On the typicality of linear Hamiltonian systems with exponential separateness

Abstract

Linear Hamiltonian systems of ordinary differential equations ,are considered, where f t is a continuous action of the group R on a complete metric space ℬ, A is an element of the space S ℬ ,consisting of the continuous bounded mappings of ℬ into the set of all pseudosymmetric matrices and endowed with the uniform convergence metric. By λ 1 (A, x) ≥⋯≥ λ 2n (A, x) we denote the Lyapunov exponents of such systems. The typicality (in the Baire sense) in the space S H ℬ × ℬ is proved for those pairs (A, x) for which one has the alternative: either λ k (A, x) = λ k+1 (A, x) or the linear subspace of the solutions of the corresponding system with exponents less than λ k (A, x) is exponentially separated from any of its algebraic complements in the space of all the solutions of the system. From here, in particular, there follows the typicality of the formulated alternative for linear Hamiltonian systems with continuous quasiperiodic coefficients (with the same frequency module) and also for linear Hamiltonian systems with continuous almost periodic coefficients.