The entropy of Lyapunov-optimizing measures of some matrix cocycles

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

The entropy of Lyapunov-optimizing measures of some matrix cocycles

Jairo Bochi ^1, and Michal Rams ^2,

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

Abstract
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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.

Keywords: Lyapunov exponents, dominated splittings, linear cocycles, joint spectral subradius, Joint spectral radius, entropy., ergodic optimization.

Mathematics Subject Classification: Primary: 15B48; Secondary: 37H15, 37D30, 93C3.

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), 813-884. 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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