# American Institute of Mathematical Sciences

February  2021, 41(2): 553-567. doi: 10.3934/dcds.2020269

## Local rigidity of certain solvable group actions on tori

 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Received  January 2020 Revised  June 2020 Published  July 2020

In this paper, we study a local rigidity property of $\mathbb Z \ltimes_\lambda \mathbb R$ affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbations close enough to an affine action is smoothly conjugate to the affine action up to constant time change.

Citation: Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
##### References:
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##### References:
 [1] M. Asaoka, Rigidity of certain solvable actions on the sphere, Geometry & Topology, 16 (2012), 1835-1857.   Google Scholar [2] M. Asaoka, Rigidity of certain solvable actions on the torus, Geometry, Dynamics, and Foliations 2013, 269-281, Adv. Stud. Pure Math., 72, Math. Soc. Japan, Tokyo, 2017. doi: 10.2969/aspm/07210269.  Google Scholar [3] L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$, Geometry & Topology, 8 (2004), 877-924.  doi: 10.2140/gt.2004.8.877.  Google Scholar [4] J. L. Dias, local conjugacy classes for analytic torus flows, Journal of Differential Equations, 245 (2008), 468-489.  doi: 10.1016/j.jde.2008.04.006.  Google Scholar [5] R. De la Llave, A tutorial on KAM theory, Proceedings of Symposia in Pure Mathematics, 69 (2001), 175-292.  doi: 10.1090/pspum/069/1858536.  Google Scholar [6] R. De la Llave, Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems, Communications in Mathematical Physics, 150 (1992), 289-320.  doi: 10.1007/BF02096662.  Google Scholar [7] R. De la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livisic cohomology equation, Annals of Mathematics, 123 (1986), 537-611.  doi: 10.2307/1971334.  Google Scholar [8] A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity of Lyapunov spectrum for toral automorphisms, preprint, arXiv: 1808.06249. Google Scholar [9] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bulletin of American Mathematical Society, 7 (1982), 65-122. doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar [10] N. Karaliolios, Local rigidity of diophantine translations in higher-dimensional tori, Regular and Chaotic Dynamics, 23 (2018), 12-25. doi: 10.1134/S1560354718010021.  Google Scholar [11] Y. Katzenlson, Ergodic automorphism of $\mathbb T^n$ are bernoulli shifts, Israel Journal of Mathematics, 10 (1971), 186-195. doi: 10.1007/BF02771569.  Google Scholar [12] R. Saghin and J. Yang, Lyapunov exponents and rigidity of Anosov automorphisms and skew products, Advance in Mathematics, 355 (2019), 106764, 45 pp. doi: 10.1016/j.aim.2019.106764.  Google Scholar [13] A. Wilkinson and J. Xue, Rigidity of some abelian-by-cyclic solvable group actions on $\mathbb T^N$, Communications in Mathematical Physics, 376 (2020), 1223-1259. doi: 10.1007/s00220-019-03658-3.  Google Scholar
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