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## Fundamenta Mathematicae

1998 | 157 | 2-3 | 235-244
Tytuł artykułu

### Ergodicity for piecewise smooth cocycles over toral rotations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let α be an ergodic rotation of the d-torus $\mathbb{T}^d = ℝ^d/ℤ^d$. For any piecewise smooth function $f: \mathbb{T}^d → ℝ$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on $L^2(\mathbb{T}^d)$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product $S_f: \mathbb{T}^{d+1} → \mathbb{T}^{d+1}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V is singular. In the case d = 1 our technique allows us to extend Pask's result on ergodicity of cylinder flows on T×ℝ to arbitrary piecewise absolutely continuous real-valued cocycles f satisfying ʃf = 0 and ʃf' ≠ 0.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
235-244
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-09-30
poprawiono
1997-12-22
Twórcy
autor
• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Bibliografia
• [1] H. Anzai, Ergodic skew product transformations on the torus, Osaka J. Math. 3 (1951), 88-99.
• [2] G. H. Choe, Products of operators with singular continuous spectra, in: Proc. Sympos. Pure Math. 51, Amer. Math. Soc., Providence, R.I., 1990, 65-68.
• [3] H. Helson, Cocycles on the circle, J. Operator Theory 16 (1986), 189-199.
• [4] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5-234.
• [5] H. Iwaniec, On a problem of Jacobsthal, Demonstratio Math. 11 (1978), 225-231.
• [6] A. Iwanik, Anzai skew products with Lebesgue component of infinite multiplicity, Bull. London Math. Soc. 29 (1997), 195-199.
• [7] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc., to appear.
• [8] A. Khintchine, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706-714.
• [9] H. A. Medina, Spectral types of unitary operators arising from irrational rotations on the circle group, Michigan Math. J. 41 (1994), 39-49.
• [10] D. A. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74.
Typ dokumentu
Bibliografia
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