Linear cocycles over hyperbolic systems and criteria of conformality

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

Linear cocycles over hyperbolic systems and criteria of conformality

Boris Kalinin ^1, and Victoria Sadovskaya ^2,

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

Abstract
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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.

Keywords: cocycle, isometry., pinching, hyperbolic system, conformality, periodic data.

Mathematics Subject Classification: Primary: 37H15; Secondary: 37D2.

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.

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