Sticking, power laws and scaling statistics in the landscape of Hamiltonian phase space.

In typical Hamiltonian systems which model physical systems phase space dynamics is mixed, in some part of phase space the it is chaotic while in other parts it is regular. For systems with two degrees of freedom, and for periodically kicked systems with one degree of freedom, transport in the chaotic regions of phase space is dominated by sticking to complicated structures. The probability to stay  in the vicinity of the initial point is a power law in time with a universal exponent for a wide class of systems. Statistical description of the transport is discussed. The description is based on scaling of different measures in phase space related to periodic and quasi periodic orbits' frequencies.  This description is used in order to calculate the sticking time exponent in the framework of the Markov Tree model proposed by Meiss and Ott in 1986. Even though many approximations are used, it predicts important results quantitatively and with agreement to direct simulations. Quantum manifestation of this phenomenon and its relevance to time correlations is discussed as well, showing different behavior for increasing Planck's constant.