# American Institute of Mathematical Sciences

2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255

## 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

Received  April 2015 Revised  April 2016 Published  July 2016

We consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.
Citation: Jairo Bochi, Michal Rams. The entropy of Lyapunov-optimizing measures of some matrix cocycles. Journal of Modern Dynamics, 2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255
##### References:
 [1] A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $SL(2,\mathbbR)$-cocycles,, Comment. Math. Helv., 85 (2010), 813. doi: 10.4171/CMH/212. Google Scholar [2] N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I,, Automat. Remote Control, 49 (1988), 152. Google Scholar [3] J. Bochi, C. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems,, Math. Z., 276 (2014), 469. doi: 10.1007/s00209-013-1209-y. Google Scholar [4] J. Bochi and N. Gourmelon, Some characterizations of domination,, Math. Z., 263 (2009), 221. doi: 10.1007/s00209-009-0494-y. Google Scholar [5] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius,, Proc. Lond. Math. Soc. (3), 110 (2015), 477. doi: 10.1112/plms/pdu058. Google Scholar [6] V. I. Bogachev, Measure Theory. Vol. II,, Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [7] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005). Google Scholar [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. doi: 10.1090/S0894-0347-01-00378-2. Google Scholar [9] H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics,, Academic Press Inc., (1953). Google Scholar [10] Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions,, Chinese J. Contemp. Math., 34 (2013), 351. Google Scholar [11] G. Contreras, Ground states are generically a periodic orbit,, Inventiones Mathematicae, (2015), 1. doi: 10.1007/s00222-015-0638-0. Google Scholar [12] D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices,, Israel J. Math., 138 (2003), 353. doi: 10.1007/BF02783432. Google Scholar [13] L. Gurvits, Stability of discrete linear inclusion,, Linear Algebra Appl., 231 (1995), 47. doi: 10.1016/0024-3795(95)90006-3. Google Scholar [14] K. G. Hare, I. D. Morris and N. Sidorov, Extremal sequences of polynomial complexity,, Math. Proc. Cambridge Philos. Soc., 155 (2013), 191. doi: 10.1017/S0305004113000157. Google Scholar [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. doi: 10.1007/BF02564647. Google Scholar [16] O. Jenkinson, Ergodic optimization,, Discrete Contin. Dyn. Syst., 15 (2006), 197. doi: 10.3934/dcds.2006.15.197. Google Scholar [17] O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures, and the finiteness conjecture,, , (). Google Scholar [18] T. Jørgensen and K. Smith, On certain semigroups of hyperbolic isometries,, Duke Math. J., 61 (1990), 1. doi: 10.1215/S0012-7094-90-06101-0. Google Scholar [19] R. Jungers, The Joint Spectral Radius. Theory and Applications,, Lecture Notes in Control and Information Sciences, (2009). doi: 10.1007/978-3-540-95980-9. Google Scholar [20] E. Garibaldi and A. O. Lopes, Functions for relative maximization,, Dyn. Syst., 22 (2007), 511. doi: 10.1080/14689360701582378. Google Scholar [21] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169. doi: 10.1007/BF02571383. Google Scholar [22] I. D. Morris, A sufficient condition for the subordination principle in ergodic optimization,, Bull. Lond. Math. Soc., 39 (2007), 214. doi: 10.1112/blms/bdl030. Google Scholar [23] ________, Maximizing measures of generic Hölder functions have zero entropy,, Nonlinearity, 21 (2008), 993. doi: 10.1088/0951-7715/21/5/005. Google Scholar [24] ________, Mather sets for sequences of matrices and applications to the study of joint spectral radii,, Proc. London Math. Soc. (3), 107 (2013), 121. doi: 10.1112/plms/pds080. Google Scholar [25] K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983). Google Scholar [26] G.-C. Rota and G. Strang, A note on the joint spectral radius,, Indag. Math., 22 (1960), 379. doi: 10.1016/S1385-7258(60)50046-1. Google Scholar [27] F. Wirth, The generalized spectral radius and extremal norms,, Linear Algebra Appl., 342 (2002), 17. doi: 10.1016/S0024-3795(01)00446-3. Google Scholar [28] J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles,, in Modern Dynamical Systems and Applications, (2004), 447. Google Scholar

