Lyapunov exponent, Liao perturbation and persistence
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https://doi.org/10.1007/s11425-019-1660-5
Lyapunov exponent, Liao perturbation and persistence Dedicated to the Memory of Professor Shantao Liao at the Centenary of His Birth
Wenxiang Sun1,∗ & Todd Young2 1School
of Mathematical Sciences, Peking University, Beijing 100871, China; of Mathematics, Ohio University, Athens, OH 45701, USA
2Department
Email: [email protected], [email protected] Received September 11, 2019; accepted March 3, 2020
Abstract
Consider a C 1 vector field together with an ergodic invariant probability that has ℓ nonzero Lya-
punov exponents. Using orthonormal moving frames along a generic orbit we construct a linear system of ℓ differential equations which is a linearized Liao standard system. We show that Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that a small perturbation to the linear system (Liao perturbation) preserves both the sign and the value of the nonzero Lyapunov exponents. Keywords MSC(2010)
Lyapunov exponent, Liao perturbation, ergodic probability, non-uniformly hyperbolicity 37B40, 37D25, 37C40
Citation: Sun W X, Young T. Lyapunov exponent, Liao perturbation and persistence. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1660-5
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Introduction
Lyapunov exponents measure the asymptotic exponential rate at which infinitesimally nearby points approach or move away from each other as time increases to infinity. Over a uniformly hyperbolic system almost all cocycles have nonvanishing Lyapunov exponents [12]. As well, understanding dynamics through Lyapunov exponents of cocycles over a non-uniformly hyperbolic system is a significant topic. We will consider Lyapunov exponents of a certain linear cocycle over a non-uniformly hyperbolic system and its perturbation called Liao perturbation in the present paper. We consider a C 1 vector field together with an ergodic probability that has ℓ nonzero Lyapunov exponents. By using moving orthonormal ℓ-frames along certain transitive orbits of the ergodic measure, and by using a characterization of the Lyapunov spectrum [2,8], we construct a linear system of differential equations whose Lyapunov exponents coincide with the nonzero exponents of the original vector field. We show that the nonzero Lyapunov exponents of the reduced standard system have certain persistence properties. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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2
Sun W X et al.
Sci China Math
Now let us describe the main theorem of the present paper. We denote by M n a compact smooth ndimensional Riemannian manifold without boundary and by S a C 1 differential system, or in other words, a C 1 vector field on M n . As usual S induces a one-parameter transformation group ϕt : M n → M n , t ∈ R on the state manifold and therefore a one-parameter transfor
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