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Billiards in the l p unit balls of the plane

M. Jeng and O. Knill
Division of Physics, Mathematics and Astronomy, Caltech Pasadena, CA, 91125, USA
Jeng's work was supported by a SURF Undergraduate Research Fellowship.
This paper appeared in the journal Chaos, Fractals and Solitons, Vol. 7, p. 543-554, 1996

Abstract

We study a one-parameter family of billiard maps tex2html_wrap_inline289 given by the convex tables tex2html_wrap_inline291 for tex2html_wrap_inline293. Using results of Mather, Hubacher and Angenent, we note that the topological entropy is positive if tex2html_wrap_inline295. We study the linear stability of some periodic orbits and observe that the stability of these orbits changes for p=2. For tex2html_wrap_inline295, there exist elliptic periodic orbits suggesting that tex2html_wrap_inline289 is not ergodic for all p. We compute numerically the metric entropy of the maps tex2html_wrap_inline289 in the interval [1,12] and the limiting behaviour near p =2.

Oliver Knill, Jul 10 1998