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

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## Abstract

For an arbitrary bounded Suslin set *S* ⊂ (0, +∞) and arbitrary parameters *m* > 1 and λ_{1} ≤ λ_{2} < 0, we construct a two-dimensional differential system ẏ = *A*(*t*)*y* + *f* (*t, y*), *y ∈* ℝ^{2}, *t* ≥ *t*_{0}, 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* ∈ ℝ^{2}, are λ_{1}(*A*) = λ_{1} ≤ λ_{2}(*A*) = λ_{2}, its coefficients are bounded on the half-line [*t*_{0}, +∞), 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* ∈ ℝ^{2}, *t* *t*_{0}.

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