Siegel discs, Herman rings and the Arnold family (Q2716156)

The authors consider the families of analytic functions \(P_{\lambda, d}(z)=\lambda z(1+z/d)^d\), \(E_\lambda(z) =\lambda ze^z\) for \(\lambda= e^{2\pi i\alpha}\), \(\alpha\in R\setminus Q\) and prove that the Brjuno condition is necessary and sufficient for the linearizability of \(P_{\lambda d}\) and \(E_\lambda\). In the second part of the paper they proceed to investigate the complex Arnold family \(F_{a,b}(z) =e^{2\pi ia}ze^{\pi b(z-1/z)}\) and its Fatou set. Finally, they study the boundaries of the Siegel discs of \(P_{\lambda, d}\) and \(E_\lambda\) and the boundaries of the Arnold-Herman rings of \(F_{a,b}\) and prove, under the assumption that the rotation number is the constant type, that the boundary of the Siegel discs are quasicircles containing a critical point of the map.