Correlation functions and generalized Lyapunov exponents

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 〈${\mu }_1^{\mathrm{-}1}$(τ)〉, for one-dimensional maps with absolutely continuous invariant measure. The decay rate γ coincides with the generalized Lyapunov exponent Λ(scrq) at scrq=2, if the sign of the multiplier does not change during the time evolution, while, in general, it is larger than Λ(2). The analysis of two-dimensional maps reveals the importance of the average second multiplier 〈${\mu }_2$(τ)〉 and of the average ratio 〈${\mu }_2$(τ)/${\mu }_1$(τ)〉 which, in some cases, can provide the leading long-time contribution.