Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 27
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Gerber, Marlies and Dauer, Thomas 2016. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete and Continuous Dynamical Systems, Vol. 36, Issue. 9, p. 4871.


    Lapidus, Michel L Miller, Robyn L and Niemeyer, Robert G 2016. Nontrivial paths and periodic orbits of theT-fractal billiard table. Nonlinearity, Vol. 29, Issue. 7, p. 2145.


    Chen, Joe P. and Niemeyer, Robert G. 2014. Periodic billiard orbits of self-similar Sierpiński carpets. Journal of Mathematical Analysis and Applications, Vol. 416, Issue. 2, p. 969.


    Frączek, Krzysztof and Ulcigrai, Corinna 2014. Non-ergodic $\mathbb{Z}$ -periodic billiards and infinite translation surfaces. Inventiones mathematicae, Vol. 197, Issue. 2, p. 241.


    KIDA, YOSHIKATA 2013. Examples of amalgamated free products and coupling rigidity. Ergodic Theory and Dynamical Systems, Vol. 33, Issue. 02, p. 499.


    Niemeyer, Robert G. and Lapidus, Michel L. 2013. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete and Continuous Dynamical Systems, Vol. 33, Issue. 8, p. 3719.


    Weiss, Barak Hubert, Pascal and Hooper, W. Patrick 2013. Dynamics on the infinite staircase. Discrete and Continuous Dynamical Systems, Vol. 33, Issue. 9, p. 4341.


    CONZE, JEAN-PIERRE and GUTKIN, EUGENE 2012. On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces. Ergodic Theory and Dynamical Systems, Vol. 32, Issue. 02, p. 491.


    Gutkin, Eugene 2012. Capillary Floating and the Billiard Ball Problem. Journal of Mathematical Fluid Mechanics, Vol. 14, Issue. 2, p. 363.


    Gutkin, Eugene 2012. Billiard dynamics: An updated survey with the emphasis on open problems. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 22, Issue. 2, p. 026116.


    Gutkin, Eugene 2010. Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Regular and Chaotic Dynamics, Vol. 15, Issue. 4-5, p. 482.


    Gutkin, Eugene 2010. Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Regular and Chaotic Dynamics, Vol. 15, Issue. 4-5, p. 482.


    Valdez, Ferrán 2009. Infinite genus surfaces and irrational polygonal billiards. Geometriae Dedicata, Vol. 143, Issue. 1, p. 143.


    Hubert, Pascal and Lelièvre, Samuel 2006. Prime arithmetic Teichmüller discs inH(2)(2). Israel Journal of Mathematics, Vol. 151, Issue. 1, p. 281.


    Zaslavsky, G. M. Carreras, B. A. Lynch, V. E. Garcia, L. and Edelman, M. 2005. Topological instability along invariant surfaces and pseudochaotic transport. Physical Review E, Vol. 72, Issue. 2,


    Hasselblatt, Boris and Katok, Anatole 2002.


    Masur, Howard and Tabachnikov, Serge 2002.


    Gutkin, Eugene 1996. Billiards in polygons: Survey of recent results. Journal of Statistical Physics, Vol. 83, Issue. 1-2, p. 7.


    Galperin, G. Krüger, T. and Troubetzkoy, S. 1995. Local instability of orbits in polygonal and polyhedral billiards. Communications in Mathematical Physics, Vol. 169, Issue. 3, p. 463.


    Shimizu, Yasushi and Shudo, Akira 1995. Polygonal billiards: Correspondence between classical trajectories and quantum eigenstates. Chaos, Solitons & Fractals, Vol. 5, Issue. 7, p. 1337.


    ×
  • Ergodic Theory and Dynamical Systems, Volume 4, Issue 4
  • December 1984, pp. 569-584

Billiards on almost integrable polyhedral surfaces

  • E. Gutkin (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700002650
  • Published online: 01 September 2008
Abstract
Abstract

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Billiards on almost integrable polyhedral surfaces
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Billiards on almost integrable polyhedral surfaces
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Billiards on almost integrable polyhedral surfaces
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]R. H. Fox & R. B. Kershner . Concerning the transitive properties of geodesies on a rationalpolyhedron. Duke Math. J. 2 (1936), 147150.

[5]H. Masur . Interval exchange transformations and measured foliations. Ann. Math. 115 (1982), 169200.

[10]P. Stäckel . Geodatische Linien auf Polyederflächen. Rend. Circ. Mat Palermo 22 (1906), 141151.

[12]W. A. Veech . Gauss measures for transformations on the space of interval exchange maps. Ann.Math. 115 (1982), 201242.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax