Projection Operator Approach to the Thermodynamical Formalism of Dynamical Systems.pdf

Projection Operator Approach to the ThermodynamicalFormalism of Dynamical SystemsWolfram Just Department of PhysicsKyushu University 33Fukuoka 812JapanJanuary 21, 1998AbstractAn analytical perturbative treatment of characteristic exponents describing the uc-tuations of temporal coarse grained quantities in the context of nonlinear dynamicalsystems is proposed. It is based on the analysis of the resolvent of the correspondingtransfer operator by a projection operator method similar to those used in statisticalmechanics. Two dierent approximation schemes are presented and tested for the caseof an exactly solvable but nontrivial model system.Keywords: Thermodynamical Formalism, Transfer Operator, Projection Operator Tech-niquePACS No.: 05.45 A part of this work was performed within a program of the Sonderforschungsbereich 185 Darmstadt{Frankfurt at the TH Darmstadt (Germany). 1 1 IntroductionThe thermodynamical formalism constitutes a program for analysing the complicated be-haviour of nonlinear dynamical systems [1, 2]. The main quantity that contains a lot ofinformation about the dynamics is given by the characteristic exponent [3]u(q) := limn!1 1n lnhexp q n1Xi=0 u(T i(x))!i (1)where to be denit we want to restrict the discussion to discrete dynamical systems xn+1 =T (xn). The expectation value in eq.(1) h: : :i is meant with respect to some distribution ofinitial points x which is usually assumed to be the (physical) invariant distribution (SRBmeasure in mathematical terms). The characteristic quantity depends on the function u(x).The case that u(x) is given by the local expansion rate is by Bowens theorem [2] of specialimportance and frequently discussed in the literature [4]. In those cases the quantity (1)is called the topological pressure. One possibility for the computation of the characteristicexponent is based on the introduction of the transfer operator [5](Huqh)(x) := Z (x T (y)) exp(qu(y))h(y) dy : (2)Its largest eigenvalue q is connected to the