2011/12/20 by Pau Rabassa, Rabassa, Pau, Angel Jorba +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.1112.4684
arxiv created 2011/12/20 · arxiv updated 2011/12/21
We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. Then we consider a two parametric family of quasi-periodically forced maps which is a unimodal one dimensional map with a full cascade of period doubling bifurcations plus a quasi-periodic perturbation. For one dimensional maps it is well known that between one period doubling and the next one there exists a parameter value where the 2n-periodic orbit is superatracting. Under appropriate hypotheses, we prove that the two parameter family has two curves of reducibility loss bifurcation around these points.