Topological Methods in Nonlinear Analysis

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

Magdalena Wysokińska

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 24, Number 2 (2004), 387-407.

Dates
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1466704925

Mathematical Reviews number (MathSciNet)
MR2114916

Zentralblatt MATH identifier
1062.37003

Citation

Wysokińska, Magdalena. A class of real cocycles over an irrational rotation for which Rokhlin cocycle extensions have Lebesgue component in the spectrum. Topol. Methods Nonlinear Anal. 24 (2004), no. 2, 387--407. https://projecteuclid.org/euclid.tmna/1466704925


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References

  • H. Anzai, Ergodic skew product transformations on the torus , Osaka J. Math., 3 (1951), 88–99 \ref\key 2
  • G. H. Choe, Spectral properties of cocycles, Ph. D. Thesis, University of California, Berkeley(1987) \ref\key 3
  • I. P. Cornfeld, S. W. Fomin and J. G. Sinai, Ergodic Theory, Springer–Verlag, Berlin–Heidelberg–New York (1982) \ref\key 4
  • M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Springer–Verlag, Berlin–Heidelberg (1997) \ref\key 5
  • B. R. Fayad, Skew products over translations on $\T^d$, $d\geq 2$ , Proc. Amer. Math. Soc., 130 (2002), 103–109 \ref\key 6
  • H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, New Jersey (1981) \ref\key 8
  • P. Gabriel, M. Lema\plsmc äczyk and P. Liardet, Ensemble d'invariants pour les produits croisés de Anzai , Mém. Bull. Soc. Math. Fr., 47 (1991) \ref\key 9
  • M. Guenais, Une majoration de la multiplicité spectrale d'opérateurs associés $\grave{a}$ des cocycles réguliers , Israel J. Math., 105 (1998), 263–283 \ref\key 10
  • H. Helson, Cocycles on the circle, J. Operator Theory, 16 (1986), 189–199 \ref\key 11
  • M. Herman, Sur la conjugaison différentiable des difféomorphismes du circle $\grave{a}$ des rotations , Publ. IHES, 49 (1979), 5–234 \ref\key 12
  • A. Iwanik, Anzai skew products with Lebesgue component of infinite multiplicity, Bull. London Math. Soc., 29 (1997), 195–199 \ref\key 13
  • A. Iwanik, M. Lema\plsmc äczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc., 59 (1999), 171–187 \ref\key 14
  • A. Iwanik, M. Lema\plsmc äczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations , Israel J. Math., 83 (1993), 73–95 \ref\key 15
  • M. Lema\plsmc äczyk and E. Lesigne, Ergodicity of Rokhlin cocycles , J. Anal. Math., 85 (2001), 43–86 \ref\key 16
  • M. Lema\plsmc äczyk and F. Parreau, Lifting mixing properties by Rokhlin cocycles, preprint \ref\key 17
  • M. Lema\plsmc äczyk, F. Parreau and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian , Fund. Math., 164 (2000), 253–293 \ref\key 18
  • W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge (1981) \ref\key 19
  • M. Queffélec, Substitution Dynamical Systems – Spectral Analysis , Lecture Notes in Mathematics, 1294 , Springer, Berlin (1974) \ref\key 20
  • V. V. Ryzhikov, Joinings, wreath products, factors and mixing properties of dynamical systems, Russian Acad. Sci. Izv. Math., 42 (1994), 91–114