Towards a renormalization theory for quasi-periodically forced one dimensional maps II. Asymptotic behavior of reducibility loss bifurcations

Abstract: In this paper we are concerned with quasi-periodic forced one dimensional maps. We consider a two parametric family of quasi-periodically forced maps such that the one dimensional map (before forcing) is unimodal and it has a full cascade of period doubling bifurcations. Between one period doubling and the next one it is known that there exist a parameter value where the $2^n$-periodic orbit is superatracting. In a previous work we proposed an extension of the one-dimensional (doubling) renormalization operator to the quasi-periodic case. We proved that, if the family satisfies suitable hypotheses, the two parameter family has two curves of reducibility loss bifurcation around these parameter values. In the present work we study the asymptotic behavior of these bifurcations when $n$ grows to infinity. We show that the asymptotic behavior depends on the Fourier expansion of the quasi-periodic coupling of the family. The theory developed here provides a theoretical explanation to the behavior that can be observed numerically.