Siegel discs, Herman rings and the Arnold family

We show that the rotation number of an analytically linearizable element of the Arnold family f(a,b) (x) = x + a + bsin(2 pix) (mod 1), a, b is an ele ment of R, 0 < b < 1/(2 pi), satisfies the Brjuno condition. Conversely, fo r every Brjuno rotation number there exists an analytically linearizable el ement of the Arnold family. Along the way we prove the necessity of the Brj uno condition for linearizability of P-lambda ,P-d(z) = lambdaz(1 + z/d)(d) and E lambda (z) = lambda ze(z), lambda = e2(pi ia), at 0. We also investi gate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of P-lambda ,P-d and E-lambda are qu asidiscs with a critical point on the boundary if the rotation number is of constant type.