Lyapunov exponent, Liao perturbation and persistence

Consider a C1 vector field together with an ergodic invariant probability that has ℓ nonzero Lyapunov 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.