Scaling properties of invariant densities of coupled Chebyshev maps

Abstract

We study one-dimensional coupled map lattices consisting of diffusively coupled Chebyshev 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 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 -dependence of the expectation value of any given observable.