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Spectral estimates for Ruelle operators with two parameters and sharp large deviations
Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps
Hua Loo-Keng Key Laboratory of Mathematics & Mathematics Institute, Academy of Mathematics and systems science, Chinese Academy of Sciences, Beijing 100190, China |
For ${\mathbb{Z}}_2-$symmetric analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of period 2, 3 caustics is that the deformation has to be an isometric transformation.
References:
[1] |
A. Avila, J. de Simoi and V. Kaloshin,
An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.
doi: 10.4007/annals.2016.184.2.5. |
[2] |
G. D. Birkhoff,
On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379.
doi: 10.1007/BF02421325. |
[3] |
G. D. Birkhoff, Dynamical Systems, With an addendum by Jurgen Moser. Amer. Math. Soc. Colloq. Publ, Vol. IX, Amer. Math. Soc., Providence, RI, 1966. |
[4] |
J. de Simoi, V. Kaloshin and Q. L. Wei (Appendix B coauthored with Hezari H.),
Deformational spectral rigidity among $ \mathbb{Z}_2-$symmetric strictly convex domains close to the circle, Annals of Mathematics, 186 (2017), 277-314.
doi: 10.4007/annals.2017.186.1.7. |
[5] |
E. Gutkin and A. Katok,
Caustics for inner and outer billiards, Comm. Math. Phys., 173 (1995), 101-133.
doi: 10.1007/BF02100183. |
[6] |
G. Huang, V. Kaloshin and A. Sorrentino,
Nearly circular domains which are integrable close to the boundary are ellipses, Geometric and Functional Analysis, 28 (2018), 334-392.
doi: 10.1007/s00039-018-0440-4. |
[7] |
V. Kaloshin and A. Sorrentino,
On the local Birkhoff Conjecture for convex billiards, Annals of Mathematics, 188 (2018), 315-380.
doi: 10.4007/annals.2018.188.1.6. |
[8] |
V. Kaloshin and A. Sorrentino, On the integrability of Birkhoff billiards, Philos. Trans. Roy. Soc. A, 376 (2018), 20170419, 16 pp.
doi: 10.1098/rsta.2017.0419. |
[9] |
V. Lazutkin,
The existence of caustics for a billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 186-216.
|
[10] |
J. D. Meiss,
Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64 (1992), 795-848.
doi: 10.1103/RevModPhys.64.795. |
[11] |
J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics ETH Zürich, Birkhauser, Verlag, Basel, 2003.
doi: 10.1007/978-3-0348-8057-2. |
[12] |
S. Pinto-de-Carvalho and R. Ramírez-Ros,
Nonpersistence of resonant caustics in perturbed elliptic billiards, Ergodic Theory and Dynamical Systems, 33 (2013), 1876-1890.
doi: 10.1017/S0143385712000417. |
[13] |
R. Ramírez-Ros,
Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables, Phys. D, 214 (2006), 78-87.
doi: 10.1016/j.physd.2005.12.007. |
[14] |
I. M. Yaglom and V. G. Boltyanskiǐ, Convex Figures, New York, Holt, Rinehart and Winston, 1960. |
show all references
References:
[1] |
A. Avila, J. de Simoi and V. Kaloshin,
An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.
doi: 10.4007/annals.2016.184.2.5. |
[2] |
G. D. Birkhoff,
On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379.
doi: 10.1007/BF02421325. |
[3] |
G. D. Birkhoff, Dynamical Systems, With an addendum by Jurgen Moser. Amer. Math. Soc. Colloq. Publ, Vol. IX, Amer. Math. Soc., Providence, RI, 1966. |
[4] |
J. de Simoi, V. Kaloshin and Q. L. Wei (Appendix B coauthored with Hezari H.),
Deformational spectral rigidity among $ \mathbb{Z}_2-$symmetric strictly convex domains close to the circle, Annals of Mathematics, 186 (2017), 277-314.
doi: 10.4007/annals.2017.186.1.7. |
[5] |
E. Gutkin and A. Katok,
Caustics for inner and outer billiards, Comm. Math. Phys., 173 (1995), 101-133.
doi: 10.1007/BF02100183. |
[6] |
G. Huang, V. Kaloshin and A. Sorrentino,
Nearly circular domains which are integrable close to the boundary are ellipses, Geometric and Functional Analysis, 28 (2018), 334-392.
doi: 10.1007/s00039-018-0440-4. |
[7] |
V. Kaloshin and A. Sorrentino,
On the local Birkhoff Conjecture for convex billiards, Annals of Mathematics, 188 (2018), 315-380.
doi: 10.4007/annals.2018.188.1.6. |
[8] |
V. Kaloshin and A. Sorrentino, On the integrability of Birkhoff billiards, Philos. Trans. Roy. Soc. A, 376 (2018), 20170419, 16 pp.
doi: 10.1098/rsta.2017.0419. |
[9] |
V. Lazutkin,
The existence of caustics for a billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 186-216.
|
[10] |
J. D. Meiss,
Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64 (1992), 795-848.
doi: 10.1103/RevModPhys.64.795. |
[11] |
J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics ETH Zürich, Birkhauser, Verlag, Basel, 2003.
doi: 10.1007/978-3-0348-8057-2. |
[12] |
S. Pinto-de-Carvalho and R. Ramírez-Ros,
Nonpersistence of resonant caustics in perturbed elliptic billiards, Ergodic Theory and Dynamical Systems, 33 (2013), 1876-1890.
doi: 10.1017/S0143385712000417. |
[13] |
R. Ramírez-Ros,
Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables, Phys. D, 214 (2006), 78-87.
doi: 10.1016/j.physd.2005.12.007. |
[14] |
I. M. Yaglom and V. G. Boltyanskiǐ, Convex Figures, New York, Holt, Rinehart and Winston, 1960. |



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