May  2013, 33(5): 2085-2104. doi: 10.3934/dcds.2013.33.2085

Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  April 2011 Revised  October 2012 Published  December 2012

We consider Hölder continuous $GL(2,\mathbb{R})$-valued cocycles over a transitive Anosov diffeomorphism. We give a complete classification up to Hölder cohomology of cocycles with one Lyapunov exponent and of cocycles that preserve two transverse Hölder continuous sub-bundles. We prove that a measurable cohomology between two such cocycles is Hölder continuous. We also show that conjugacy of periodic data for two such cocycles does not always imply cohomology, but a slightly stronger assumption does. We describe examples that indicate that our main results do not extend to general $GL(2,\mathbb{R})$-valued cocycles.
Citation: Victoria Sadovskaya. Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2085-2104. doi: 10.3934/dcds.2013.33.2085
References:
[1]

A. Gogolev, On diffeomorphismsHölder conjugate to Anosov ones,, Ergodic Theory Dynam. Systems, 30 (2010), 441. doi: 10.1017/S0143385709000169. Google Scholar

[2]

M. Guysinsky, Some results about Livšic theorem for $2\times 2$ matrix valued cocycles,, Preprint., (). Google Scholar

[3]

B. Kalinin, Livšic theorem for matrix cocycles,, Annals of Mathematics, 173 (2011), 1025. doi: 10.4007/annals.2011.173.2.11. Google Scholar

[4]

B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems andcriteria of conformality,, Journal of Modern Dynamics, 4 (2010), 419. doi: 10.3934/jmd.2010.4.419. Google Scholar

[5]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Math. and Its Applications, 54 (1995). Google Scholar

[6]

R. de la Llave and A. Windsor, Livšic theorem for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures,, Ergodic Theory Dynam. Systems, 30 (2010), 1055. doi: 10.1017/S014338570900039X. Google Scholar

[7]

A. N. Livšic, Homology properties of Y-systems,, Math. Zametki, 10 (1971), 758. Google Scholar

[8]

A. N. Livšic, Cohomology of dynamical systems,, Math. USSR Izvestija, 6 (1972), 1278. Google Scholar

[9]

V. Niţică and A. Török, Regularity of the transfer map for cohomologous cocycles,, Ergodic Theory Dynam. Systems, 18 (1998), 1187. doi: 10.1017/S0143385798117480. Google Scholar

[10]

M. Nicol and M. Pollicott, Measurable cocycle rigidity for some non-compact groups,, Bull. London Math. Soc., 31 (1999), 592. doi: 10.1112/S0024609399005937. Google Scholar

[11]

W. Parry, The Livšic periodic point theorem for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 687. doi: 10.1017/S0143385799146789. Google Scholar

[12]

M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups,, Trans. Amer. Math. Soc., 353 (2001), 2879. doi: 10.1090/S0002-9947-01-02708-8. Google Scholar

[13]

K. Schmidt, Remarks on Livšic theory for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 703. doi: 10.1017/S0143385799146790. Google Scholar

show all references

References:
[1]

A. Gogolev, On diffeomorphismsHölder conjugate to Anosov ones,, Ergodic Theory Dynam. Systems, 30 (2010), 441. doi: 10.1017/S0143385709000169. Google Scholar

[2]

M. Guysinsky, Some results about Livšic theorem for $2\times 2$ matrix valued cocycles,, Preprint., (). Google Scholar

[3]

B. Kalinin, Livšic theorem for matrix cocycles,, Annals of Mathematics, 173 (2011), 1025. doi: 10.4007/annals.2011.173.2.11. Google Scholar

[4]

B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems andcriteria of conformality,, Journal of Modern Dynamics, 4 (2010), 419. doi: 10.3934/jmd.2010.4.419. Google Scholar

[5]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Math. and Its Applications, 54 (1995). Google Scholar

[6]

R. de la Llave and A. Windsor, Livšic theorem for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures,, Ergodic Theory Dynam. Systems, 30 (2010), 1055. doi: 10.1017/S014338570900039X. Google Scholar

[7]

A. N. Livšic, Homology properties of Y-systems,, Math. Zametki, 10 (1971), 758. Google Scholar

[8]

A. N. Livšic, Cohomology of dynamical systems,, Math. USSR Izvestija, 6 (1972), 1278. Google Scholar

[9]

V. Niţică and A. Török, Regularity of the transfer map for cohomologous cocycles,, Ergodic Theory Dynam. Systems, 18 (1998), 1187. doi: 10.1017/S0143385798117480. Google Scholar

[10]

M. Nicol and M. Pollicott, Measurable cocycle rigidity for some non-compact groups,, Bull. London Math. Soc., 31 (1999), 592. doi: 10.1112/S0024609399005937. Google Scholar

[11]

W. Parry, The Livšic periodic point theorem for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 687. doi: 10.1017/S0143385799146789. Google Scholar

[12]

M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups,, Trans. Amer. Math. Soc., 353 (2001), 2879. doi: 10.1090/S0002-9947-01-02708-8. Google Scholar

[13]

K. Schmidt, Remarks on Livšic theory for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 703. doi: 10.1017/S0143385799146790. Google Scholar

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