American Institute of Mathematical Sciences

Advanced
February  2008, 22(1&2): 183-200. doi: 10.3934/dcds.2008.22.183

$C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus

Andrey Gogolev 1, and  Misha Guysinsky 1,

1. 

109 McAllister Bldg., University Park, PA 16802, United States, United States

Received  August 2007 Revised  October 2007 Published  June 2008

We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.
Keywords: Anosov diffeomorphism, smooth conjugacy, periodic data..
Mathematics Subject Classification: Primary: 37C15, 37D20; Secondary: 34D1.
Citation: Andrey Gogolev, Misha Guysinsky. $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 183-200. doi: 10.3934/dcds.2008.22.183
[1]

Andrey Gogolev. Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori. Journal of Modern Dynamics, 2008, 2 (4) : 645-700. doi: 10.3934/jmd.2008.2.645
[2]

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 & Continuous Dynamical Systems, 2011, 29 (3) : 1141-1154. doi: 10.3934/dcds.2011.29.1141
[3]

Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017
[4]

Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687
[5]

Oliver Butterley, Carlangelo Liverani. Smooth Anosov flows: Correlation spectra and stability. Journal of Modern Dynamics, 2007, 1 (2) : 301-322. doi: 10.3934/jmd.2007.1.301
[6]

Sylvain Ervedoza, Enrique Zuazua. A systematic method for building smooth controls for smooth data. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1375-1401. doi: 10.3934/dcdsb.2010.14.1375
[7]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
[8]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[9]

José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653
[10]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[11]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[12]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719
[13]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063
[14]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355
[15]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073
[16]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[17]

A. Yu. Ol'shanskii and M. V. Sapir. The conjugacy problem for groups, and Higman embeddings. Electronic Research Announcements, 2003, 9: 40-50.

[18]

Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 867-888. doi: 10.3934/dcds.2018037
[19]

Christian Bonatti, Sylvain Crovisier and Amie Wilkinson. The centralizer of a $C^1$-generic diffeomorphism is trivial. Electronic Research Announcements, 2008, 15: 33-43. doi: 10.3934/era.2008.15.33
[20]

Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy