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In this article, we study the one-parameter family of billiard maps tex2html_wrap_inline289 on the tex2html_wrap_inline287 -unit ball in tex2html_wrap_inline315 , tex2html_wrap_inline291 , for tex2html_wrap_inline293 . In Fig. 1 we see examples of such tables. For p=1,2 or tex2html_wrap_inline323 , the map can be understood completely - for p=1, or tex2html_wrap_inline327 , the table is a square, while for p=2, the table is a circle. The family contains smooth algebraic curves for p=2n with tex2html_wrap_inline333 . However, in general the curves are not differentiable. We prove that this family of billiard maps has positive topological entropies for the nonintegrable cases tex2html_wrap_inline335 , and tex2html_wrap_inline323 , and we measure positive metric entropies, which indicates the coexistence of stable and random motions on a set of positive measure.

A few one-parameter families of billiard maps at convex tables have been studied already:

tex2html_wrap_inline339 The Robnik billiard tables [18] are given by the convex curves tex2html_wrap_inline341 for parameter values tex2html_wrap_inline343 . This one-parameter family of billiard maps contains the integrable case p=0, where the table is a circle. It is known [8] that for p=1/4, the map is not ergodic due to stable periodic orbits even so there are no invariant curves. tex2html_wrap_inline349. Measurements show that the map has positive metric entropy for some parameter values.
tex2html_wrap_inline339 The Bunimovich stadium tables are obtained by joining two unit half circles with two parallel straight lines of length tex2html_wrap_inline353 . Again, the case p=0 gives the circle. It is known that this billiard has positive Lyapunov exponents almost everywhere for p;SPMgt;0 and Wojtkowsky [23] showed quantitatively that for sufficiently small p, the metric entropy is tex2html_wrap_inline361.
tex2html_wrap_inline339 The Benettin-Strelcyn Billiard [2],[9] is a one parameter family of billiards where the curve consists of four arcs interpolating a circle.
tex2html_wrap_inline339 The string construction billiards is a one-parameter family of billiard curves obtained by making the string construction on a strictly convex curve of length 1 [7]. For every string length tex2html_wrap_inline369 , one obtains a billiard table. In the (limiting) case tex2html_wrap_inline327 the table is a circle. The pictures below show the string construction when the given convex curve is an equilateral triangle.

In general, dynamical systems are neither integrable nor hyperbolic and one has to deal with maps exhibiting both stable and unstable behavior in the phase space. Coexistence of chaos and order seems to be the rule for smooth convex billiards: only the ellipses are known to be integrable and an old conjecture of Birkhoff states that this should be the only case (compare [4] [3]). No uniformly hyperbolic convex billiard is known. However, the weaker condition of nonuniform hyperbolicity (positive Lyapunov exponent almost everywhere) can be achieved for convex billiards if one does not require the table to be smooth.

A routine investigation of a family of monotone twist maps can contain the following steps:

tex2html_wrap_inline339 Establish symmetries and smoothness of the map.
tex2html_wrap_inline339 Prove the existence or absence or destruction of invariant curves or estimate the size of regions without invariant curves [6].
tex2html_wrap_inline339 Investigate the (linear) stability of the periodic orbits in dependence of the parameter and find bifurcations. Numerically investigate the stability of some orbits.
tex2html_wrap_inline339 Determine the length spectra (=set of lengths of periodic orbits) and, related, the motion of the Dirichlet eigenvalues in dependence of the tables.
tex2html_wrap_inline339 Measure or estimate the metric entropy by numerical calculations of the Lyapunov exponents.
tex2html_wrap_inline339 Find horseshoes or homoclinic points in some cases.
tex2html_wrap_inline339 Investigate the limiting behavior near integrable maps.

We followed a part of this program in the case of our family given by the tex2html_wrap_inline287 - unit balls. Except for the trivial integrable cases, we can prove that the topological entropy is positive, and measure positive metric entropies. Measurements of the metric entropy suggests that the Kolmogorov entropy tex2html_wrap_inline389 behaves roughly like a piecewise convex function and that near p=2, the entropy function tex2html_wrap_inline393 has a discontinuity in the first derivative.

Next: Symmetry and regularity of Up: Billiards in the unit Previous: Billiards in the unit

Oliver Knill, Jul 10 1998