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Global rigidity of smooth ${\mathbb{Z}}\ltimes _{\unicode{x3bb}}{\mathbb{R}}$-actions on ${\mathbb{T}}^{2}$

Part of: Smooth dynamical systems: general theory

Published online by Cambridge University Press:  14 October 2024

CHANGGUANG DONG  and
YI SHI
CHANGGUANG DONG*
Affiliation:
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, Tianjin, China
YI SHI
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610065, Sichuan, China (e-mail: shiyi@scu.edu.cn)
*
e-mail: dongchg@nankai.edu.cn
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Abstract

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.

Keywords

global rigiditysolvable group actionAnosov diffeomorphismsmooth conjugacy

MSC classification

Primary: 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 4 , April 2025 , pp. 1161 - 1176
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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