Journal of Modern Dynamics (JMD)

The entropy of Lyapunov-optimizing measures of some matrix cocycles
Pages: 255 - 286, Volume 10, 2016

doi:10.3934/jmd.2016.10.255      Abstract        References        Full text (332.4K)           Related Articles

Jairo Bochi - Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile (email)
Michal Rams - Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8. 00-956 Warsaw, Poland (email)

1 A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $SL(2,\mathbbR)$-cocycles, Comment. Math. Helv., 85 (2010), 813-884.       
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.       
4 J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231.       
5 J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509.       
6 V. I. Bogachev, Measure Theory. Vol. II, Springer-Verlag, Berlin, 2007.       
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.       
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.
12 D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices, Israel J. Math., 138 (2003), 353-376.       
13 L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85.       
14 K. G. Hare, I. D. Morris and N. Sidorov, Extremal sequences of polynomial complexity, Math. Proc. Cambridge Philos. Soc., 155 (2013), 191-205.       
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.       
16 O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.       
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.       
19 R. Jungers, The Joint Spectral Radius. Theory and Applications, Lecture Notes in Control and Information Sciences, 385, Springer-Verlag, Berlin, 2009.       
20 E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dyn. Syst., 22 (2007), 511-528.       
21 J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.       
22 I. D. Morris, A sufficient condition for the subordination principle in ergodic optimization, Bull. Lond. Math. Soc., 39 (2007), 214-220.       
23 ________, Maximizing measures of generic Hölder functions have zero entropy, Nonlinearity, 21 (2008), 993-1000.       
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.       
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.       
27 F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40.       
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.       

Go to top