On the baire classification of Sergeev frequencies of zeros and roots of solutions of linear differential equations

Abstract

We show that the upper and lower characteristic frequencies of zeros and the upper frequency of roots of a solution of a linear differential equation treated as functions on the direct product of the space of equations with the compact-open topology by the space of initial vectors of solutions belong to the third Baire class and that the lower characteristic frequency of roots belongs to the second Baire class. As a corollary, we show that the ranges of the considered frequencies on the solutions of a given equation are Suslin (analytic) sets. In addition, we prove the Lebesgue measurability and the Baire property of the extreme characteristic frequencies of zeros and roots of an equation treated as functions of a real parameter on which the coefficients of the equation depend continuously.