A renormalization operator for 1D maps under quasi-periodic perturbations

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 gα 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 αn such that gαn has a superstable periodic orbit of period 2. Consider a quasi-periodic perturbation (with only one frequency) of the one dimensional family of maps, and let us call ε the perturbing parameter. For ε small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on α and ε) of the perturbed system. Under suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value (αn, 0), which can be locally expressed as (α + n (ε), ε) and (α− n (ε), ε). 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 d dεα ± n (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 { d dεα ± n (0)}n≥0 is governed by the dynamics of the proposed quasi-periodic renormalization operator. AMS classification scheme numbers: 37C55, 37E20, 37G35