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

Uniform hyperbolicity in nonflat billiards

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.

Citation: Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 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
Figure 2.  A grazing collision on a dispersing billiard in $\mathbb T^2$. The flow stops being defined after this time
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
Figure 4.  Each $A_k$ maps the cone $xy > 0$ (in grey) into the smaller cone $C_\epsilon$ (in dark grey)
Figure 5.  $u$ is not well-defined at the convergence point
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