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

This paper concerns the reducibility loss of (periodic) invariant curves of quasi-periodically forced one-dimensional maps and its relationship with the renormalization operator. Let g_α 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 α_n such that $g_{\alpha_n}$ has a superstable periodic orbit of period 2ⁿ. 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 a suitable hypothesis, it is known that there exist two reducibility loss bifurcation curves around each parameter value (α_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 $\frac{\rmd}{\rmd\varepsilon} \alpha_n^\pm(0)$ 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 $\{\frac{\rmd}{\rmd\varepsilon} \alpha_n^\pm(0)\}_{n\geq 0}$ is governed by the dynamics of the proposed quasi-periodic renormalization operator.