American Institute of Mathematical Sciences

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August  2018, 38(8): 3977-3991. doi: 10.3934/dcds.2018173

Continuity of spectral radius over hyperbolic systems

Rui Zou 1, , Yongluo Cao 1,2,*, and  Gang Liao 1,,

1. 

School of Mathematical Sciences, Soochow University, Suzhou 215006, China
2. 

Department of Mathematics, East China Normal University, Shanghai 200062, China

** Corresponding author: Gang Liao was partially supported by NSFC (11701402, 11790274), BK 20170327 and Jiangsu province "Double Plan"

*Yongluo Cao was partially supported by NSFC (11771317, 11790274), Science and Technology Commission of Shanghai Municipality (18dz22710000)

Received  October 2017 Revised  February 2018 Published  May 2018

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.

Keywords: Joint spectral radius, generalized spectral radius, Lyapunov exponents, Berger-Wang formula, hyperbolic systems.
Mathematics Subject Classification: 15A60, 34D08, 37C50, 37D25.
Citation: Rui Zou, Yongluo Cao, Gang Liao. Continuity of spectral radius over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3977-3991. doi: 10.3934/dcds.2018173
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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