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Mosc. Math. J., 2006, Volume 6, Number 4, Pages 673–701 (Mi mmj265)  

This article is cited in 11 scientific papers (total in 11 papers)

Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature

E. A. Gutkinab, S. L. Tabachnikovc

a Instituto Nacional de Matemática Pura e Aplicada
b University of California, Los Angeles
c Department of Mathematics, Pennsylvania State University

Abstract: We introduce piecewise convex transformations, and develop geometric tools to study their complexity. We apply the results to the complexity of polygonal inner and outer billiards on surfaces of constant curvature.

Key words and phrases: Geodesic polygon, constant curvature, complexity, inner billiard, outer billiard.

DOI: https://doi.org/10.17323/1609-4514-2006-6-4-673-701

Full text: http://www.ams.org/.../abst6-4-2006.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 53D25, 37E99, 37B10
Received: April 29, 2006; in revised form October 16, 2006
Language:

Citation: E. A. Gutkin, S. L. Tabachnikov, “Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature”, Mosc. Math. J., 6:4 (2006), 673–701

Citation in format AMSBIB
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\by E.~A.~Gutkin, S.~L.~Tabachnikov
\paper Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 4
\pages 673--701
\mathnet{http://mi.mathnet.ru/mmj265}
\crossref{https://doi.org/10.17323/1609-4514-2006-6-4-673-701}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2291158}
\zmath{https://zbmath.org/?q=an:1123.53043}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208596000004}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Genin D., “Hyperbolic Outer Billiards: a First Example”, Nonlinearity, 19:6 (2006), 1403–1413  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Tabachnikov S., “A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards”, Geom. Dedicata, 129:1 (2007), 83–87  crossref  mathscinet  zmath  isi  scopus
    3. Gutkin E., Rams M., “Growth rates for geometric complexities and counting functions in polygonal billiards”, Ergodic Theory Dynam. Systems, 29:4 (2009), 1163–1183  crossref  mathscinet  zmath  isi  scopus
    4. Castle S., Peyerimhoff N., Siburg K.F., “Billiards in ideal hyperbolic polygons”, Discrete Contin. Dyn. Syst., 29:3 (2011), 893–908  crossref  mathscinet  zmath  isi  scopus
    5. E. A. Gutkin, “Dinamika billiarda: obzornaya statya s aktsentom na nereshennye zadachi”, Nelineinaya dinam., 7:3 (2011), 489–512  mathnet
    6. Bedaride N., Cassaigne J., “Outer billiard outside regular polygons”, J. Lond. Math. Soc. (2), 84:2 (2011), 303–324  crossref  mathscinet  zmath  isi  scopus
    7. Gutkin E., “Billiard Dynamics: an Updated Survey with the Emphasis on Open Problems”, Chaos, 22:2 (2012), 026116  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Blumen V., Kim K.Y., Nance J., Zharnitsky V., “Three-Period Orbits in Billiards on the Surfaces of Constant Curvature”, Int. Math. Res. Notices, 2012, no. 21, 5014–5024  crossref  mathscinet  zmath  isi  elib  scopus
    9. Scheglov D., “Growth of Periodic Orbits and Generalized Diagonals for Typical Triangular Billiards”, J. Mod. Dyn., 7:1 (2013), 31–44  crossref  mathscinet  zmath  isi  scopus
    10. Del Magno G., Gaivao J.P., Gutkin E., “Dissipative Outer Billiards: a Case Study”, Dynam. Syst., 30:1 (2015), 45–69  crossref  mathscinet  zmath  isi  scopus
    11. dos Santos L.C., Pinto-de-Carvalho S., “Periodic Orbits of Oval Billiards on Surfaces of Constant Curvature”, Dynam. Syst., 32:2 (2017), 283–294  crossref  mathscinet  zmath  isi  scopus
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