Scaling properties of invariant densities of coupled Tchebyscheff maps

Abstract

We study 1-dimensional coupled map lattices consisting of diffusively coupled Tchebyscheff maps of N-th order. For small coupling constants a we determine the invariant 1-point and 2-point densities of these nonhyperbolic systems in a perturbative way. For arbitrarily small couplings a>0 the densities exhibit a selfsimilar cascade of patterns, which we analyse in detail. We prove that there are log-periodic oscillations of the density both in phase space as well as in parameter space. We show that expectations of arbitrary observables scale with \sqrt{a} in the low-coupling limit, contrasting the case of hyperbolic maps where one has scaling with a. Moreover we prove that there are log-periodic oscillations of period \log N^2 modulating the \sqrt{a}-dependence of the expectation value of any given observable.