The critical renormalization fixed point for commuting pairs of area-preserving maps (Q982433)

This paper studies the renormalization operator \(\mathcal R\) which acts on pairs of area-preserving maps \((F,G)\) of the plane such that \[ \mathcal R(F,G)=(\Lambda^{-1}G \Lambda, \Lambda^{-1}FG \Lambda) \] where \(\Lambda(x,z)=(\lambda x, \mu z)\) (\(\lambda\) and \(\mu\) are determined by \(\Lambda^{-1}FG \Lambda(0,0)=(-1,-1)\)). The main result proves that \(\mathcal R\) has a fixed point \((F,G)\). These maps \(F\) and \(G\) are area-preserving, real analytic, reversible, satisfy a twist condition, and commute. \textit{R. S. MacKay} [Renormalisation in area-preserving maps. Advanced Series in Nonlinear Dynamics, 6, Singapore: World Scientific (1993; Zbl 0791.58002)] conjectured the existence of this fixed point in his 1982 Princeton Thesis. Partial results were given by \textit{A. Stirnemann} [Commun. Math. Phys. 188, No. 3, 723--735 (1997; Zbl 0888.58057)].