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In this paper we are concerned with quasi-periodic forced one dimensionalmaps. We consider a two parametric family of quasi-periodically forced mapssuch that the one dimensional map (before forcing) is unimodal and it has afull cascade of period doubling bifurcations. Between one period doubling andthe 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 anextension of the one-dimensional (doubling) renormalization operator to thequasi-periodic case. We proved that, if the family satisfies suitablehypotheses, the two parameter family has two curves of reducibility lossbifurcation around these parameter values. In the present work we study theasymptotic behavior of these bifurcations when $n$ grows to infinity. We showthat the asymptotic behavior depends on the Fourier expansion of thequasi-periodic coupling of the family. The theory developed here provides atheoretical explanation to the behavior that can be observed numerically.

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