Abstract
The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.
Similar content being viewed by others
References
L. Auslander and F. Hahn, “Real functions coming from flows on compact spaces of almost periodicity,“Trans. Amer. Math. Soc.,106, 415–426 (1963).
I. U. Bronshtein,Extensions of Minimal Transformations of Groups [in Russian], Stiinta, Kishinev (1975).
B. M. Levitan and V. V. Zhikov,Almost Periodic Functions and Differential Equations [in Russian], Moscow State University, Moscow (1978).
R. J. Sakers and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems, III,“J. Diff. Equal.,22, 497–522 (1976).
V.A. Glavan, “The structure of linear extensions with Favard-type conditions,“ Ukr. Mat. Zh.,44, 596–604 (1992).
Y. Hino, “Recurrent solutions for linear almost periodic systems,“ Func. Ekvac.,28, 117–119 (1985).
V. A. Glavan, “Linear systems of differential equations with the additivity property of the recurrent solutions,“ in:Asymptotic Solutions of Linear Equations with Small Parameters [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991).
V. A. Glavan, “Structure of linear extensions with the Favard-type conditions. I. Linear extensions with additivity property for recurrent motions,“Ukr. Mat. Zh.,45, No. 4, 466–471, (1993).
V. A. Glavan, “Favard-type theorems without the Favard condition,“ in:Discontinuous Dynamical Systems. Abstracts of Scientific Workshop (September 1991, Uzhgorod) [in Russian], Obshchestvo “Znanie“ Ukrainy, Kiev (1991).
V. A. Glavan, “On nilpotent-like linear extensions of minimal flows,“ in:Simpozionul al VI-lea Tiraspolean de Topologie Generala si Aplicatiile ei (September, (1991)), Stiinta, Kishinev (1991).
M. Atiyah,Lectures on K-Theory [Russian translation], Mir, Moscow (1967).
I. U. Bronshtein,Nonautonomous Dynamical Systems [in Russian], Stiinta, Kishinev (1984).
M. Rees, “Tangential distal flows,“Israel J. Math.,35, 9–31 (1980).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 45, No. 2, pp. 233–238, February, 1993.
Rights and permissions
About this article
Cite this article
Glavan, V.A. On equicontinuous factors of linear extensions of minimal dynamical systems. Ukr Math J 45, 249–254 (1993). https://doi.org/10.1007/BF01060980
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01060980