# American Institute of Mathematical Sciences

July  2010, 4(3): 571-584. doi: 10.3934/jmd.2010.4.571

## Lipschitz continuous invariant forms for algebraic Anosov systems

 1 Institut de Recherche Mathematique Avancée, UMR 7501 du Centre National de la Recherche Scientifique, 7 Rue René Descartes, 67084, Strasbourg Cedex, France 2 Department of Mathematics, Tufts University, Medford, MA 02155, United States

Received  May 2010 Revised  September 2010 Published  October 2010

We prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz continuous invariant $1$-form. This has corollaries for rigidity of time-changes, and we give a particular application to geometric rigidity of quasiconformal Anosov flows.
Several features of the reasoning are interesting; namely, the use of exterior calculus for Lipschitz continuous forms, the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
Citation: Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571
##### References:

show all references

##### References:
 [1] Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80 [2] Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377 [3] Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873 [4] Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 [5] Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98. [6] Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271 [7] A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133. [8] 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 [9] Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471 [10] Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97. [11] Michael Hochman. Smooth symmetries of $\times a$-invariant sets. Journal of Modern Dynamics, 2018, 13: 187-197. doi: 10.3934/jmd.2018017 [12] Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397 [13] Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669 [14] Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931 [15] 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 [16] Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667 [17] Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006 [18] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [19] Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1 [20] 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 - A, 2011, 29 (3) : 1141-1154. doi: 10.3934/dcds.2011.29.1141

2018 Impact Factor: 0.295

## Metrics

• HTML views (0)
• Cited by (2)

• on AIMS