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

Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps

Jianlu Zhang

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.

Keywords: Convex billiard map, caustic, trigonometric polynomial, analytic function, nearly circular deformation, generating function.
Mathematics Subject Classification: 37E40, 70H09.
Citation: Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278
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.  Google Scholar

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G. D. Birkhoff, Dynamical Systems, With an addendum by Jurgen Moser. Amer. Math. Soc. Colloq. Publ, Vol. IX, Amer. Math. Soc., Providence, RI, 1966.  Google Scholar

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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.  Google Scholar

[5]

E. Gutkin and A. Katok, Caustics for inner and outer billiards, Comm. Math. Phys., 173 (1995), 101-133.  doi: 10.1007/BF02100183.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

J. D. Meiss, Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64 (1992), 795-848.  doi: 10.1103/RevModPhys.64.795.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

I. M. Yaglom and V. G. Boltyanskiǐ, Convex Figures, New York, Holt, Rinehart and Winston, 1960.  Google Scholar

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.  Google Scholar

[2]

G. D. Birkhoff, On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379.  doi: 10.1007/BF02421325.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. Gutkin and A. Katok, Caustics for inner and outer billiards, Comm. Math. Phys., 173 (1995), 101-133.  doi: 10.1007/BF02100183.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

J. D. Meiss, Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64 (1992), 795-848.  doi: 10.1103/RevModPhys.64.795.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

I. M. Yaglom and V. G. Boltyanskiǐ, Convex Figures, New York, Holt, Rinehart and Winston, 1960.  Google Scholar

Figure 1.  The reflective angle equals the incident angle for every rebound
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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
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Figure 3.  Two deformations of the circle $ \gamma_0 $ may be essentially the same, by certain rigid transformation
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