October  2017, 22(8): 3113-3126. doi: 10.3934/dcdsb.2017166

On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems

1. 

Institute of Mathematics, Vietnam Academy of Science and Technology, Viet Nam

2. 

Department of Mathematics, Hokkaido University, Japan

Received  March 2016 Revised  December 2016 Published  June 2017

Fund Project: This work is supported by the JSPS International Fellowship for Research in Japan (P15320).

In this paper, we construct an open and dense set in the space of bounded linear random dynamical systems (both discrete and continuous time) equipped with the essential sup norm such that the Lyapunov exponents depend analytically on the coefficients in this set. As a consequence, analyticity for Lyapunov exponents of bounded linear random dynamical systems is a generic property.

Citation: Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166
References:
[1]

W. Ambrose, Representation of ergodic flows, Annals of Mathematics, 42 (1941), 723-739.  doi: 10.2307/1969259.  Google Scholar

[2]

L. ArnoldV. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, The Annals of Applied Probability, 4 (1994), 859-901.  doi: 10.1214/aoap/1177004975.  Google Scholar

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L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

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J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, Unpublished, http://www.mat.uc.cl/~jairo.bochi/docs/discont.pdf. Google Scholar

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N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergodic Theory and Dynamical Systems, 25 (2005), 1775-1797.  doi: 10.1017/S0143385705000337.  Google Scholar

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N. D. Cong and T. S. Doan, On integral separation of bounded linear random differential equations, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 995-1007.  doi: 10.3934/dcdss.2016038.  Google Scholar

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I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory Grundlehren der Mathematischen Wissenschaften, 245. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

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H. Crauel, Lyapunov exponents of random dynamical systems on Grassmannians, Lyapunov Exponents (Oberwolfach, 1990), 38–50, Lecture Notes in Math. , 1486, Springer, Berlin, 1991. doi: 10.1007/BFb0086656.  Google Scholar

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L. Dubois, Real cone contractions and analyticity properties of the characteristic exponents, Nonlinearity, 21 (2008), 2519-2536.  doi: 10.1088/0951-7715/21/11/003.  Google Scholar

[10]

G. FroylandC. González-Tokman and A. Quas, Stochastic stability of Lyapunov exponents and Oseledets splitting for semi-invertible matrix cocycles, Comm. Pure Appl. Math., 68 (2015), 2052-2081.  doi: 10.1002/cpa.21569.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[12]

O. Knill, The upper Lyapunov exponent of Sl(2, R) cocycles: Discontinuity and the problem of positivity, Lecture Notes in Mathematics, 1486 (1990), 86-97.  doi: 10.1007/BFb0086660.  Google Scholar

[13]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Transactions of the Moscow Mathematical Society, 19 (1968), 179-210.   Google Scholar

[14]

D. Ruelle, Analyticity properties of the characteristic exponents of random matrix products, Advances in Mathematics, 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[15]

E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Am. Math. Soc., 16 (1965), 1077-1083.  doi: 10.1090/S0002-9939-1965-0184092-2.  Google Scholar

show all references

References:
[1]

W. Ambrose, Representation of ergodic flows, Annals of Mathematics, 42 (1941), 723-739.  doi: 10.2307/1969259.  Google Scholar

[2]

L. ArnoldV. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, The Annals of Applied Probability, 4 (1994), 859-901.  doi: 10.1214/aoap/1177004975.  Google Scholar

[3]

L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[4]

J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, Unpublished, http://www.mat.uc.cl/~jairo.bochi/docs/discont.pdf. Google Scholar

[5]

N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergodic Theory and Dynamical Systems, 25 (2005), 1775-1797.  doi: 10.1017/S0143385705000337.  Google Scholar

[6]

N. D. Cong and T. S. Doan, On integral separation of bounded linear random differential equations, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 995-1007.  doi: 10.3934/dcdss.2016038.  Google Scholar

[7]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory Grundlehren der Mathematischen Wissenschaften, 245. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[8]

H. Crauel, Lyapunov exponents of random dynamical systems on Grassmannians, Lyapunov Exponents (Oberwolfach, 1990), 38–50, Lecture Notes in Math. , 1486, Springer, Berlin, 1991. doi: 10.1007/BFb0086656.  Google Scholar

[9]

L. Dubois, Real cone contractions and analyticity properties of the characteristic exponents, Nonlinearity, 21 (2008), 2519-2536.  doi: 10.1088/0951-7715/21/11/003.  Google Scholar

[10]

G. FroylandC. González-Tokman and A. Quas, Stochastic stability of Lyapunov exponents and Oseledets splitting for semi-invertible matrix cocycles, Comm. Pure Appl. Math., 68 (2015), 2052-2081.  doi: 10.1002/cpa.21569.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[12]

O. Knill, The upper Lyapunov exponent of Sl(2, R) cocycles: Discontinuity and the problem of positivity, Lecture Notes in Mathematics, 1486 (1990), 86-97.  doi: 10.1007/BFb0086660.  Google Scholar

[13]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Transactions of the Moscow Mathematical Society, 19 (1968), 179-210.   Google Scholar

[14]

D. Ruelle, Analyticity properties of the characteristic exponents of random matrix products, Advances in Mathematics, 32 (1979), 68-80.  doi: 10.1016/0001-8708(79)90029-X.  Google Scholar

[15]

E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Am. Math. Soc., 16 (1965), 1077-1083.  doi: 10.1090/S0002-9939-1965-0184092-2.  Google Scholar

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