Non-autonomous parabolic bifurcation

Abstract:

Let $f(z) = z+z^2+O(z^3)$ and $f_\epsilon (z) = f(z) + \epsilon ^2$. A classical result in parabolic bifurcation in one complex variable is the following: if $N-\frac {\pi }{\epsilon }\to 0$ we obtain $(f_\epsilon )^{N} \to \mathcal {L}_f$, where $\mathcal {L}_f$ is the Lavaurs map of $f$. In this paper we study a non-autonomous parabolic bifurcation. We focus on the case of $f_0(z)=\frac {z}{1-z}$. Given a sequence $\{\epsilon _i\}_{1\leq i\leq N}$, we denote $f_n(z) = f_0(z) + \epsilon _n^2$. We give sufficient and necessary conditions on the sequence $\{\epsilon _i\}$ that imply that $f_{N}\circ \ldots f_{1} \to \operatorname {Id}$ (the Lavaurs map of $f_0$). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.