Measure-preserving diffeomorphisms with an arbitrary spectral multiplicity (Q1310477)

Several authors have constructed, for each natural number \(n\), an ergodic transformation whose maximal spectral multiplicity is equal to \(n\). The first such construction is due to \textit{E. A. Robinson jun.} [Invent. Math. 72, 299-314 (1983; Zbl 0519.28008)]. In this paper the authors show that it is possible to give a smooth version of this result. Specifically, for each natural number \(n\), an analytic diffeomorphism whose maximal spectral multiplicity is equal to \(n\) is constructed. These examples are skew products of the form \(T_ \phi: \mathbb{T}^{n+1}\to \mathbb{T}^{n+1}\) (where \(\mathbb{T}\) is the unit circle in \(\mathbb{C}\)), \[ T_ \phi(z,(w_ 1,w_ 2,\dots,w_ n))= (Tz,\phi(z)\cdot (w_ 1,w_ 2,\dots,w_ n)), \] where \(T: \mathbb{T}\to \mathbb{T}\), \(Tz= z\cdot e^{2\pi i\alpha}\), with \(\alpha\) irrational, and \(\phi: \mathbb{T}=\mathbb{T}^ n\) is an analytic cocycle.