Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

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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

Abstract
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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 and 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.
[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.
[3]	D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society (EMS), Zurich, 2004. doi: 10.4171/003.
[12]	M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc., 353 (2001), 2879-2895. doi: 10.1090/S0002-9947-01-02708-8.
[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.
[14]	K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790.
[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.
[16]	A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Asterisque, 358 (2013), 75-165.

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.
[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.
[3]	D. Dolgopyat, Livsic theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society (EMS), Zurich, 2004. doi: 10.4171/003.
[12]	M. Pollicott and C. P. Walkden, Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc., 353 (2001), 2879-2895. doi: 10.1090/S0002-9947-01-02708-8.
[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.
[14]	K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790.
[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.
[16]	A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Asterisque, 358 (2013), 75-165.

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