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Global rigidity of smooth
${\mathbb{Z}}\ltimes _{\unicode{x3bb}}{\mathbb{R}}$-actions on 
${\mathbb{T}}^{2}$
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 14 October 2024, pp. 1-16
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- Article
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For
$\unicode{x3bb}>1$, we consider the locally free 
${\mathbb Z}\ltimes _\unicode{x3bb} \mathbb R$ actions on 
$\mathbb T^2$. We show that if the action is 
$C^r$ with 
$r\geq 2$, then it is 
$C^{r-\epsilon }$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along the expanding eigen-direction of the automorphism. In contrast, there exists a 
$C^{1+\alpha }$-action which is semi-conjugate, but not topologically conjugate to an affine action.
