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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

Boris Kalinin 1, and  Victoria Sadovskaya 1,

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.
Keywords: Cocycle, cohomology, holonomy, partially hyperbolic diffeomorphism..
Mathematics Subject Classification: Primary: 37D30, 37C15, 37H0.
Citation: Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 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), 13-74.  Google Scholar

[2]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. 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), 55-67.  Google Scholar

[4]

A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210. 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, Cambridge, 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), 167-188. 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), 1055-1100. 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), 289-320. 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), 592-600. 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), 1187-1209. doi: 10.1017/S0143385798117480.  Google Scholar

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society (EMS), Zurich, 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), 2879-2895. 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, Available on CJO 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), 703-721. 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), 643-680. doi: 10.4007/annals.2008.167.643.  Google Scholar

[16]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Asterisque, 358 (2013), 75-165.  Google Scholar

show all references

References:
[1]

A. Avila, J. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents, Asterisque, 358 (2013), 13-74.  Google Scholar

[2]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. 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), 55-67.  Google Scholar

[4]

A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210. 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, Cambridge, 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), 167-188. 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), 1055-1100. 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), 289-320. 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), 592-600. 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), 1187-1209. doi: 10.1017/S0143385798117480.  Google Scholar

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society (EMS), Zurich, 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), 2879-2895. 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, Available on CJO 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), 703-721. 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), 643-680. doi: 10.4007/annals.2008.167.643.  Google Scholar

[16]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Asterisque, 358 (2013), 75-165.  Google Scholar

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