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 $L^2({\mathcal B}/{\mathbb Z})\ni g \mapsto e^{2\pi i c f}\cdot g\circ T$ 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.