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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Recurrent-proximal linear differential systems with almost periodic coefficients


Author: Mahesh G. Nerurkar
Journal: Proc. Amer. Math. Soc. 100 (1987), 739-743
MSC: Primary 58F27; Secondary 34C28, 54H20
MathSciNet review: 894447
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Abstract: We consider a system of linear differential equations, $ \dot x = A(\omega \cdot t)x$, parametrized by a point $ \omega \in {{\mathbf{T}}^2}$, the $ 2$-torus, where $ (\omega ,t) \to \omega \cdot t$ denotes an irrational rotation flow on $ {{\mathbf{T}}^2}$. We show that if the rotation number of this flow is well approximable by rationals, then residually many equations (with respect to the $ {C^k}$-topology on a certain class of matrix valued maps $ A(\omega )$ on $ {{\mathbf{T}}^2}$) exhibit recurrent-proximal behavior. Also the order of differentiability $ k$ of the class in which this generic result holds is related to the "speed" of approximation by rationals.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0894447-0
Article copyright: © Copyright 1987 American Mathematical Society