# American Institute of Mathematical Sciences

July  2010, 4(3): 419-441. doi: 10.3934/jmd.2010.4.419

## Linear cocycles over hyperbolic systems and criteria of conformality

 1 Department of Mathematics and statistics, University of South Alabama, Mobile, AL 36688, United States 2 Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, AL 36688

Received  August 2009 Published  October 2010

In this paper, we study Hölder-continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant subbundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Hölder-continuous Riemannian metric.
Citation: Boris Kalinin, Victoria Sadovskaya. Linear cocycles over hyperbolic systems and criteria of conformality. Journal of Modern Dynamics, 2010, 4 (3) : 419-441. doi: 10.3934/jmd.2010.4.419
##### References:
 [1] A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, preprint. [2] A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, to appear in Inventiones Mathematicae. [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. [4] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. of Math. (2), 126 (1987), 221-275. doi: doi:10.2307/1971401. [5] M. Guysinsky, Some results about Livšic Theorem for $2\times 2$ matrix-valued cocycles, preprint. [6] B. Kalinin, Livšic Theorem for matrix cocycles, to appear in Annals of Mathematics. [7] B. Kalinin and V. Sadovskaya, On local and global rigidity of quasiconformal Anosov diffeomorphisms, Journal of the Institute of Math. of Jussieu, 2 (2003), 567-582. doi: doi:10.1017/S1474748003000161. [8] B. Kalinin and V. Sadovskaya, On Anosov diffeomorphisms with asymptotically conformal periodic data, Ergodic Theory Dynam. Systems, 29 (2009), 117-136. doi: doi:10.1017/S0143385708000357. [9] M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30. [10] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Math. and its Applications, 54, Cambridge University Press, London-New York, 1995. [11] R. de la Llave, Rigidity of higher-dimensional conformal Anosov systems, Ergodic Theory Dynam. Systems, 22 (2002), 1845-1870. doi: doi:10.1017/S0143385702000871. [12] R. de la Llave, Further rigidity properties of conformal Anosov systems, Ergodic Theory Dynam. Systems, 24 (2004), 1425-1441. doi: doi:10.1017/S0143385703000725. [13] R. de la Llave and V. Sadovskaya, On regularity of integrable conformal structures invariant under Anosov systems, Discrete and Continuous Dynamical Systems, 12 (2005), 377-385. [14] R. de la Llave and A. Windsor, Livšic Theorems for noncommutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, to appear in Ergodic Theory Dynam. Systems. [15] A. N. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 6 (1972), 1278-1301. doi: doi:10.1070/IM1972v006n06ABEH001919. [16] H. Maass, "Siegel's Modular Forms and Dirichlet Series," Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. [17] W. Parry, The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: doi:10.1017/S0143385799146789. [18] W. Parry and M. Pollicott, The Livšic cocycle equation for compact Lie group extensions of hyperbolic systems, J. London Math. Soc. (2), 56 (1997), 405-416. doi: doi:10.1112/S0024610797005474. [19] F. R. Hertz, Global rigidity of certain abelian actions by toral automorphisms, Journal of Modern Dynamics, 1 (2007), 425-442. [20] V. Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Research Letters, 12 (2005), 425-441. [21] K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: doi:10.1017/S0143385799146790. [22] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in "Riemann Surfaces and Related Topics," Annals of Math. Studies, 97 (1981), 465-497. [23] P. Tukia, On quasiconformal groups, Jour. d'Analyse Mathe., 46 (1986), 318-346. [24] C. Yue, Qasiconformality in the geodesic flow of negatively curved manifolds, Geometric and Functional Analysis, 6 (1996), 740-750. doi: doi:10.1007/BF02247120.

