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March  2018, 38(3): 1145-1160. doi: 10.3934/dcds.2018048

Uniform hyperbolicity in nonflat billiards

Mickaël Kourganoff

Institut Fourier, Université Grenoble Alpes, 100, rue des mathématiques, 38610 Gières, France

Received  February 2017 Revised  July 2017 Published  December 2017

Fund Project: The author is supported by ERC advanced grant 320939.

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian surface with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.

Keywords: Billiard, Sinai billiard, Riemannian geometry, geodesic flow, Anosov geodesic flow.
Mathematics Subject Classification: 37D20, 53D25, 70E55.
Citation: Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
References:
[1]

V. I. Arnold and A. Avez, Problémes Ergodiques de la Mécanique Classique, Monographies Internationales de Mathématiques Modernes, No. 9, Gauthier-Villars, Éditeur, Paris, 1967.  Google Scholar

[2]

V. Baladi, M. Demers and C. Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Inventiones mathematicae, (2017), 1-139, arXiv: 1506.02836. doi: 10.1007/s00222-017-0745-1.  Google Scholar

[3]

M. Bauer and A. Lopes, A billiard in the hyperbolic plane with decay of correlation of type n-2, Discrete Contin. Dynam. Systems, 3 (1997), 107-116.   Google Scholar

[4]

N. Chernov and R. Markarian, Chaotic Billiards, vol. 127 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006, URL http://dx.doi.org/10.1090/surv/127.  Google Scholar

[5]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390, URL http://dx.doi.org/10.2307/121012.  Google Scholar

[6]

V. J. Donnay and C. Pugh, Anosov geodesic flows for embedded surfaces, Asterisque, 287 (2003), 61-69.   Google Scholar

[7]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, Ⅱ, J. Differential Geometry, 8 (1973), 437-463; Ibid., 8 (1973), 565-577. doi: 10.4310/jdg/1214431801.  Google Scholar

[8]

B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90, URL http://dx.doi.org/10.1007/s002200050748.  Google Scholar

[9]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésique, J. Math. pures appl., 4 (1898), 27-73.   Google Scholar

[10]

T. J. Hunt and R. S. MacKay, Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity, 16 (2003), 1499-1510, URL http://dx.doi.org/10.1088/0951-7715/16/4/318.  Google Scholar

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995, URL http://dx.doi.org/10.1017/CBO9780511809187.  Google Scholar

[12]

W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.  doi: 10.2307/1971011.  Google Scholar

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[14]

M. Kourganoff, Anosov geodesic flows, billiards and linkages, Comm. Math. Phys. , 344 (2016), 831-856, URL http://dx.doi.org/10.1007/s00220-016-2646-3.  Google Scholar

[15]

M. Kourganoff, Embedded surfaces with Anosov geodesic flows, approximating spherical billiards, arXiv preprint. Google Scholar

[16]

A. Krámli, N. Simányi and D. Szász, Dispersing billiards without focal points on surfaces are ergodic, Comm. Math. Phys., 125 (1989), 439-457, URL http://projecteuclid.org/euclid.cmp/1104179528.  Google Scholar

[17]

C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312, URL http://dx.doi.org/10.4007/annals.2004.159.1275.  Google Scholar

[18]

M. Magalhães and M. Pollicott, Geometry and dynamics of planar linkages, Communications in Mathematical Physics, 317 (2013), 615-634.  doi: 10.1007/s00220-012-1521-0.  Google Scholar

[19]

G. P. Paternain, Geodesic Flows, vol. 180 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1999, URL http://dx.doi.org/10.1007/978-1-4612-1600-1.  Google Scholar

[20]

C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23.  doi: 10.1007/BF01418639.  Google Scholar

[21]

J. G. Sinaĭ, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25 (1970), 141-192.   Google Scholar

[22]

A. Vetier, Sinaĭ billiard in potential field (construction of stable and unstable fibers), in Limit theorems in probability and statistics, Vol. Ⅰ, Ⅱ (Veszprém, 1982), vol. 36 of Colloq. Math. Soc. J´anos Bolyai, North-Holland, Amsterdam, 1984, 1079-1146.  Google Scholar

[23]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 145-161, URL http://dx.doi.org/10.1017/S0143385700002807.  Google Scholar

[24]

P. Zhang, Convex billiards on convex spheres, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 793-816, URL http://dx.doi.org/10.1016/j.anihpc.2016.07.001.  Google Scholar

show all references

References:
[1]

