Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy

We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis’ non-extensive statistics at these critical points. We study the map xn+1 = xn + u |xn| , 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 SQ associated to the fixed-point map is maximum with respect to that of the other maps in its basin of attraction. The degree of non-extensivity the index Q in SQ and the degree of nonlinearity z are equivalent and the generalized Lyapunov exponent λq, q = 2 − Q−1, is the leading map expansion coefficient u. The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results.