American Institute of Mathematical Sciences

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

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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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