Construction of an Arbitrary Suslin Set of Positive Characteristic Exponents in the Perron Effect

Abstract

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.