Abstract
We prove that if the Lyapunov exponent of the norm of the coefficient matrix of a linear differential system is nonpositive, then the supremum of Perron exponents of the solutions issuing from any given affine subspace is attained and the set of initial vectors of solutions with the maximum Perron exponent has full Lebesgue measure in the subspace.
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Perron, O., Die Ordnungszahlen linearer Differentialgleichungssysteme, Math. Zeitschr., 1930, vol. 31, no. 5, pp. 748–766.
Izobov, N.A., Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to the Theory of Lyapunov Exponents), Minsk: Belarus. Gos. Univ., 2006.
Sergeev, I.N., Metrically typical and essential values of exponents of linear systems, Differ. Uravn., 2011, vol. 47, no. 11, pp. 1661–1662.
Izobov, N.A., The set of lower exponents of a linear differential system, Differ. Uravn., 1965, vol. 1, no. 4, pp. 469–477.
Izobov, N.A., The measure of the solution set of a linear system with the largest lower exponent, Differ. Uravn., 1988, vol. 24, no. 12, pp. 2168–2170.
Gargyants, A.G., A remark on the problem of the typicality and essentiality of values of the Perron exponent of unbounded linear systems, Differ. Uravn., 2013, vol. 49, no. 11, pp. 1505–1506.
Gargyants, A.G., On the metric typicality of the Perron exponent on solutions of unbounded systems, Differ. Equations, 2014, vol. 50, no. 11, pp. 1557–1558.
Sergeev, I.N., Topologically typical and essential values of exponents of linear systems, Differ. Uravn., 2012, vol. 48, no. 11, pp. 1567–1568.
Gargyants, A.G., On the topological typicality and essentiality of values of the Perron exponent of a linear system, Differ. Uravn., 2013, vol. 49, no. 6, pp. 808–809.
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Original Russian Text © A.G. Gargyants, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 8, pp. 1011–1017.
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Gargyants, A.G. Supremum of the Perron Exponent on the Solutions of a Linear System with Slowly Growing Coefficients is Metrically Typical. Diff Equat 54, 993–999 (2018). https://doi.org/10.1134/S0012266118080013
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DOI: https://doi.org/10.1134/S0012266118080013