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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 almosteverywhere
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 crosssectional 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, almostevery 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 almosteverywhere" (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 2dimensional 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 semicircle of angles pointing into the table.
All our spaces are metric spaces. A measurespace ( ; ) 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 continuoustime 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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This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.
 Fall '09
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