The Existence of Siegel Disks for the Cremona Map

Abstract

This paper is concerned with the existence of Siegel disks of the Cremona map $F_\alpha(x,y)=(x\cos\alpha -(y-x^2)\sin\alpha,\,\, x\sin\alpha+(y-x^2)\cos\alpha)$ with the parameter $\alpha\in [0,2\pi)$. This problem is reduced to the existence of local invertible analytic solutions to a functional equation with small divisors $\lambda^n+\lambda^{-n}-\lambda-\lambda^{-1}$. The main aim of this paper is to investigate whether this equation with $|\lambda|=1$ has such a solution under the Brjuno condition.