A class of real cocycles over an irrational rotation for which Rokhlin cocycle extensions have Lebesgue component in the spectrum

Abstract

We describe a class of functions $f\colon \mathcal B/\mathbb Z \to \mathcal B$ such that for each irrational rotation $Tx=x+\alpha$, where $\alpha$ has the property that the sequence of aritmethical means of its partial quotients is bounded, the corresponding weighted unitary operators $L^2({\mathbb{R} / \mathbb{Z})\ni g \mapsto e^{2 \pi icf} \cdot g \circ T$ have a Lebesgue spectrum for each $c\in \mathbb R\setminus\{0\}$. We show that for such $f$ and $T$ and for an arbitrary ergodic $\mathcal B$-action $\mathcal S=(S_t)_{t\in\mathcal B}$ on $(Y,\mathcal C,\nu)$ the corresponding Rokhlin cocycle extension $T_{f,\mathcal S}(x,y)=(Tx,S_{f(x)}y)$ acting on $(\mathcal B/\mathbb Z\times Y,\mu \otimes \nu)$ has also a Lebesgue spectrum in the orthogonal complement of $L^2(\mathcal B/\mathbb Z,\mu)$ and moreover the weak closure of powers of $T_{f,\mathcal S}$ in the space of self-joinings consists of ergodic elements.