TITLE: A renormalization operator for 1D maps under quasi-periodic perturbations AUTHORS: Angel Jorba^(1), Pau Rabassa^(2) and Joan Carles Tatjer^(1) (1) Departament de Matematica Aplicada i Analisi Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain (2) School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom E-mails: angel@maia.ub.es, p.sans@qmul.ac.uk, jcarles@maia.ub.es ABSTRACT: This paper concerns with the reducibility loss of (periodic) invariant curves of quasi-periodically forced one dimensional maps and its relationship with the renormalization operator. Let $g_\alpha$ be a one-parametric family of one dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, there exists a parameter value $\alpha_n$ such that $g_{\alpha_n}$ has a superstable periodic orbit of period $2^n$. Consider a quasi-periodic perturbation (with only one frequency) of the one dimensional family of maps, and let us call $\varepsilon$ the perturbing parameter. For $\varepsilon$ small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. Under suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value $(\alpha_n,0)$, which can be locally expressed as $(\alpha_n^+(\varepsilon), \varepsilon)$ and $(\alpha_n^-(\varepsilon), \varepsilon)$. We propose an extension of the classic one-dimensional (doubling) renormalization operator to the quasi-periodic case. We show that this extension is well defined and the operator is differentiable. Moreover, we show that the slopes of reducibility loss bifurcation $\frac{d}{d\varepsilon} \alpha_n^\pm(0)$ can be written in terms of the tangent map of the new quasi-periodic renormalization operator. In particular, our result applies to the families of quasi-periodic forced perturbations of the Logistic Map typically encountered in the literature. We also present a numerical study that demonstrates that the asymptotic behaviour of $\{\frac{d}{d\varepsilon} \alpha_n^\pm(0)\}_{n\geq 0}$ is governed by the dynamics of the proposed quasi-periodic renormalization operator.