# Ergodicity for piecewise smooth cocycles over toral rotations

Fundamenta Mathematicae (1998)

- Volume: 157, Issue: 2-3, page 235-244
- ISSN: 0016-2736

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topIwanik, Anzelm. "Ergodicity for piecewise smooth cocycles over toral rotations." Fundamenta Mathematicae 157.2-3 (1998): 235-244. <http://eudml.org/doc/212288>.

@article{Iwanik1998,

abstract = {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.},

author = {Iwanik, Anzelm},

journal = {Fundamenta Mathematicae},

keywords = {skew product; ergodicity of cylinder flows},

language = {eng},

number = {2-3},

pages = {235-244},

title = {Ergodicity for piecewise smooth cocycles over toral rotations},

url = {http://eudml.org/doc/212288},

volume = {157},

year = {1998},

}

TY - JOUR

AU - Iwanik, Anzelm

TI - Ergodicity for piecewise smooth cocycles over toral rotations

JO - Fundamenta Mathematicae

PY - 1998

VL - 157

IS - 2-3

SP - 235

EP - 244

AB - 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.

LA - eng

KW - skew product; ergodicity of cylinder flows

UR - http://eudml.org/doc/212288

ER -

## References

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- [6] A. Iwanik, Anzai skew products with Lebesgue component of infinite multiplicity, Bull. London Math. Soc. 29 (1997), 195-199. Zbl0865.28012
- [7] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc., to appear. Zbl0931.28015
- [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. Zbl0999.47024
- [10] D. A. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. Zbl0703.28009

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