Periodic orbit theory for classical hyperbolic system is very significant
matter of how we can interpret spectral statistics in terms of semiclassical
theory. Although pruning is significant and generic property for almost all
hyperbolic systems, pruning-proof property for the correlation among the
periodic orbits which gains a resurgence of second term of the random matrix
form factor remains open problem. In the light of the semiclassical form
factor, our attention is paid to statistics for the pairs of periodic orbits.
Also in the context of pruning, we investigated statistical properties of the
"actual" periodic orbits in 4-disk billiard system. This analysis presents some
universality for pair-orbits' statistics. That is, even if the pruning
progresses, there remains the periodic peak structure in the statistics for
periodic orbit pairs. From that property, we claim that if the periodic peak
structure contributes to the correlation, namely the off-diagonal part of the
semiclassical form factor, then the correlation must remain while pruning
progresse.Comment: 30 pages, 58 figure
