Ergodicity for piecewise smooth cocycles over toral rotations
Iwanik, Anzelm
Fundamenta Mathematicae, Tome 158 (1998), p. 235-244 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let α be an ergodic rotation of the d-torus 𝕋d=ℝd/ℤd. For any piecewise smooth function f:𝕋d→ℝ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on L2(𝕋d) is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product Sf:𝕋d+1→𝕋d+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.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:212288 
@article{bwmeta1.element.bwnjournal-article-fmv157i2p235bwm,
     author = {Anzelm Iwanik},
     title = {Ergodicity for piecewise smooth cocycles over toral rotations},
     journal = {Fundamenta Mathematicae},
     volume = {158},
     year = {1998},
     pages = {235-244},
     zbl = {0918.28015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p235bwm}
}

Iwanik, Anzelm. Ergodicity for piecewise smooth cocycles over toral rotations. Fundamenta Mathematicae, Tome 158 (1998) pp. 235-244. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p235bwm/

[00000] [1] H. Anzai, Ergodic skew product transformations on the torus, Osaka J. Math. 3 (1951), 88-99. | Zbl 0043.11203

[00001] [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.

[00002] [3] H. Helson, Cocycles on the circle, J. Operator Theory 16 (1986), 189-199. | Zbl 0644.43003

[00003] [4] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5-234.

[00004] [5] H. Iwaniec, On a problem of Jacobsthal, Demonstratio Math. 11 (1978), 225-231.

[00005] [6] A. Iwanik, Anzai skew products with Lebesgue component of infinite multiplicity, Bull. London Math. Soc. 29 (1997), 195-199. | Zbl 0865.28012

[00006] [7] A. Iwanik, M. Lemańczyk and C. Mauduit, Piecewise absolutely continuous cocycles over irrational rotations, J. London Math. Soc., to appear. | Zbl 0931.28015

[00007] [8] A. Khintchine, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706-714.

[00008] [9] H. A. Medina, Spectral types of unitary operators arising from irrational rotations on the circle group, Michigan Math. J. 41 (1994), 39-49. | Zbl 0999.47024

[00009] [10] D. A. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. | Zbl 0703.28009