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Continuity of spectral radius over hyperbolic systems
1. | School of Mathematical Sciences, Soochow University, Suzhou 215006, China |
2. | Department of Mathematics, East China Normal University, Shanghai 200062, China |
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.
References:
[1] |
L. Backes,
On the periodic approximation of Lyapunov exponents for semi-invertible cocycles, Discrete Contin. Dyn. Syst., 37 (2017), 6353-6368.
doi: 10.3934/dcds.2017275. |
[2] |
N. E. Barabanov,
On the Lyapunov exponent of discrete inclusions. I, Avtomat. i Telemekh., 49 (1988), 40-46.
|
[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.
doi: 10.1017/CBO9781107326026. |
[4] |
M. A. Berger and Y. Wang,
Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27.
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. |
[6] |
X. Dai,
On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23 (2010), 513-528.
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 (2011), 321-347.
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 (2012), 1099-1113.
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 (2014), 2910-2937.
doi: 10.1016/j.jfranklin.2014.01.010. |
[10] |
X. Dai, T. Huang and Y. Huang,
Exponential stability of matrix-valued Markov chains via nonignorable periodic data, Trans. Amer. Math. Soc., 369 (2017), 5271-5292.
doi: 10.1090/tran/6912. |
[11] |
D. Dragičević and G. Froyland,
Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory Dynam. Systems, 38 (2018), 961-981.
doi: 10.1017/etds.2016.55. |
[12] |
G. Froyland, S. Lloyd and A. Quas,
Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[13] |
B. Kalinin,
Livšic theorem for matrix cocycles, Ann. of Math. (2), 173 (2011), 1025-1042.
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.
doi: 10.1017/etds.2017.43. |
[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.
doi: 10.1017/CBO9780511809187. |
[16] |
V. Kozyakin,
An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18.
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.
doi: 10.1007/BFb0061433. |
[18] |
B. E. Moision, A. Orlitsky and P. H. Siegel,
On codes that avoid specified differences, IEEE Trans. Inform. Theory, 47 (2001), 433-442.
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 (2012), 811-824.
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.(3), 107 (2013), 121-150.
doi: 10.1112/plms/pds080. |
[21] |
V. I. Oseledets,
A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.
|
[22] |
C. Pugh and M. Shub,
Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.
doi: 10.1090/S0002-9947-1989-0983869-1. |
[23] |
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. |
[24] |
M. Viana and K. Oliveira,
Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 151, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316422601. |
[25] |
Y. Wang and J. You,
Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.
doi: 10.1215/00127094-2371528. |
[26] |
Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
[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. |
show all references
References:
[1] |
L. Backes,
On the periodic approximation of Lyapunov exponents for semi-invertible cocycles, Discrete Contin. Dyn. Syst., 37 (2017), 6353-6368.
doi: 10.3934/dcds.2017275. |
[2] |
N. E. Barabanov,
On the Lyapunov exponent of discrete inclusions. I, Avtomat. i Telemekh., 49 (1988), 40-46.
|
[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.
doi: 10.1017/CBO9781107326026. |
[4] |
M. A. Berger and Y. Wang,
Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27.
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. |
[6] |
X. Dai,
On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23 (2010), 513-528.
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 (2011), 321-347.
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 (2012), 1099-1113.
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 (2014), 2910-2937.
doi: 10.1016/j.jfranklin.2014.01.010. |
[10] |
X. Dai, T. Huang and Y. Huang,
Exponential stability of matrix-valued Markov chains via nonignorable periodic data, Trans. Amer. Math. Soc., 369 (2017), 5271-5292.
doi: 10.1090/tran/6912. |
[11] |
D. Dragičević and G. Froyland,
Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergodic Theory Dynam. Systems, 38 (2018), 961-981.
doi: 10.1017/etds.2016.55. |
[12] |
G. Froyland, S. Lloyd and A. Quas,
Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[13] |
B. Kalinin,
Livšic theorem for matrix cocycles, Ann. of Math. (2), 173 (2011), 1025-1042.
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.
doi: 10.1017/etds.2017.43. |
[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.
doi: 10.1017/CBO9780511809187. |
[16] |
V. Kozyakin,
An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18.
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.
doi: 10.1007/BFb0061433. |
[18] |
B. E. Moision, A. Orlitsky and P. H. Siegel,
On codes that avoid specified differences, IEEE Trans. Inform. Theory, 47 (2001), 433-442.
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 (2012), 811-824.
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.(3), 107 (2013), 121-150.
doi: 10.1112/plms/pds080. |
[21] |
V. I. Oseledets,
A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.
|
[22] |
C. Pugh and M. Shub,
Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.
doi: 10.1090/S0002-9947-1989-0983869-1. |
[23] |
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. |
[24] |
M. Viana and K. Oliveira,
Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 151, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316422601. |
[25] |
Y. Wang and J. You,
Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.
doi: 10.1215/00127094-2371528. |
[26] |
Z. Wang and W. Sun,
Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc., 362 (2010), 4267-4282.
doi: 10.1090/S0002-9947-10-04947-0. |
[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. |
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