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Analysing the results, we note that the asymptotic form factors (denoted in Fig. 24 as `full classical form factora) approach constant values, which are indeed close to 2n, as predicted. More signi"cant are the deviations from the constant (Poissonian) result at low k, which demonstrate unambiguously the existence of correlations in the classical spectra. The structure of the form factor indicates that the classical spectrum is rigid on the scale of a correlation length (n;R), which can be de"ned as the inverse of the k value at which the form factor makes its approach to the asymptotic value [11].
69). We also show the asymptotic prediction (66). The sum-rule (66) which formed the basis of the previous analysis is an expression of the ergodic nature of the billiards dynamics. , taking the surface of the sphere and the tangent velocity vector as the PoincareH section. The resulting return-map excludes the bouncing-ball manifolds since they do not intersect the section. ect is noticed because between successive collisions with the sphere the trajectory may re#ect o! the planar faces of the billiard an arbitrary number of times.
Being the shortest bouncing ball, it is isolated from the lengths of other bouncing balls. 6S. 1. Since other periodic orbits are fairly distant, this shortest bouncing ball is an ideal test-ground of the length spectrum. k )" exp ! S/(3) /2] . U (6 ) (105) Due to its isolation, one expects that the shortest bouncing ball gives the dominant contribution to the length spectrum near its length. Thus, for l+S/(3, one has "D "+"D U ". The latter is U independent of k. erent values of k, and compared with (105).