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Unformatted text preview: e ow. The proposition below is certainly plausible on
physical grounds; in any case it follows from standard approximation arguments applied to (11a),
and so we omit its proof.
Flux Proposition, 13. Suppose that L has an everywhere positive rstreturn function RL .
Then
(a) ux( ) is a measure on the subsets of L.
Printed: February 14, 1995 Filename is Article/18Billiards/billiards.ams.tex 11 Billiards inside a cusp (b) Volumemeasure locally near L is the productmeasure of ux cross Lebesguemeasure on
\time". Speci cally, suppose U L is a subset whose rstreturn is uniformly positive,
that is, the number := liminf v2U RU (v) is positive. Then
For any S ( ;0] (U): vol(S) = Z ux(U \ t S) dt : 0 Inducing a transformation. For an arbitrary cueball set , its rstreturn function tells us
when a cueball returns to . The induced map , T , tells us where. It is de ned on the subset of
those v 2 which actually return to at time R (v),
T (v) := R (v)(v) :
is a closed subset of then the domain of T is simply where
is nite.) As an illustration,
were the union of the two meshes at the ends of our submerged tube, then T would be de ned
just on the upstream surface and map it in a 1to1 fashion to the downstream surface.
As suggested by the physical situation of water owing through a tube, the induced map preserves ux.
(c) For an arbitrary set of cueballs, the induced map T is measurepreserving wherever it
is de ned: If B is included in the range of T then (Of course if R ux T 1 (B) = ux B : This is too is an approximation argument, achieved by splitting into countably many pieces
whose rstreturn functions are nearly constant and then using that ows at constant speed. x4 Conservativity on an Infinite Cusp
A particular case where the induced map T is everywhere de ned is when consists of all cueballs
on the boundary @ . A symbiosis exists between this induced transformation and the ow: Transformation T@ is conservative i is conservative. Even though Poincare's recurrence theorem does not apply to this transformation {the measure
it preserves being in nite because @ has in nite length{ nonetheless, on a nitearea table, T@
inherits conservativity from the associated billiard ow .
We conclude this article by turning the implication around, in that we will use an induced
transformation to prove conservativity of the ow.
Pinchedcusp Theorem, 14. The billiard ow under a pinched cusp, even one of in nite area,
is conservative.
A ow is pinched if, arbitrarily far out the cusp, there are crosssections L of arbitrarily small
ux. So our billiard ow is pinched exactly when
liminf f (x) = 0 ;
x!1
Math. Intelligencer, vol.17 no.1, (1995), 8{16. 12 J.L. King since the value f (x) is proportional to the ux of the set of cueballs with footpoint on the vertical
linesegment going from x; 0 up to x; f (x) .
As an example of a pinched cusp of in nite area, consider
1
f (x) := x 1 sin(x) + x+1 : Even though for this cushion the supremum of f (x) is in nite, nonetheless the theorem asserts that
a cueball placed at a random location and then hit in a random direction will pass arbitrarily near
to its starting position and direction. In contrast, it would seem di cult to show by means of the
calculus technique of the Introduction that for this cushion there is even a single (nonperiodic)
recurrent trajectory.
Squeeze Play on an In nite Cusp. Intuitively, conservativity on a nitearea table came from
being unable to squeeze a gallon into a pintsized bottle. This time, our bottle has in nite volume
but, being vague for a moment, it still has in some sense a pintsized neck. Our gallon of water
will not be abl...
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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.
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