Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik

Fundamenta Mathematicae (1998)

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

Abstract

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Let α be an ergodic rotation of the d-torus 𝕋 d = ℝ d / ℤ d . For any piecewise smooth function f : 𝕋 d → ℝ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on L 2 ( 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 : 𝕋 d + 1 → 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.

How to cite

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Iwanik, 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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  1. [1] H. Anzai, Ergodic skew product transformations on the torus, Osaka J. Math. 3 (1951), 88-99. Zbl0043.11203
  2. [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. [3] H. Helson, Cocycles on the circle, J. Operator Theory 16 (1986), 189-199. Zbl0644.43003
  4. [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. [5] H. Iwaniec, On a problem of Jacobsthal, Demonstratio Math. 11 (1978), 225-231. 
  6. [6] A. Iwanik, Anzai skew products with Lebesgue component of infinite multiplicity, Bull. London Math. Soc. 29 (1997), 195-199. Zbl0865.28012
  7. [7] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc., to appear. Zbl0931.28015
  8. [8] A. Khintchine, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706-714. 
  9. [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. [10] D. A. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. Zbl0703.28009

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