaos 6, 43–58 (1996).
[21] C. Foltin, "Billiards with positive topological entropy", Nonlinearity 15, 2053–2076 (2002).
[22] G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, "Periodic orbits in Hamiltonian chaos of the annular billiard", Phys. Rev. E 65, 016212 (2001).
[23] A. Grigo, "Billiards and statistical mechanics", Ph.D. Thesis (2009).
[24] A. Hayli, "Numerical exploration of a family of strictly convex billiards with boundary of class C2", J. Stat. Phys. 83, 71–79 (1996).
[25] E. Heller and S. Tomsovic, "Postmodern quantum mechanics", Physics Today 46, 38–46 (1993).
[26] M. Hénon and J. Wisdom, "The Benettin-Strelcyn oval billiard revisited", Physica D 8, 157–169 (1983).
[27] M. Hentschel and K. Richter, "Quantum chaos in optical systems: The annular billiard", Physical Review E 66, 056207 (2002).
[28] R. Markarian, "Billiards with Pesin region of measure one", Commun. Math. Phys. 118, 87–97 (1988).
[29] R. Markarian, "Billiards with polynomial decay of correlations", Ergodic Theory Dynam. Systems 24, 177–197(2004).
[30] N. Saito, H. Hirooka, J. Ford, F. Vivaldi and G. H. Walker, "Numerical study of billiard motion in an annulus bounded by non-concentric circles", Physica D 5, 273–86 (1982).
[31] Z. Sándor, B. érdi, A. Széll, and B. Funk. "The relative Lyapunov indicator: an efficient method of chaos detection". Celest. Mech. Dyn. Astron. 90, 127–?138(2004).
[32] Y. Sinaˇ?, "Dynamical systems with elastic reflections", Russ. Math. Surveys 25, 137–191 (1970).
[33] H.-J. Stockmann, "Quantum Chaos: An Introduction", Cambridge University Press, Cambridge, (1999).
[34] M. Wojtkowski, "Principles for the design of billiards with nonvanishing Lyapunov exponents", Commun. Math. Phys. 105, 391–414 (1986).