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##### References:
 [1] A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $SL(2,\mathbbR)$-cocycles,, Comment. Math. Helv., 85 (2010), 813. doi: 10.4171/CMH/212. Google Scholar [2] N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I,, Automat. Remote Control, 49 (1988), 152. Google Scholar [3] J. Bochi, C. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems,, Math. Z., 276 (2014), 469. doi: 10.1007/s00209-013-1209-y. Google Scholar [4] J. Bochi and N. Gourmelon, Some characterizations of domination,, Math. Z., 263 (2009), 221. doi: 10.1007/s00209-009-0494-y. Google Scholar [5] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius,, Proc. Lond. Math. Soc. (3), 110 (2015), 477. doi: 10.1112/plms/pdu058. Google Scholar [6] V. I. Bogachev, Measure Theory. Vol. II,, Springer-Verlag, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [7] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005). Google Scholar [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. doi: 10.1090/S0894-0347-01-00378-2. Google Scholar [9] H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics,, Academic Press Inc., (1953). Google Scholar [10] Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions,, Chinese J. Contemp. Math., 34 (2013), 351. Google Scholar [11] G. Contreras, Ground states are generically a periodic orbit,, Inventiones Mathematicae, (2015), 1. doi: 10.1007/s00222-015-0638-0. Google Scholar [12] D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. I. Positive matrices,, Israel J. Math., 138 (2003), 353. doi: 10.1007/BF02783432. Google Scholar [13] L. Gurvits, Stability of discrete linear inclusion,, Linear Algebra Appl., 231 (1995), 47. doi: 10.1016/0024-3795(95)90006-3. Google Scholar [14] K. G. Hare, I. D. Morris and N. Sidorov, Extremal sequences of polynomial complexity,, Math. Proc. Cambridge Philos. Soc., 155 (2013), 191. doi: 10.1017/S0305004113000157. Google Scholar [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. doi: 10.1007/BF02564647. Google Scholar [16] O. Jenkinson, Ergodic optimization,, Discrete Contin. Dyn. Syst., 15 (2006), 197. doi: 10.3934/dcds.2006.15.197. Google Scholar [17] O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures, and the finiteness conjecture,, , (). Google Scholar [18] T. Jørgensen and K. Smith, On certain semigroups of hyperbolic isometries,, Duke Math. J., 61 (1990), 1. doi: 10.1215/S0012-7094-90-06101-0. Google Scholar [19] R. Jungers, The Joint Spectral Radius. Theory and Applications,, Lecture Notes in Control and Information Sciences, (2009). doi: 10.1007/978-3-540-95980-9. Google Scholar [20] E. Garibaldi and A. O. Lopes, Functions for relative maximization,, Dyn. Syst., 22 (2007), 511. doi: 10.1080/14689360701582378. Google Scholar [21] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169. doi: 10.1007/BF02571383. Google Scholar [22] I. D. Morris, A sufficient condition for the subordination principle in ergodic optimization,, Bull. Lond. Math. Soc., 39 (2007), 214. doi: 10.1112/blms/bdl030. Google Scholar [23] ________, Maximizing measures of generic Hölder functions have zero entropy,, Nonlinearity, 21 (2008), 993. doi: 10.1088/0951-7715/21/5/005. Google Scholar [24] ________, Mather sets for sequences of matrices and applications to the study of joint spectral radii,, Proc. London Math. Soc. (3), 107 (2013), 121. doi: 10.1112/plms/pds080. Google Scholar [25] K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983). Google Scholar [26] G.-C. Rota and G. Strang, A note on the joint spectral radius,, Indag. Math., 22 (1960), 379. doi: 10.1016/S1385-7258(60)50046-1. Google Scholar [27] F. Wirth, The generalized spectral radius and extremal norms,, Linear Algebra Appl., 342 (2002), 17. doi: 10.1016/S0024-3795(01)00446-3. Google Scholar [28] J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles,, in Modern Dynamical Systems and Applications, (2004), 447. Google Scholar
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