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

Abstract

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 $x_{n+1}=x_{n}+u| x_{n}| ^{z}$, $z>1$, 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 non-extensive 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 non-extensivity - the index $Q$ in $S_{Q}$ - and the degree of nonlinearity $z$ are equivalent and the generalized Lyapunov exponent $\lambda_{q}$, $q=2-Q^{-1}$, is the leading map expansion coefficient $u$. The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results.