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0 which forces since f is convex that xn 1 thus yn

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Unformatted text preview: : n=1 Consequently n ! 0, which forces {since f ( ) is convex{ that xn ! 1. Thus yn & 0. P Since tan( )= ! 1 as ! 0, summation (3a) can be restated as 1 f 0 (xn ) < 1. Letting n=1 sn be the absolute-value of the slope of the line joining point Pn with Pn+1 , the convexity of f implies that sn f 0 (xn ) . Consequently, 1 X n=1 Printed: February 14, 1995 sn < 1: (3b) Filename is Article/18Billiards/billiards.ams.tex 3 Billiards inside a cusp Computing the slope sn . We have used the fact, when the cueball hits the upper cushion f , that the angles of incidence and re ection are equal. But somewhere our argument had better use that these angles are equal when the ball bounces up o the oor! Here it is: tan( ) = yn + yn+1 : n The upshot is that xn+1 = ytan(y n ) xn+1 xn n + n+1 1 As a consequence, xn tan( 1 ) : yn + yn+1 tan( 1 ) yn + yn+1 : yn yn+1 So our summation condition mutates one last time, to become note sn === yn yn+1 xn+1 xn 1 X yn yn+1 y + yn+1 n=1 n < 1; with yn & 0. (3c) But this cannot be|such a sum as this last one must always be in nite. Its M th partial sum is PM 1 X yn yn+1 = y + yn+1 n=M n 1 X yn yn+1 y + yM n=M M 1 1 yM nl!1 yn+1 = : im yM + yM 2 Since the partial sums fPM g1=1 do not go to zero, conditions (3c,b,a) were all impossible, as was M gure 2. Any cueball shot out the cusp must turn around. Post mortem re ection. This proof can be readily shown to a second-semester calculus class and gives a non-traditional and curious use for a series-divergence test. All that is used about the upper cushion y = f (x) of the table is that f is an eventually-convex di erentiable function which is asymptotic to the x-axis. However, the argument is unsatisfactory from the point of view of understanding \why" the cueball had to turn around. One test of the strength of a method of argument is whether it can be used on related questions. Suppose we remove the convexity condition and allow the upper-cushion to have wiggles. Can cueballs wander monotonically out the cusp for the (4a) table determined by, say, f (x) = 3 + sin(px) (x + 1)2 ? While one could possibly use a series-divergence argument to exhibit a speci c cueball which fails to escape, such an approach might require real delicacy to make a substantial general assertion. Yet another natural question for which the series-divergence approach looks ill-adapted focuses on a stronger sense in which cueballs might fail to escape. Do cueballs return arbitrarily close (in both (4b) position and direction) to where they started? By the way, a cueball which in nitely-often returns arbitrarily near to its initial state is called recurrent . Having developed more powerful tools, we will come back to recurrence later. Math. Intelligencer, vol.17 no.1, (1995), 8{16. 4 J.L. King Philosophy. Answering questions such as (4a,b) for an individual cueball may be di cult. Yet nearby cueballs have nearby trajectories {for a while{ and so it may be pro table to make assertions about collections of cueballs. This suggests nding a useful measure on the space of cueballs|a measure which is preserved under the action of \rolling" and \bouncing o the cushion". It turns out {this is well-known to those who study dynamical systems but is not a commonplace among mathematicians in general{ that the \billiard ow" on any billiard table has a natural invariant volume. The theme of this article is the tool of an invariant measure hidden inside a problem which, on the surface, has no mention of measures. Along the way we will encounter a few elementary but usefu...
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