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arXiv · 2403.10060

Global rigidity of smooth ${\mathbb Z}\ltimes_\lambda{\mathbb R}$-actions on ${\mathbb T}^2$

Changguang Dong·Yi Shi
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Abstract

For $\lambda>1$, we consider the locally free ${\mathbb Z}\ltimes_\lambda{\mathbb R}$ actions on ${\mathbb T}^2$. We show that, if the action is $C^r$ with $r\geq2$, then it is $C^{r-\epsilon}$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along 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.

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Changguang Dong, Yi Shi. 2024-03-15. Global rigidity of smooth ${\mathbb Z}\ltimes_\lambda{\mathbb R}$-actions on ${\mathbb T}^2$. https://arxiv.org/abs/2403.10060

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