Slow Decay of Correlations in Non-hyperbolic Dynamical Systems(Perspectives of Nonequilibrium Statistical Physics-The Memory of Professor Shuichi Tasaki-)

Abstract

In this paper, we reviewed a piecewise linear map we have proposed in [24]. This model is a modified version of the map introduced by Tasaki and Gaspard [34, 35]. A main difference between our model and Tasaki-Gaspard model is the normalizability of invariant densities. The invariant density of the Tasaki-Gaspard model [34, 35] is not normalizable for a range of the parameter values. This is a typical property of dynamical systems with MUFPs [2, 1], and is caused by sticking motion in neighborhoods of the MUFPs. On the other hand, the uniform density is invariant for the map φ(x) presented in this paper; therefore the invariant density is normalizable for any values of the system parameters, even though our system has also a MUFP. This is because the present model has the mechanism suppressing injections of the orbits into neighborhoods of the MUFP and this property prevents divergences of the invariant density at the MUFP. As a consequence of the normalizability, the present model does not exhibit non-stationarity, which is generically observed in maps with MUFPs [1, 2, 34, 35]. The spectral properties of the FP operator of the present model is similar to those of the Tasaki-Gaspard model [34, 35]. There are two simple eigenvalues 1 and λ_d∈(-1, 0); the former corresponds to the invariant eigenstate and the latter to the oscillating one. The eigenstate associated to λ_d, however, does not contribute to the long time behavior of the correlation functions because it decays exponentially fast. There is also a continuous spectrum on the real interval [0, 1]; this continuous spectrum leads to the power law decay of correlation functions. Furthermore, the piecewise linear map φ(x) has been extended to an area-preserving invertible map ψ(x, y) on the unit square. In contrast to the baker transformation, which is hyperbolic and shows exponential decays of correlation functions, this area-preserving map ψ(x, y) is non-hyperbolic and displays a power law decay of correlations. As is well known, the mixed type Hamiltonian systems also exhibit a power law decay of correlation functions [20, 11, 22, 23, 15, 38, 26]. Thus, the area-preserving map ψ(x, y) might well be considered as an abstract model of the mixed type Hamiltonian systems in the following sense. Instabilities of the orbits of the map ψ(x, y) [Eq.(6)] is weak in neighborhoods of the line x=0, and the escape time from the left part x<b to the right part x>b diverges as x→0. In other words, the orbits stick to the line x=0 for long times. This property seems to be similar to dynamics of Hamiltonian systems near torus, cantorus, and marginally unstable periodic orbits, where chaotic orbits become stagnant for long times. In fact, similar dynamics is observed in a Poincare map of the mushroom billiard, which is a mixed type systems with sharply divided phase spaces [3, 4, 10, 9, 33], as reviewed in the second part of the present paper. This relation between the map ψ(x, y) and a billiard system is similar to the relation between the baker map and the Lorentz gas [37].