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  February 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
References:
[1]

M. Asaoka, Rigidity of certain solvable actions on the sphere, Geometry & Topology, 16 (2012), 1835-1857. 

[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.

[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.

[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.

[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.

[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.

[7]

R. De la LlaveJ. 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.

[8]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity of Lyapunov spectrum for toral automorphisms, preprint, arXiv: 1808.06249.

[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.

[10]

N. Karaliolios, Local rigidity of diophantine translations in higher-dimensional tori, Regular and Chaotic Dynamics, 23 (2018), 12-25. doi: 10.1134/S1560354718010021.

[11]

Y. Katzenlson, Ergodic automorphism of $\mathbb T^n$ are bernoulli shifts, Israel Journal of Mathematics, 10 (1971), 186-195. doi: 10.1007/BF02771569.

[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.

[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.

show all references

References:
[1]

M. Asaoka, Rigidity of certain solvable actions on the sphere, Geometry & Topology, 16 (2012), 1835-1857. 

[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.

[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.

[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.

[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.

[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.

[7]

R. De la LlaveJ. 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.

[8]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity of Lyapunov spectrum for toral automorphisms, preprint, arXiv: 1808.06249.

[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.

[10]

N. Karaliolios, Local rigidity of diophantine translations in higher-dimensional tori, Regular and Chaotic Dynamics, 23 (2018), 12-25. doi: 10.1134/S1560354718010021.

[11]

Y. Katzenlson, Ergodic automorphism of $\mathbb T^n$ are bernoulli shifts, Israel Journal of Mathematics, 10 (1971), 186-195. doi: 10.1007/BF02771569.

[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.

[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.

[1]

Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017

[2]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[3]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251

[4]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130

[5]

Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271

[6]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

[7]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[8]

Rafael de la Llave, A. Windsor. Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1141-1154. doi: 10.3934/dcds.2011.29.1141

[9]

Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471

[10]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[11]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

[12]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[13]

Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040

[14]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453

[15]

Bertuel Tangue Ndawa. Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures. Journal of Geometric Mechanics, 2022, 14 (3) : 409-426. doi: 10.3934/jgm.2022006

[16]

Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003

[17]

Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639

[18]

Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125

[19]

Danijela Damjanovic, Anatole Katok. Local rigidity of homogeneous parabolic actions: I. A model case. Journal of Modern Dynamics, 2011, 5 (2) : 203-235. doi: 10.3934/jmd.2011.5.203

[20]

Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (217)
  • HTML views (269)
  • Cited by (0)

Other articles
by authors

[Back to Top]