American Institute of Mathematical Sciences

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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

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.  Google Scholar

[2]

N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I, Automat. Remote Control, 49 (1988), 152-157.  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), 469-503. doi: 10.1007/s00209-013-1209-y.  Google Scholar

[4]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. 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), 477-509. doi: 10.1112/plms/pdu058.  Google Scholar

[6]

V. I. Bogachev, Measure Theory. Vol. II, Springer-Verlag, Berlin, 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 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), 77-111. 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., New York, N. Y., 1953.  Google Scholar

[10]

Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions, Chinese J. Contemp. Math., 34 (2013), 351-360.  Google Scholar

[11]

G. Contreras, Ground states are generically a periodic orbit, Inventiones Mathematicae, (2015), 1-30. 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), 353-376. doi: 10.1007/BF02783432.  Google Scholar

[13]

L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. 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), 191-205. 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), 453-502. doi: 10.1007/BF02564647.  Google Scholar

[16]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. 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), 1-10. 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, 385, Springer-Verlag, Berlin, 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), 511-528. doi: 10.1080/14689360701582378.  Google Scholar

[21]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. 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), 214-220. doi: 10.1112/blms/bdl030.  Google Scholar

[23]

________, Maximizing measures of generic Hölder functions have zero entropy, Nonlinearity, 21 (2008), 993-1000. 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), 121-150. doi: 10.1112/plms/pds080.  Google Scholar

[25]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge University Press, Cambridge, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I, Automat. Remote Control, 49 (1988), 152-157.  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), 469-503. doi: 10.1007/s00209-013-1209-y.  Google Scholar

[4]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. 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), 477-509. doi: 10.1112/plms/pdu058.  Google Scholar

[6]

V. I. Bogachev, Measure Theory. Vol. II, Springer-Verlag, Berlin, 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 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), 77-111. 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., New York, N. Y., 1953.  Google Scholar

[10]

Y. Y. Chen and Y. Zhao, Ergodic optimization for a sequence of continuous functions, Chinese J. Contemp. Math., 34 (2013), 351-360.  Google Scholar

[11]

G. Contreras, Ground states are generically a periodic orbit, Inventiones Mathematicae, (2015), 1-30. 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), 353-376. doi: 10.1007/BF02783432.  Google Scholar

[13]

L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. 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), 191-205. 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), 453-502. doi: 10.1007/BF02564647.  Google Scholar

[16]

O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224. 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), 1-10. 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, 385, Springer-Verlag, Berlin, 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), 511-528. doi: 10.1080/14689360701582378.  Google Scholar

[21]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. 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), 214-220. doi: 10.1112/blms/bdl030.  Google Scholar

[23]

________, Maximizing measures of generic Hölder functions have zero entropy, Nonlinearity, 21 (2008), 993-1000. 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), 121-150. doi: 10.1112/plms/pds080.  Google Scholar

[25]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge University Press, Cambridge, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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