Outer Billiards with Contraction - Brown University
Transcript of Outer Billiards with Contraction - Brown University
![Page 1: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/1.jpg)
Outer Billiards with Contraction
Presented on August 6th
(modified on August 9th)
In-Jee Jeong, Brown University
Francisc Bozgan, UCLA
Julienne Lachance, Rensselaer
![Page 2: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/2.jpg)
Summary
• In section I, we discuss stability issues, shedding light on a striking difference between the regular pentagon and the regular septagon.
![Page 3: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/3.jpg)
Summary
• In section II, we discuss trapezoidal outer billiards with contraction. We can prove that for any trapezoid there are infinitely many stable degenerate periodic orbits (SDPOs).
![Page 4: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/4.jpg)
Summary
• In section III, we outline a proof that for certain choice of the polygon and the contraction, attracting Cantor sets exist.
![Page 5: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/5.jpg)
A Comparison
Square / Triangle
Trapezoid Reg. Pentagon
Reg. Septagon
Finiteness in bounded ball
Yes Yes (?) No No (?)
Exotic PeriodicOrbits
None Infinitelymany
None (?) ???
Stability of Periodic Domains
Yes Yes (?) Yes No !
Possible Approach
Hierarchy of the tiling
Horizontal Slicing /
quasihierarchy
Renor-malization /
quasihierarchy
???????
![Page 6: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/6.jpg)
I. Stability
![Page 7: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/7.jpg)
• Theorem. (Stability Criterion)
• Given a periodic domain Q for P, look at reflected images of P determined by the combinatorics of Q. Then Q is λ-stable if and only if the barycenter of all the images of P lies in the interior of Q.
![Page 8: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/8.jpg)
![Page 9: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/9.jpg)
Illustration of the Stability
![Page 10: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/10.jpg)
Period 10 orbit
![Page 11: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/11.jpg)
Period 30 orbit
![Page 12: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/12.jpg)
• Corollary. Symmetric periodic domains are stable
• Corollary. Odd periodic domains are stable
![Page 13: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/13.jpg)
• Corollary. When n = 3, 4, 5, 6, 8, all periodic domains for the regular n-gonare stable
• Corollary. If P is centrally symmetric, all Culter periodic domains are stable
![Page 14: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/14.jpg)
Stability of theRegular PentagonalPeriodic Domains
![Page 15: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/15.jpg)
S. Tabachnikov Adv in Math (1995)
![Page 16: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/16.jpg)
![Page 17: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/17.jpg)
S. Tabachnikov Adv in Math (1995)
![Page 18: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/18.jpg)
Regular Septagon: Unstable Pentagon
• Period 57848, Radius around 0.0001
![Page 19: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/19.jpg)
Schwartz’s Zoo of Exotic Domains
• Diameter: 0.002, Period: up to 500000
![Page 20: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/20.jpg)
II. Trapzoids
![Page 21: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/21.jpg)
3-5-7 Conjecture.
• All the exotic periodic orbits have periods ending with either 3, 5, or 7.
• Finiteness follows.
![Page 22: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/22.jpg)
3-5-7 Weaker Conjecture.
• For any positive integer congruent to either 3, 5, or 7, there exists a SDPO with that integer as its period.
• Infinitely many exotic periodic orbits!
![Page 23: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/23.jpg)
3-5-7 Weaker Conjecture.
• Weaker conjecture is proved.
• Boils down to studying fixed points of orientation-reversing interval exchange transforms
![Page 24: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/24.jpg)
The “Pinwheel” Dynamics
![Page 25: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/25.jpg)
Interval Exchange Transform
![Page 26: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/26.jpg)
III. Cantor Set
![Page 27: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/27.jpg)
Affine Contractions
• H. Bruin, J Deane Proceedings of the AMS (2008)• Coloring Scheme by P. Hooper
![Page 28: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/28.jpg)
Extending the Rotation Theory
• F. Rhodes, C. ThompsonRotation number for monotone functions on the circle (1985)
Topologies and rotation numbers for families of monotone functions on the circle (1989)
• Summary– Well-defined and nice with respect to strictly
increasing monotone maps of the circle
– Continuous with respect to the Hausdorff metric on graphs
![Page 29: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/29.jpg)
Extending the Rotation Theory (2)
• R. BretteRotation numbers of discontinuous orientation-
preserving circle maps (2003)
• Summary
– If rational, all orbits are asymptotically periodic.
– If irrational, the limit set is either the whole circle or a unique Cantor set
![Page 30: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/30.jpg)
Triangular Transition on Quadrilaterals
![Page 31: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/31.jpg)
An Invariant Region
![Page 32: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/32.jpg)
Dynamics of the Return Map
![Page 33: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/33.jpg)
• Semiconjugacy
![Page 34: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/34.jpg)
Proving existence of attracting Cantor set: Ingredients
• Rotation theory
• Bijection between periodic orbits
• Continuity of the rotation number
![Page 35: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/35.jpg)
Attracting Cantor set Exists!
![Page 36: Outer Billiards with Contraction - Brown University](https://reader033.fdocuments.net/reader033/viewer/2022051804/6281d61d30958f295b01be2a/html5/thumbnails/36.jpg)
• We sincerely thank our advisors Prof. P. Hooper, Prof. S. Tabachnikov, Diana and Tarik.
• We also thank Prof. R. Schwartz. Many figures were generated by his program.
• We also thank all ICERM staff.