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 and Continuous Dynamical Systems, 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. 

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

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. 

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[1]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985

[2]

Alejandro Adem and Jeff H. Smith. On spaces with periodic cohomology. Electronic Research Announcements, 2000, 6: 1-6.

[3]

Andi Kivinukk, Anna Saksa. On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Mathematical Foundations of Computing, 2022, 5 (3) : 197-218. doi: 10.3934/mfc.2021030

[4]

Hannelore Lisei, Radu Precup, Csaba Varga. A Schechter type critical point result in annular conical domains of a Banach space and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3775-3789. doi: 10.3934/dcds.2016.36.3775

[5]

C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935

[6]

H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50

[7]

Shuying He, Rumei Zhang, Fukun Zhao. A note on a superlinear and periodic elliptic system in the whole space. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1149-1163. doi: 10.3934/cpaa.2011.10.1149

[8]

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

[9]

Ciprian Preda, Petre Preda, Adriana Petre. On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637

[10]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129

[11]

Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593

[12]

Yūki Naito, Takasi Senba. Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1861-1880. doi: 10.3934/cpaa.2013.12.1861

[13]

Victoria Sadovskaya. Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2085-2104. doi: 10.3934/dcds.2013.33.2085

[14]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91

[15]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081

[16]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89

[17]

Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure and Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421

[18]

Yanheng Ding, Fukun Zhao. On a diffusion system with bounded potential. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1073-1086. doi: 10.3934/dcds.2009.23.1073

[19]

Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313

[20]

N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (131)
  • HTML views (60)
  • Cited by (2)

Other articles
by authors

[Back to Top]