November  2019, 39(11): 6419-6440. doi: 10.3934/dcds.2019278

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

Received  November 2018 Revised  May 2019 Published  August 2019

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.

Citation: Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278
References:
[1]

A. AvilaJ. 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 SimoiV. 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. HuangV. 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. AvilaJ. 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 SimoiV. 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. HuangV. 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.

Figure 1.  The reflective angle equals the incident angle for every rebound
Figure 2.  The Reuleaux triangle is formed from the intersection of three circular disks, each having its center on the boundary of the other two
Figure 3.  Two deformations of the circle $ \gamma_0 $ may be essentially the same, by certain rigid transformation
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