Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

Abstract

We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.

Keywords:

• Outer billiards
• Gutkin billiards
• total integrability
• chaotic behavior

Acknowledgments

This work was started during the XXXVII Workshop on Geometric Methods in Physics, BIAŁOWIEŻA, POLAND. We would like to thank the organizers for this opportunity.

Additional information

Funding

M.B. and L.S were supported in part by ISF grant 162/15 and A.E.M. was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). It is our pleasure to thank these funds for the support.