Construction of an Arbitrary Suslin Set of Positive Characteristic Exponents in the Perron Effect
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For an arbitrary bounded Suslin set S ⊂ (0, +∞) and arbitrary parameters m > 1 and λ1 ≤ λ2 < 0, we construct a two-dimensional differential system ẏ = A(t)y + f (t, y), y ∈ ℝ2, t ≥ t0, with infinitely differentiable matrix A(t) and with vector function f (t,y) infinitely differentiable with respect to its arguments such that all of its nonzero solutions are infinitely extendable to the right and S is their set of characteristic exponents. Further, the characteristic exponents of the linear approximation system ẋ = A(t)x, x ∈ ℝ2, are λ1(A) = λ1 ≤ λ2(A) = λ2, its coefficients are bounded on the half-line [t0, +∞), and the perturbation f (t, y)is of order m > 1 in a neighborhood of the origin y = 0 and of an admissible order of growth outside it: ‖ f (t,y)‖ ≤ const ‖y‖m, y ∈ ℝ2, t t0.
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