September  2017, 37(9): 4959-4972. doi: 10.3934/dcds.2017213

Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  December 2016 Revised  April 2017 Published  June 2017

Fund Project: The author is supported in part by NSF grant DMS-1301693

We consider Hölder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Hölder continuously cohomologous if and only if they have Hölder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.

Citation: Victoria Sadovskaya. Fiber bunching and cohomology for Banach cocycles over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4959-4972. doi: 10.3934/dcds.2017213
References:
[1]

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

[2]

L. Backes, Rigidity of fiber bunched cocycles, Bul. Brazilian Math. Soc., 46 (2015), 163-179. doi: 10.1007/s00574-015-0089-7. Google Scholar

[3]

L. Backes and A. Kocsard, Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 36 (2016), 1703-1722. doi: 10.1017/etds.2014.149. Google Scholar

[4]

G. Grabarnik and M. Guysinsky, Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390. doi: 10.3934/dcds.2017187. Google Scholar

[5]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. Google Scholar

[6]

B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188. doi: 10.1007/s10711-012-9808-z. Google Scholar

[7]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, To appear in Ergodic Theory Dynam. Systems.Google Scholar

[8]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54 Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[9]

A. Katok and V. Nitica, Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem, Cambridge University Press, 2011. doi: 10.1017/CBO9780511803550. Google Scholar

[10]

A. N. Livšic, Homology properties of Y-systems, Math. Zametki, 10 (1971), 555-564. Google Scholar

[11]

A. N. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 36 (1972), 1296-1320. Google Scholar

[12]

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

[13]

V. Nitica and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810. doi: 10.1215/S0012-7094-95-07920-4. Google Scholar

[14]

V. Nitica 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

[15]

W. Parry, The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789. Google Scholar

[16]

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

[17]

V. Sadovskaya, Cohomology of GL(2, $\mathbb{R} $)-valued cocycles over hyperbolic systems, Discrete and Continuous Dynamical Systems, 33 (2013), 2085-2104. doi: 10.3934/dcds.2013.33.2085. Google Scholar

[18]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 35 (2015), 2669-2688. doi: 10.1017/etds.2014.43. Google Scholar

[19]

K. Schmidt, Remarks on Livšic theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790. Google Scholar

[20]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Math., 167 (2008), 643-680. doi: 10.4007/annals.2008.167.643. Google Scholar

show all references

References:
[1]

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

[2]

L. Backes, Rigidity of fiber bunched cocycles, Bul. Brazilian Math. Soc., 46 (2015), 163-179. doi: 10.1007/s00574-015-0089-7. Google Scholar

[3]

L. Backes and A. Kocsard, Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 36 (2016), 1703-1722. doi: 10.1017/etds.2014.149. Google Scholar

[4]

G. Grabarnik and M. Guysinsky, Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017), 4379-4390. doi: 10.3934/dcds.2017187. Google Scholar

[5]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math., 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. Google Scholar

[6]

B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188. doi: 10.1007/s10711-012-9808-z. Google Scholar

[7]

B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, To appear in Ergodic Theory Dynam. Systems.Google Scholar

[8]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54 Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[9]

A. Katok and V. Nitica, Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem, Cambridge University Press, 2011. doi: 10.1017/CBO9780511803550. Google Scholar

[10]

A. N. Livšic, Homology properties of Y-systems, Math. Zametki, 10 (1971), 555-564. Google Scholar

[11]

A. N. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 36 (1972), 1296-1320. Google Scholar

[12]

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

[13]

V. Nitica and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995), 751-810. doi: 10.1215/S0012-7094-95-07920-4. Google Scholar

[14]

V. Nitica 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

[15]

W. Parry, The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: 10.1017/S0143385799146789. Google Scholar

[16]

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

[17]

V. Sadovskaya, Cohomology of GL(2, $\mathbb{R} $)-valued cocycles over hyperbolic systems, Discrete and Continuous Dynamical Systems, 33 (2013), 2085-2104. doi: 10.3934/dcds.2013.33.2085. Google Scholar

[18]

V. Sadovskaya, Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 35 (2015), 2669-2688. doi: 10.1017/etds.2014.43. Google Scholar

[19]

K. Schmidt, Remarks on Livšic theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: 10.1017/S0143385799146790. Google Scholar

[20]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Math., 167 (2008), 643-680. doi: 10.4007/annals.2008.167.643. Google Scholar

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