## Topological Methods in Nonlinear Analysis

### Completely squashable smooth ergodic cocycles over irrational rotations

Dalibor Volný

#### Abstract

Let $\alpha$ be an irrational number and the trasformation $$Tx \mapsto x+\alpha\,{\rm mod}\,1, \quad x\in [0,1),$$ represent an irrational rotation of the unit circle. We construct an ergodic and completely squashable smooth real extension, i.e. we find a real analytic or $k$ time continuously differentiable real function $F$ such that for every $\lambda\neq 0$ there exists a commutor $S_\lambda$ of $T$ such that $F\circ S_\lambda$ is $T$-cohomologous to $\lambda\varphi$ and the skew product $T_F(x,y) = (Tx, y+F(x))$ is ergodic.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 2 (2003), 331-344.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266340

Mathematical Reviews number (MathSciNet)
MR2036380

Zentralblatt MATH identifier
1057.37003

#### Citation

Volný, Dalibor. Completely squashable smooth ergodic cocycles over irrational rotations. Topol. Methods Nonlinear Anal. 22 (2003), no. 2, 331--344. https://projecteuclid.org/euclid.tmna/1475266340

#### References

• J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50 , Amer. Math. Soc., Providence, R.I, U.S. (1997). (Corrections page http://www.math.tau.ac.il/$\sim$aaro/book) \ref\key 2 ––––, The asymptotic distributional behaviour of transformations preserving infinite measures , J. Anal. Math., 39 (1981), 203–234 \ref\key 3 ––––, The intrinsic normalising constants of transformations preserving infinite measures , J. Anal. Math., 49 (1987), 239–270 \ref\key 4
• J. Aaronson, M. Lemańczyk, C. Mauduit and H. Nakada, Koksma'a inequality and group extensions of Kronecker transformations , Algorithms, Fractals and Dynamics, Proceedings of the Hayashibara Forum '92, Okayama, Japan and the Kyoto symposium (Y. Takahashi, ed.), Plenum Publishing Company, New York (1995), 27–50 \ref\key 5
• J. Aaronson, M.Lemańczyk and D. Volný, A cut salad of cocycles , Fund. Math., 158 , 99–119 (1998) \ref\key 6
• L. Baggett, K. Merrill, Smooth cocycles for an irrational rotation , Israel J. Math., 79 (1992), 281–288 \ref\key 7
• P. Billingsley, Ergodic Theory and Information, Wiley (1965) \ref\key 8
• A. B. Hajian, Y. Ito and S. Kakutani, Invariant measures and orbits of dissipative transformations , Adv. Math., 9 (1972), 52–65 \ref\key 9
• J. Kwiatkowski, M. Lemańczyk, D. Rudolph, Weak isomorphisms of measure-preserving diffeomorphisms , Israel J. Math., 80 (1992), 33–64 \ref\key10 ––––, A class of real cocycles having an analytic coboundary modification , Israel J. Math., 87 (1994), 337–360
• \ref \key 11 P. Liardet and D. Volný, Sums of continuous and differenctiable functions in dynamical systems , Israel J. Math., 98 , 29–60 (1997) \ref\key 12
• K. Schmidt, Cocycles of Ergodic Transformation Groups, Lect. Notes in Math., 1 , Mac Millan Co. of India(1977) \ref \key 13
• D. Volný, Constructions of smooth and analytic cocycles over irrational circle rotations , Comment. Math. Univ. Carolin., 36 , 745–764 (1995)