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January  2016, 36(1): 245-259. doi: 10.3934/dcds.2016.36.245

Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

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

Received  August 2014 Revised  February 2015 Published  June 2015

We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Hölder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a continuous conjugacy between two cocycles.
Citation: Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245
References:
 [1] A. Avila, J. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents,, Asterisque, 358 (2013), 13. Google Scholar [2] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451. doi: 10.4007/annals.2010.171.451. Google Scholar [3] D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. Google Scholar [4] A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems,, Math. Res. Letters, 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. Google Scholar [5] A. Katok and V. Nitică, Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem,, Cambridge University Press, (2011). doi: 10.1017/CBO9780511803550. Google Scholar [6] B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems,, Geom. Dedicata, 167 (2013), 167. doi: 10.1007/s10711-012-9808-z. Google Scholar [7] 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 [8] R. de la Llave, Smooth conjugacy and SRB measures for uniformly and nonuniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. Google Scholar [9] 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 [10] V. Nitică and A. Török, Regularity of the transfer map for cohomologous cocycles,, Ergodic Theory Dynam. Systems, 18 (1998), 1187. doi: 10.1017/S0143385798117480. Google Scholar [11] Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (2004). doi: 10.4171/003. 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] V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems,, Ergodic Theory Dynam. Systems, (2014). doi: 10.1017/etds.2014.43. Google Scholar [14] K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 703. doi: 10.1017/S0143385799146790. Google Scholar [15] M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math., 167 (2008), 643. doi: 10.4007/annals.2008.167.643. Google Scholar [16] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Asterisque, 358 (2013), 75. Google Scholar

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References:
 [1] A. Avila, J. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents,, Asterisque, 358 (2013), 13. Google Scholar [2] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451. doi: 10.4007/annals.2010.171.451. Google Scholar [3] D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. Google Scholar [4] A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems,, Math. Res. Letters, 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. Google Scholar [5] A. Katok and V. Nitică, Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem,, Cambridge University Press, (2011). doi: 10.1017/CBO9780511803550. Google Scholar [6] B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems,, Geom. Dedicata, 167 (2013), 167. doi: 10.1007/s10711-012-9808-z. Google Scholar [7] 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 [8] R. de la Llave, Smooth conjugacy and SRB measures for uniformly and nonuniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. Google Scholar [9] 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 [10] V. Nitică and A. Török, Regularity of the transfer map for cohomologous cocycles,, Ergodic Theory Dynam. Systems, 18 (1998), 1187. doi: 10.1017/S0143385798117480. Google Scholar [11] Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (2004). doi: 10.4171/003. 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] V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems,, Ergodic Theory Dynam. Systems, (2014). doi: 10.1017/etds.2014.43. Google Scholar [14] K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles,, Ergodic Theory Dynam. Systems, 19 (1999), 703. doi: 10.1017/S0143385799146790. Google Scholar [15] M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents,, Ann. of Math., 167 (2008), 643. doi: 10.4007/annals.2008.167.643. Google Scholar [16] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Asterisque, 358 (2013), 75. Google Scholar
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