Stagnant motion in hamiltonian dynamics mushroom billiard case with smooth outermost KAM tori

The detailed phase space structure near the smooth outermost KAM surface is studied carrying out with the mushroom billiard. When the rotation number of the outermost surface α is the rational Fibonacci ratio (α_k = F_k+2/F_k+4, k ≥ 0), some singular points always exist on the outermost surface, but the stagnant layers sharply disappear when k increases. On the other hand, when the rotation number approaches to the limit irrational number (k → ∞), some singular periodic points remain in chaotic sea far from the outermost KAM surface, and that the stagnant layers are formed arround each singular point. The phase volume of the stagnant layers is theoretically evaluated, and it is shown that the trapping time in each stagnant layer obeys the inverse power distribution. This explains the universal aspect of the slow dynamics in the mushroom billiard, where the power spectral density reveals log ω (ω < 1) for 0.5 ≲ r/R indifferent from the rotation number α of the outermost KAM surface. Indeed, even in other cases for irrational α, for instance, algebraic irrationals and transcendental irrationals, the power spectrum reveals the same scaling. But in other conditions for r/R ∼ 0:5, the power spectrum obeys the ω^-v-scaling originated from the bouncing motion in the stark area, and the spectral transition is numerically determined.