Existence of the FS-type renormalisation fixed point for unidirectionally-coupled pairs of maps

Abstract

We give the first proof of the existence of a renormalisation fixed-point for period-doubling in pairs of maps of two variables lying in the so-called Feigenbaum-Summation (FS) universality class. The first map represents a subsystem that is unimodal with an extremum of degree two. The dynamics of the second map accumulates an integral characteristic of the dynamics of the first, via a particular form of unidirectional coupling. We prove the existence of the corresponding renormalisation fixed point by rigorous computer-assisted means and gain tight rigorous bounds on the associated universal constants. Our work provides the first step in establishing rigorously the picture conjectured by Kuznetsov et al of the birth, from the FS-type fixed point, of the so-called C-type two-cycle via a period doubling in the dynamics of the renormalisation group transformation itself.