The origin of 1/f spectrum fluctuation in the hamiltonian dynamics for lattice systems.

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The classical and quantum mechanical aspects of hamiltonian chaos have been studied in relation to the 1/f spectral fluctuations in the long time behaviors. The recent results are the followings :1.The spectral structure of the most stable KAM torus was determined under the nontwist condition.2.The universality of the 1/f^<mu> spectrum (1<less than or equal>mu<less than or equal>2) was obtained by the computer simulations for the symplectic map in 4 dimensional3.The relaxation mechanisms of resonace states were theoretically elucidated by the geodesics representation and the subdynamics method.4.The level statistics for the l/f quantum billiard was obtained by computer simulations, but the mechanism of level repulsions has not yet been elucidated completely.The preliminary results have been already reported in severeal journals.