Physica A 314 (2002) 437–441 www.elsevier.com/locate/physa The renormalization group and optimization of non-extensive entropy: criticality in non-linear one-dimensional maps A. Robledo Instituto de F  isica, Universidad Nacional Aut onoma de M exico, Apartado Postal 20-364, M exico, 01000 D.F., Mexico Dedicated to Gene Stanley on the occasion of his 60th birthday Abstract We examine the pitchfork and tangent bifurcations in unimodal maps to illustrate a connection between renormalization group (RG) xed 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 xed-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. c  2002 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear maps; Period doubling; Intermittency; Renormalization group; Entropy; Non-extensivity Here we describe further evidence for a possible connection between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scale invariance properties [1]. In this theory the useful property of a variational approach has been noticeably absent, but if pertinent, the technique of entropy optimization may be of practical importance to the RG applications. As an example we select the bifurcation points of unimodal (one dimensional with one maximum at say x =0 and monotonic for x¡ 0 and x¿ 0) maps known as the pitchfork and tangent bifurcations, and interestingly, we nd they possess universal properties of non-extensivity and entropy extrema. E-mail address: robledo@fenix.isicacu.unam.mx (A. Robledo). 0378-4371/02/$-see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0378-4371(02)01177-9