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Math. Intelligencer, vol.17 no.1, (1995), 8{16. 14 J.L. King generalization for ows on an arbitrary pinched cusp. Although it was illustrated here with a billiard ow, the
conservativity result holds mutatis mutandis for any measurepreserving continuous ow whose induced measure is
pinched. The illustrations in this article were drawn with the excellent computer facilities at the Mathematical
Sciences Research Institute, whom I thank for its hospitality.
(A1) In the setting of Lemma 9, under a continuous ow transformation the set of recurrent points {in addition
to being a fullmeasure set{ must be residual (must include a dense G set), once one adds the natural assumption
that gives positive measure to every nonempty open set.
In contrast, if there is no such conservative invariant measure , then the set of recurrent points need not be
residual. Nonetheless, Birkho established that there is at least one recurrent point under any continuous ow or
transformation on a compact space. The transformation x x + 1 on the topological circle R
shows that no
more can be guaranteed.
(A2) An unexplained coincidence occurs for ux measure in the special case where our billiard table is bounded
by an elliptical cushion C := @ . It turns out that the set C of inward pointing cueballs breaks up into TCinvariant
subsets; one for each ellipse E which is inside of, and has the same foci as, C . The invariant set consists of those
cueballs v C whose ow trajectory will pass tangent to E before it again hits C .
This invariant decomposition of C implies that ux( ), on C, breaks up into measures parameterized by confocal
ellipses E . When suitably normalized, each of these measures turns out to the the \Poncelet CE measure" of 1],
which arises from what appears to be an entirely unrelated construction.
(A3) Billiard ows are a kind of geodesic ow on surfaces of only zero and in nite curvature. A stronger result
(see, for example, Donnay 1988) is known for the geodesic ow on the surfaceofrevolution around the xaxis
generated by a di erentiable f : 0; ) R+. If the surface is \pinched", liminfx!1 f (x) = 0, then every geodesic
orbit is bounded {except for the obvious ones which ow directly out the cusp.
(A4) An open and probably di cult research question is suggested by Sullivan's 1982 result on the geodesic
ow on a cusp of constant negative curvature. Letting dist(v) denote the distance of the footpoint of v to some
chosen point on the surface, Sullivan gives an explicit speed function D(t) such that
7! f1g 2 1 ! limsup dist((t) v) = 1 ;
D
t !1 t for a.e. v. Paul Shields raised the tantalizing question of characterizing the nitearea cuspidal billiard tables which have such
a speed function. References
V.J. Donnay, Geodesic ow on the twosphere, Part I: Positive measure entropy, Ergodic Theory and Dynamical
Systems 8 (1988), 531{553.
J.L. King, Three Problems in search of a Measure, Amer. Math. Monthly 101 # 7 (1994), 609{628.
D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,
Acta Math. 149 (1982), 215{237. Printed: February 14, 1995 Filename is Article/18Billiards/billiards.ams.tex...
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