- Tytuł:
- Ergodicity for piecewise smooth cocycles over toral rotations
- Autorzy:
- Iwanik, Anzelm
- Powiązania:
- https://bibliotekanauki.pl/articles/1205331.pdf
- Data publikacji:
- 1998
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Opis:
- 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.
- Źródło:
-
Fundamenta Mathematicae; 1998, 157, 2-3; 235-244
0016-2736 - Pojawia się w:
- Fundamenta Mathematicae
- Dostawca treści:
- Biblioteka Nauki
Artykuł