A class of real cocycles over an irrational rotation for which Rokhlin cocycle extensions have Lebesgue component in the spectrum
Keywords
Ergodicity, Rokhlin cocycle extension, Lebesgue spectrum, mixing, joining, ELFAbstract
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.Downloads
Published
2004-12-01
How to Cite
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WYSOKIŃSKA, Magdalena. A class of real cocycles over an irrational rotation for which Rokhlin cocycle extensions have Lebesgue component in the spectrum. Topological Methods in Nonlinear Analysis. Online. 1 December 2004. Vol. 24, no. 2, pp. 387 - 407. [Accessed 28 March 2024].
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