Quasi-monte Carlo Studies of Spatial Averages of Quadratic Maps

Using Quasi-Monte Carlo integration, we found numerically that the spatial averages of chaotic quadratic maps either uctuate periodically or converge to constants. 1. Introduction A series of papers 3] 4] 5] 7] motivated us to study the dynamics of spatial averages of dynamical systems, especially, those with chaotic behaviors. The problem arises naturally when one considers the collective behavior of a large lattice system with subsystems being identical and chaotic. The main interest of the study is to investigate the dynamics of the thermodynamic limit of some global quantities such as spatial averages as the lattice size tends to innnity. In numerical simulations , the problem is often approached by taking a large lattice size. However such approach has major diiculties when the dimension of the lattice is even moderately high. In a previous paper 2], we rst proved the existence of the thermodynamic limit under very general assumptions. As an easy consequence of Ergodic Theorem, we then obtained an explicit formula for the thermodynamic limit of spatial averages. This enables us to study the dynamics of spatial averages independent of the lattice size. We showed that the dynamics of spatial averages is closely related to the existence of an asymptotic invariant measure (SRB-measure) for the local subsystem when the interactions among subsystems are weak. If such measure exists, the spatial average converges to a constant.