### Completely squashable smooth ergodic cocycles over irrational rotations

DOI: http://dx.doi.org/10.12775/TMNA.2003.043

#### Abstract

Let $\alpha$ be an irrational number and the trasformation

$$

Tx \mapsto x+\alpha \bmod 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.

$$

Tx \mapsto x+\alpha \bmod 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.

#### Keywords

Cocycles over irrational rotations; squashable cocycles

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