Correlation functions and generalized Lyapunov exponents.

Abstract

Correlation functions of one- and two-dimensional piecewise linear maps are analytically investigated. The asymptotic time behavior is shown to be given by the average inverse multiplier 〈${\ensuremath{\mu}}_{1}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\tau})〉, for one-dimensional maps with absolutely continuous invariant measure. The decay rate \ensuremath{\gamma} coincides with the generalized Lyapunov exponent \ensuremath{\Lambda}(scrq) at scrq=2, if the sign of the multiplier does not change during the time evolution, while, in general, it is larger than \ensuremath{\Lambda}(2). The analysis of two-dimensional maps reveals the importance of the average second multiplier 〈${\ensuremath{\mu}}_{2}$(\ensuremath{\tau})〉 and of the average ratio 〈${\ensuremath{\mu}}_{2}$(\ensuremath{\tau})/${\ensuremath{\mu}}_{1}$(\ensuremath{\tau})〉 which, in some cases, can provide the leading long-time contribution.