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The billiard reflection law
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On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces
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 |
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.
Mathematics Subject Classification:
37D20, 53D25, 70E55.
Citation:
Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems,
2018, 38
(3)
: 1145-1160.
doi: 10.3934/dcds.2018048
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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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [6] |
V. J. Donnay and C. Pugh,
Anosov geodesic flows for embedded surfaces, Asterisque, 287 (2003), 61-69.
|
| [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. |
| [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. |
| [9] |
J. Hadamard,
Les surfaces à courbures opposées et leurs lignes géodésique, J. Math. pures appl., 4 (1898), 27-73.
|
| [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. |
| [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. |
| [12] |
W. Klingenberg,
Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.
doi: 10.2307/1971011. |
| [13] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
| [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. |
| [15] |
M. Kourganoff, Embedded surfaces with Anosov geodesic flows, approximating spherical billiards, arXiv preprint. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
C. Pugh and M. Shub,
Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23.
doi: 10.1007/BF01418639. |
| [21] |
J. G. Sinaĭ,
Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25 (1970), 141-192.
|
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [6] |
V. J. Donnay and C. Pugh,
Anosov geodesic flows for embedded surfaces, Asterisque, 287 (2003), 61-69.
|
| [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. |
| [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. |
| [9] |
J. Hadamard,
Les surfaces à courbures opposées et leurs lignes géodésique, J. Math. pures appl., 4 (1898), 27-73.
|
| [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. |
| [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. |
| [12] |
W. Klingenberg,
Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.
doi: 10.2307/1971011. |
| [13] |
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. |
| [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. |
| [15] |
M. Kourganoff, Embedded surfaces with Anosov geodesic flows, approximating spherical billiards, arXiv preprint. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
C. Pugh and M. Shub,
Ergodicity of Anosov actions, Invent. Math., 15 (1972), 1-23.
doi: 10.1007/BF01418639. |
| [21] |
J. G. Sinaĭ,
Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25 (1970), 141-192.
|
| [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. |
| [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. |
| [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. |
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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