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Effective equidistribution of translates of maximal horospherical measures in the space of lattices
The entropy of Lyapunov-optimizing measures of some matrix cocycles
1. | Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile |
2. | Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8. 00-956 Warsaw, Poland |
References:
[1] |
A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $SL(2,\mathbbR)$-cocycles, Comment. Math. Helv., 85 (2010), 813-884.
doi: 10.4171/CMH/212. |
[2] |
N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I, Automat. Remote Control, 49 (1988), 152-157. |
[3] |
J. Bochi, C. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems, Math. Z., 276 (2014), 469-503.
doi: 10.1007/s00209-013-1209-y. |
[4] |
J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231.
doi: 10.1007/s00209-009-0494-y. |
[5] |
J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509.
doi: 10.1112/plms/pdu058. |
[6] |
V. I. Bogachev, Measure Theory. Vol. II, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[7] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[8] |
T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111.
doi: 10.1090/S0894-0347-01-00378-2. |
[9] |
H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics, Academic Press Inc., New York, N. Y., 1953. |
[10] |
Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions, Chinese J. Contemp. Math., 34 (2013), 351-360. |
[11] |
G. Contreras, Ground states are generically a periodic orbit, Inventiones Mathematicae, (2015), 1-30.
doi: 10.1007/s00222-015-0638-0. |
[12] |
D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices, Israel J. Math., 138 (2003), 353-376.
doi: 10.1007/BF02783432. |
[13] |
L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85.
doi: 10.1016/0024-3795(95)90006-3. |
[14] |
K. G. Hare, I. D. Morris and N. Sidorov, Extremal sequences of polynomial complexity, Math. Proc. Cambridge Philos. Soc., 155 (2013), 191-205.
doi: 10.1017/S0305004113000157. |
[15] |
M. R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502.
doi: 10.1007/BF02564647. |
[16] |
O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.
doi: 10.3934/dcds.2006.15.197. |
[17] |
O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures, and the finiteness conjecture, arXiv:1501.03419. |
[18] |
T. Jørgensen and K. Smith, On certain semigroups of hyperbolic isometries, Duke Math. J., 61 (1990), 1-10.
doi: 10.1215/S0012-7094-90-06101-0. |
[19] |
R. Jungers, The Joint Spectral Radius. Theory and Applications, Lecture Notes in Control and Information Sciences, 385, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-95980-9. |
[20] |
E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dyn. Syst., 22 (2007), 511-528.
doi: 10.1080/14689360701582378. |
[21] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[22] |
I. D. Morris, A sufficient condition for the subordination principle in ergodic optimization, Bull. Lond. Math. Soc., 39 (2007), 214-220.
doi: 10.1112/blms/bdl030. |
[23] |
________, Maximizing measures of generic Hölder functions have zero entropy, Nonlinearity, 21 (2008), 993-1000.
doi: 10.1088/0951-7715/21/5/005. |
[24] |
________, Mather sets for sequences of matrices and applications to the study of joint spectral radii, Proc. London Math. Soc. (3), 107 (2013), 121-150.
doi: 10.1112/plms/pds080. |
[25] |
K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge University Press, Cambridge, 1989. |
[26] |
G.-C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math., 22 (1960), 379-381.
doi: 10.1016/S1385-7258(60)50046-1. |
[27] |
F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40.
doi: 10.1016/S0024-3795(01)00446-3. |
[28] |
J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 447-458. |
show all references
References:
[1] |
A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $SL(2,\mathbbR)$-cocycles, Comment. Math. Helv., 85 (2010), 813-884.
doi: 10.4171/CMH/212. |
[2] |
N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I, Automat. Remote Control, 49 (1988), 152-157. |
[3] |
J. Bochi, C. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems, Math. Z., 276 (2014), 469-503.
doi: 10.1007/s00209-013-1209-y. |
[4] |
J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231.
doi: 10.1007/s00209-009-0494-y. |
[5] |
J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509.
doi: 10.1112/plms/pdu058. |
[6] |
V. I. Bogachev, Measure Theory. Vol. II, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[7] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[8] |
T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111.
doi: 10.1090/S0894-0347-01-00378-2. |
[9] |
H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics, Academic Press Inc., New York, N. Y., 1953. |
[10] |
Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions, Chinese J. Contemp. Math., 34 (2013), 351-360. |
[11] |
G. Contreras, Ground states are generically a periodic orbit, Inventiones Mathematicae, (2015), 1-30.
doi: 10.1007/s00222-015-0638-0. |
[12] |
D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices, Israel J. Math., 138 (2003), 353-376.
doi: 10.1007/BF02783432. |
[13] |
L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85.
doi: 10.1016/0024-3795(95)90006-3. |
[14] |
K. G. Hare, I. D. Morris and N. Sidorov, Extremal sequences of polynomial complexity, Math. Proc. Cambridge Philos. Soc., 155 (2013), 191-205.
doi: 10.1017/S0305004113000157. |
[15] |
M. R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502.
doi: 10.1007/BF02564647. |
[16] |
O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.
doi: 10.3934/dcds.2006.15.197. |
[17] |
O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures, and the finiteness conjecture, arXiv:1501.03419. |
[18] |
T. Jørgensen and K. Smith, On certain semigroups of hyperbolic isometries, Duke Math. J., 61 (1990), 1-10.
doi: 10.1215/S0012-7094-90-06101-0. |
[19] |
R. Jungers, The Joint Spectral Radius. Theory and Applications, Lecture Notes in Control and Information Sciences, 385, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-95980-9. |
[20] |
E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dyn. Syst., 22 (2007), 511-528.
doi: 10.1080/14689360701582378. |
[21] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[22] |
I. D. Morris, A sufficient condition for the subordination principle in ergodic optimization, Bull. Lond. Math. Soc., 39 (2007), 214-220.
doi: 10.1112/blms/bdl030. |
[23] |
________, Maximizing measures of generic Hölder functions have zero entropy, Nonlinearity, 21 (2008), 993-1000.
doi: 10.1088/0951-7715/21/5/005. |
[24] |
________, Mather sets for sequences of matrices and applications to the study of joint spectral radii, Proc. London Math. Soc. (3), 107 (2013), 121-150.
doi: 10.1112/plms/pds080. |
[25] |
K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge University Press, Cambridge, 1989. |
[26] |
G.-C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math., 22 (1960), 379-381.
doi: 10.1016/S1385-7258(60)50046-1. |
[27] |
F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40.
doi: 10.1016/S0024-3795(01)00446-3. |
[28] |
J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 447-458. |
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