V. I. Arnold and A. Avez, Problémes Ergodiques de la Mécanique Classique, Monographies Internationales de Mathématiques Modernes, No. 9, Gauthier-Villars, Éditeur, Paris, 1967.  Google Scholar

[2]

V. Baladi, M. Demers and C. Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Inventiones mathematicae, (2017), 1-139, arXiv: 1506.02836. doi: 10.1007/s00222-017-0745-1.  Google Scholar

[3]

M. Bauer and A. Lopes, A billiard in the hyperbolic plane with decay of correlation of type n-2, Discrete Contin. Dynam. Systems, 3 (1997), 107-116.   Google Scholar

[4]

N. Chernov and R. Markarian, Chaotic Billiards, vol. 127 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006, URL http://dx.doi.org/10.1090/surv/127.  Google Scholar

[5]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357-390, URL http://dx.doi.org/10.2307/121012.  Google Scholar

[6]

V. J. Donnay and C. Pugh, Anosov geodesic flows for embedded surfaces, Asterisque, 287 (2003), 61-69.   Google Scholar

[7]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, Ⅱ, J. Differential Geometry, 8 (1973), 437-463; Ibid., 8 (1973), 565-577. doi: 10.4310/jdg/1214431801.  Google Scholar

[8]

B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90, URL http://dx.doi.org/10.1007/s002200050748.  Google Scholar

[9]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésique, J. Math. pures appl., 4 (1898), 27-73.   Google Scholar

[10]

T. J. Hunt and R. S. MacKay, Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity, 16 (2003), 1499-1510, URL http://dx.doi.org/10.1088/0951-7715/16/4/318.  Google Scholar

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995, URL http://dx.doi.org/10.1017/CBO9780511809187.  Google Scholar

[12]

W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.  doi: 10.2307/1971011.  Google Scholar

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[14]

M. Kourganoff, Anosov geodesic flows, billiards and linkages, Comm. Math. Phys. , 344 (2016), 831-856, URL http://dx.doi.org/10.1007/s00220-016-2646-3.  Google Scholar

[15]

M. Kourganoff, Embedded surfaces with Anosov geodesic flows, approximating spherical billiards, arXiv preprint. Google Scholar

[16]

A. Krámli, N. Simányi and D. Szász, Dispersing billiards without focal points on surfaces are ergodic, Comm. Math. Phys., 125 (1989), 439-457, URL http://projecteuclid.org/euclid.cmp/1104179528.  Google Scholar

[17]

C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312, URL http://dx.doi.org/10.4007/annals.2004.159.1275.  Google Scholar

[18]

M. Magalhães and M. Pollicott, Geometry and dynamics of planar linkages, Communications in Mathematical Physics, 317 (2013), 615-634.  doi: 10.1007/s00220-012-1521-0.  Google Scholar

[19]

G. P. Paternain, Geodesic Flows, vol. 180 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1999, URL http://dx.doi.org/10.1007/978-1-4612-1600-1.  Google Scholar

[20]

C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23.  doi: 10.1007/BF01418639.  Google Scholar

[21]

J. G. Sinaĭ, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25 (1970), 141-192.   Google Scholar

[22]

A. Vetier, Sinaĭ billiard in potential field (construction of stable and unstable fibers), in Limit theorems in probability and statistics, Vol. Ⅰ, Ⅱ (Veszprém, 1982), vol. 36 of Colloq. Math. Soc. J´anos Bolyai, North-Holland, Amsterdam, 1984, 1079-1146.  Google Scholar

[23]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 145-161, URL http://dx.doi.org/10.1017/S0143385700002807.  Google Scholar

[24]

P. Zhang, Convex billiards on convex spheres, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 793-816, URL http://dx.doi.org/10.1016/j.anihpc.2016.07.001.  Google Scholar

Figure 1.  The billiard reflection law
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Figure 2.  A grazing collision on a dispersing billiard in $\mathbb T^2$. The flow stops being defined after this time
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Figure 3.  On the left, a dispersing billiard in $\mathbb T^2$ with infinite horizon. On the right, a dispersing billiard in $\mathbb T^2$ with finite horizon
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Figure 4.  Each $A_k$ maps the cone $xy > 0$ (in grey) into the smaller cone $C_\epsilon$ (in dark grey)
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Figure 5.  $u$ is not well-defined at the convergence point
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