Gibbsian Distribution on the Lorenz Attractor

Abstract

A representation of the Lorenz attractor by a 1-dimensional Isingsystem is constructed. Gibbsian distribution function which describesthe irregular motion of this dissipative dynamical system is investigatedwith the help of this representation. Relation between measure-theoreticalentropy and positive Lyapunov characteristic exponent is also investigated.The following conclusions are obtained: (1) The statistical propertiesof the Lorenz system can be reduced to those of 1-dimensional Isingsystem with short-range interaction, in other words, the time correlationfunction of the Lorenz system shows no singular long-time behaviour.(2) The positive Lyapunov characteristic exponent of the Lorenz systemis almost equal to its measure-theoretical entropy.