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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.

[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.

[3]

D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (2004). doi: 10.4171/003.

[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.

[13]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems,, Ergodic Theory Dynam. Systems, (2014). doi: 10.1017/etds.2014.43.

[14]

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

[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.

[16]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Asterisque, 358 (2013), 75.

show all references

References:
[1]

A. Avila, J. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents,, Asterisque, 358 (2013), 13.

[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.

[3]

D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (2004). doi: 10.4171/003.

[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.

[13]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems,, Ergodic Theory Dynam. Systems, (2014). doi: 10.1017/etds.2014.43.

[14]

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

[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.

[16]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Asterisque, 358 (2013), 75.

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