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On billiards that share a common convex caustic

O. Knill, Jan 1996

Division of Physics, Mathematics and Astronomy Caltech, Pasadena, CA, 91125, USA.
Some mathematical results in this document have been obtained in collaboration with Eugene Gutkin from the Department of Mathematics at USC, Los Angeles in the time from February 1994 until December 1995. This version is dated January 3, 1996 and proves some of the claims in our proceedings paper . The responsibility for this HTML document is by the signed author.

Abstract

We consider the one-parameter family of billiard tables tex2html_wrap_inline1370 which have a common caustic tex2html_wrap_inline1372 and study the corresponding family of billiard maps tex2html_wrap_inline1374 . The billiard tables tex2html_wrap_inline1376 are constructed geometrically by the string construction, where the length tex2html_wrap_inline1378 of the string is the parameter. We study the family of circle homeomorphisms tex2html_wrap_inline1380 obtained by restricting the billiard map tex2html_wrap_inline1374 to the canonical invariant circle tex2html_wrap_inline1384 belonging to the caustic and the rotation function tex2html_wrap_inline1386 . We prove that for a dense tex2html_wrap_inline1388 set of curves tex2html_wrap_inline1372 which includes curves with flat points and polygons, the function tex2html_wrap_inline1392 is a devil-staircase. We analyze the motion of Aubry-Mather sets tex2html_wrap_inline1394 passing the invariant curve tex2html_wrap_inline1384 as tex2html_wrap_inline1378 is increasing. The interesting case are Mather sets tex2html_wrap_inline1400 with rational rotation number p/q, which consists of Birkhoff periodic orbits. The passage of such orbits through the canonical invariant circle tex2html_wrap_inline1384 as the parameter changes is accompanied by bifurcations which can be described by index methods.

Oliver Knill, HTML translation made July 8, 1998