show all references

##### References:
 [1] A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, preprint. [2] A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, to appear in Inventiones Mathematicae. [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. [4] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. of Math. (2), 126 (1987), 221-275. doi: doi:10.2307/1971401. [5] M. Guysinsky, Some results about Livšic Theorem for $2\times 2$ matrix-valued cocycles, preprint. [6] B. Kalinin, Livšic Theorem for matrix cocycles, to appear in Annals of Mathematics. [7] B. Kalinin and V. Sadovskaya, On local and global rigidity of quasiconformal Anosov diffeomorphisms, Journal of the Institute of Math. of Jussieu, 2 (2003), 567-582. doi: doi:10.1017/S1474748003000161. [8] B. Kalinin and V. Sadovskaya, On Anosov diffeomorphisms with asymptotically conformal periodic data, Ergodic Theory Dynam. Systems, 29 (2009), 117-136. doi: doi:10.1017/S0143385708000357. [9] M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30. [10] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Math. and its Applications, 54, Cambridge University Press, London-New York, 1995. [11] R. de la Llave, Rigidity of higher-dimensional conformal Anosov systems, Ergodic Theory Dynam. Systems, 22 (2002), 1845-1870. doi: doi:10.1017/S0143385702000871. [12] R. de la Llave, Further rigidity properties of conformal Anosov systems, Ergodic Theory Dynam. Systems, 24 (2004), 1425-1441. doi: doi:10.1017/S0143385703000725. [13] R. de la Llave and V. Sadovskaya, On regularity of integrable conformal structures invariant under Anosov systems, Discrete and Continuous Dynamical Systems, 12 (2005), 377-385. [14] R. de la Llave and A. Windsor, Livšic Theorems for noncommutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, to appear in Ergodic Theory Dynam. Systems. [15] A. N. Livšic, Cohomology of dynamical systems, Math. USSR Izvestija, 6 (1972), 1278-1301. doi: doi:10.1070/IM1972v006n06ABEH001919. [16] H. Maass, "Siegel's Modular Forms and Dirichlet Series," Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. [17] W. Parry, The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 687-701. doi: doi:10.1017/S0143385799146789. [18] W. Parry and M. Pollicott, The Livšic cocycle equation for compact Lie group extensions of hyperbolic systems, J. London Math. Soc. (2), 56 (1997), 405-416. doi: doi:10.1112/S0024610797005474. [19] F. R. Hertz, Global rigidity of certain abelian actions by toral automorphisms, Journal of Modern Dynamics, 1 (2007), 425-442. [20] V. Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Research Letters, 12 (2005), 425-441. [21] K. Schmidt, Remarks on Livšic' theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999), 703-721. doi: doi:10.1017/S0143385799146790. [22] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in "Riemann Surfaces and Related Topics," Annals of Math. Studies, 97 (1981), 465-497. [23] P. Tukia, On quasiconformal groups, Jour. d'Analyse Mathe., 46 (1986), 318-346. [24] C. Yue, Qasiconformality in the geodesic flow of negatively curved manifolds, Geometric and Functional Analysis, 6 (1996), 740-750. doi: doi:10.1007/BF02247120.
 [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] 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 [3] Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 [4] Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487 [5] Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911 [6] Anna-Lena Trautmann. Isometry and automorphisms of constant dimension codes. Advances in Mathematics of Communications, 2013, 7 (2) : 147-160. doi: 10.3934/amc.2013.7.147 [7] William A. Veech. The Forni Cocycle. Journal of Modern Dynamics, 2008, 2 (3) : 375-395. doi: 10.3934/jmd.2008.2.375 [8] Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 391-412. doi: 10.3934/dcds.2013.33.391 [9] Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94. [10] Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615 [11] Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems and Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035 [12] Moulay-Tahar Benameur, Alan L. Carey. On the analyticity of the bivariant JLO cocycle. Electronic Research Announcements, 2009, 16: 37-43. doi: 10.3934/era.2009.16.37 [13] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure and Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [14] Sergio Frigeri. Asymptotic behavior of a hyperbolic system arising in ferroelectricity. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1393-1414. doi: 10.3934/cpaa.2008.7.1393 [15] Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763 [16] Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333 [17] Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246 [18] Manas Bhatnagar, Hailiang Liu. Sharp critical thresholds in a hyperbolic system with relaxation. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5851-5869. doi: 10.3934/dcds.2021098 [19] Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001 [20] Cheng-Feng Hu, Hsiao-Fan Wang, Tingyang Liu. Measuring efficiency of a recycling production system with imprecise data. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 79-91. doi: 10.3934/naco.2021052

2020 Impact Factor: 0.848

## Metrics

• HTML views (0)
• Cited by (13)

• on AIMS