Siegel discs, Herman rings and the Arnold family

Abstract:

We show that the rotation number of an analytically linearizable element of the Arnold family $f_{a,b}(x)=x+a+b\sin (2\pi x)\pmod 1$, $a,b\in \mathbb {R}$, $0<b<1/(2\pi )$, satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of $P_{\lambda ,d}(z)=\lambda z(1+z/d)^d$ and $E_\lambda (z)=\lambda z e^z$, $\lambda =e^{2\pi i\alpha }$, at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of $P_{\lambda ,d}$ and $E_\lambda$ are quasidiscs with a critical point on the boundary if the rotation number is of constant type.