The renormalization group and optimization of non-extensive entropy: criticality in non-linear one-dimensional maps

We examine the pitchfork and tangent bifurcations in unimodal maps to illustrate a connection between renormalization group (RG) fixed points and entropy extremal properties. We observe that the exact RG solution for the tangent bifurcation is also applicable to the period-doubling cascade and assess its physical meaning. Since the expression for the fixed-point map can be put into the form of the non-extensive expressions for the temporal evolution of phase-space volume and sensitivity of initial conditions, we conclude that the map critical points possess the properties of this formalism. The universality of the RG solution makes this interpretation inclusive to all one-dimensional maps of non-linearity z>1.