×
Zou, Rui; Cao, Yongluo; Liao, Gang

Continuity of spectral radius over hyperbolic systems. (English) Zbl 1396.37061

Discrete Contin. Dyn. Syst. 38, No. 8, 3977-3991 (2018).
Summary: The continuity of joint and generalized spectral radius is proved for Hölder continuous cocycles over hyperbolic systems. We also prove the periodic approximation of Lyapunov exponents for non-invertible non-uniformly hyperbolic systems, and establish the Berger-Wang formula for general dynamical systems.

Cited in 3 Documents

MSC:

37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

Keywords:

joint spectral radius; generalized spectral radius; Lyapunov exponent; Berger-Wang formula; hyperbolic system
Cite Review PDF
Full Text: DOI

References:

[1] L. Backes, On the periodic approximation of Lyapunov exponents for semi-invertible cocycles, Discrete Contin. Dyn. Syst., 37, 6353-6368 (2017) · Zbl 1379.37098 · doi:10.3934/dcds.2017275
[2] N. E. Barabanov, On the Lyapunov exponent of discrete inclusions. I, Avtomat. i Telemekh., 49, 40-46 (1988) · Zbl 0665.93043
[3] L. Barrira and Ya. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. · Zbl 1144.37002
[4] M. A. Berger; Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166, 21-27 (1992) · Zbl 0818.15006 · doi:10.1016/0024-3795(92)90267-E
[5] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, revised ed., Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. · Zbl 1172.37001
[6] X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23, 513-528 (2010) · Zbl 1189.37093 · doi:10.1088/0951-7715/23/3/004
[7] X. Dai, Exponential closing property and approximation of Lyapunov exponents of linear cocycles, Forum Math., 23, 321-347 (2011) · Zbl 1237.37056 · doi:10.1515/FORM.2011.011
[8] X. Dai, A Gel’fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl., 436, 1099-1113 (2012) · Zbl 1237.15010 · doi:10.1016/j.laa.2011.07.029
[9] X. Dai, Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations, J. Franklin Inst., 351, 2910-2937 (2014) · Zbl 1372.93209 · doi:10.1016/j.jfranklin.2014.01.010
[10] X. Dai; T. Huang; Y. Huang, Exponential stability of matrix-valued Markov chains via nonignorable periodic data, Trans. Amer. Math. Soc., 369, 5271-5292 (2017) · Zbl 1361.93064 · doi:10.1090/tran/6912
[11] D. Dragičević; G. Froyland, Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory Dynam. Systems, 38, 961-981 (2018) · Zbl 1390.37093 · doi:10.1017/etds.2016.55
[12] G. Froyland; S. Lloyd; A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30, 729-756 (2010) · Zbl 1205.37015 · doi:10.1017/S0143385709000339
[13] B. Kalinin, Livšic theorem for matrix cocycles, Ann. of Math. (2), 173, 1025-1042 (2011) · Zbl 1238.37008 · doi:10.4007/annals.2011.173.2.11
[14] B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles Cambridge University Press, 2017, P43, arXiv: 1608. 05757. · Zbl 1394.37086
[15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020
[16] V. Kozyakin, An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433, 12-18 (2010) · Zbl 1198.15006 · doi:10.1016/j.laa.2010.01.028
[17] R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, 522-577. · Zbl 0522.58030
[18] B. E. Moision; A. Orlitsky; P. H. Siegel, On codes that avoid specified differences, IEEE Trans. Inform. Theory, 47, 433-442 (2001) · Zbl 0998.94563 · doi:10.1109/18.904557
[19] I. D. Morris, The generalised Berger-Wang formula and the spectral radius of linear cocycles, J. Funct. Anal., 262, 811-824 (2012) · Zbl 1254.47006 · doi:10.1016/j.jfa.2011.09.021
[20] I. D. Morris, Mather sets for sequences of matrices and applications to the study of joint spectral radii, Proc. Lond. Math. Soc., 107, 121-150 (2013) · Zbl 1277.15009 · doi:10.1112/plms/pds080
[21] V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19, 179-210 (1968) · Zbl 0236.93034
[22] C. Pugh; M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312, 1-54 (1989) · Zbl 0684.58008 · doi:10.1090/S0002-9947-1989-0983869-1
[23] G.-C. Rota; G. Strang, A note on the joint spectral radius, Indag. Math., 22, 379-381 (1960) · Zbl 0095.09701 · doi:10.1016/S1385-7258(60)50046-1
[24] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 151, Cambridge University Press, Cambridge, 2016. · Zbl 1369.37001
[25] Y. Wang; J. You, Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162, 2363-2412 (2013) · Zbl 1405.37032 · doi:10.1215/00127094-2371528
[26] Z. Wang; W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362, 4267-4282 (2010) · Zbl 1201.37025 · doi:10.1090/S0002-9947-10-04947-0
[27] F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342, 17-40 (2002) · Zbl 0996.15020 · doi:10.1016/S0024-3795(01)00446-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.