• Previous Article
    Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation
  • DCDS Home
  • This Issue
  • Next Article
    On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points
January  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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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.  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.  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.  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.  doi: 10.1017/S0143385798117480.  Google Scholar

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (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.  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, (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.  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.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[16]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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.  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.  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.  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.  doi: 10.1017/S0143385798117480.  Google Scholar

[11]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, European Mathematical Society (EMS), (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.  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, (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.  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.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[16]

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

[1]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[2]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158

[3]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[4]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[5]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[6]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[7]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[8]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014

[9]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[10]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[11]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[12]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226

[13]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[14]

Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006

[15]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]