Anatomy section 1 de nes the billiard ow and gives a

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Unformatted text preview: l tools from dynamical systems. Anatomy. Section 1 de nes the billiard ow and gives a pictorial proof that billiard measure, which is a type of volume, is indeed invariant under the ow. Using this measure, x2 presents Weiss's solution to Feldman's problem when the cusp has nite area and gives an almost-everywhere solution to questions (4a,b). In order to handle cusps with in nite area it is advantageous to view the billiard measure di erently, and for that reason x3 introduces the notion of the cross-sectional measure \induced" by the billiard ow. The article culminates in x4 by using this induced measure , a type of area, to prove that on a \pinched" table, even a table of in nite area, almost-every cueball rolls recurrently. This result, which is illustrated in gure 17, appears to be new. The Appendix contains brief connections to ergodic theory, and ends with an open problem. History. Originally, Feldman's Billiard Problem was part of a longer article with the same theme of hidden invariant measures. The other problems have been split o into a companion paper, Three Problems in search of a Measure , 1], which applies the tool of invariant measures to Poncelet's Theorem, Tarski's Plank Problem, and Gelfand's Question. The Appendix of the current article describes a connection, in the case of an elliptical billiard table, between the induced measure of x3 and the \Poncelet measure" of 1]. Idiosyncrasy. Use \a := b" to mean \a is de ned to be b". Symbol F1 Bk indicates the k=1 sets fBk gk in the union happen to be disjoint. For a measure of \area" or \volume", a nullset will be a set which has zero area or volume. When a statement \holds almost-everywhere" (a.e.), this means that it holds except for a nullset of points. Re ection problems such as David Feldman's are called \billiard problems"; some curve or collection of curves form the boundary, the cushions , and the closed 2-dimensional region that they bound is the billiard table . A mathematical cueball v = hv; i will be a point v 2 on the table together with a direction . If v {sometimes called the \footpoint" of v{ is on the boundary of the table, then is restricted to the semi-circle of angles pointing into the table. All our spaces are metric spaces. A measure-space ( ; ) means that is a Borel measure on space ; all sets and functions are tacitly Borel measurable. A transformation T : ! is a measurable map; we think of T n (!) := T (T ( n T (!) )) as the location of ! at time n. A measure is T -invariant , or T preserves , if (T 1 S ) = (S ) for each set S . After a cueball v has rolled for t seconds, let t (v) denote the resulting cueball. This mapping is called a \ ow" and satis es that if one ows for s seconds followed by t seconds, the same result is obtained by owing (t + s) seconds. Speci cally, a ow {which is a continuous-time analogue of a transformation{ on a space is a measurable map :R Printed: February 14, 1995 ! satisfying t s (! ) = t+s(!) Filename is Article/18Billiards/billiards.ams.tex 5 Billiards inside a cusp such that each t is a transformation of , and 0 is the identity. Saying the ow is measurepreserving means that each t is a -preserving transformation. We use boldface lowercase letters for individual cueballs, e.g., v, w, e, and boldface uppercase letters for sets of cueballs, e.g., S, B, , . We use slanted lowercase letters for (foot)points in the plane v, w, e, and slanted uppercase letters for sets of points, S , B, , . x1 The Billia...
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