Criticality in nonlinear one-dimensional maps: RG universal map and nonextensive entropy

Abstract

We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis’ nonextensive statistics at these critical points. We study the map x n+1 = x n + u| x n | z , z>1, as it describes generically the neighborhood of all of these transitions. The exact renormalization group (RG) fixed-point map and perturbation static expressions match the corresponding expressions for the dynamics of iterates. The time evolution is universal in the RG sense and the nonextensive entropy S Q associated to the fixed-point map is maximum with respect to that of the other maps in its basin of attraction. The degree of nonextensivity—the index Q in S Q —and the degree of nonlinearity z are equivalent and the generalized Lyapunov exponent λ q , q=2− Q −1, is the leading map expansion coefficient u. The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results.