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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 |
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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