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.

[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.

[3]

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

[4]

J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, Unpublished, http://www.mat.uc.cl/~jairo.bochi/docs/discont.pdf.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

J. Bochi, Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, Unpublished, http://www.mat.uc.cl/~jairo.bochi/docs/discont.pdf.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

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