On a generic property of vector subspaces defined by Lyapunov exponents

It is proved that the dependence on a Cauchy problem of the subspace consisting of the initial values of those solutions of the linearized Cauchy problem whose Lyapunov exponents do not exceed the $k$th Lyapunov exponent of the linearized system is generically continuous, provided that the $k$th exponent of the linearized system is different from its $(k-1)$st exponent.
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