The Oseledec and SackerSell spectra for almost periodic linear systems: an example
Author:
Russell A. Johnson
Journal:
Proc. Amer. Math. Soc. 99 (1987), 261267
MSC:
Primary 34C35; Secondary 54H20, 58F19, 58F27
MathSciNet review:
870782
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We give an example illustrating the relation between the Oseledec spectrum (roughly speaking, the set of Lyapunov exponents) and the SackerSell (or continuous) spectrum for Bohr almost periodic linear systems.
 [1]
A. Coppel, Dichotomies and stabilitity theory, Lecture Notes in Math., vol. 629, SpringerVerlag, Berlin and New York, 1978.
 [2]
Ju.
L. Dalec′kiĭ and M.
G. Kreĭn, Stability of solutions of differential equations
in Banach space, American Mathematical Society, Providence, R.I.,
1974. Translated from the Russian by S. Smith; Translations of Mathematical
Monographs, Vol. 43. MR 0352639
(50 #5126)
 [3]
Robert
Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc.,
New York, 1969. MR 0267561
(42 #2463)
 [4]
Robert
Ellis and Russell
A. Johnson, Topological dynamics and linear differential
systems, J. Differential Equations 44 (1982),
no. 1, 21–39. MR 651685
(83c:54058), http://dx.doi.org/10.1016/00220396(82)900237
 [5]
Harry
Furstenberg, Harvey
Keynes, and Leonard
Shapiro, Prime flows in topological dynamics, Israel J. Math.
14 (1973), 26–38. MR 0321055
(47 #9588)
 [6]
Russell
A. Johnson, The recurrent Hill’s equation, J.
Differential Equations 46 (1982), no. 2,
165–193. MR
675906 (84m:34037), http://dx.doi.org/10.1016/00220396(82)901140
 [7]
R. Johnson, K. Palmer and G. Sell, Ergodic theory of linear dynamical systems, SIAM J. Appl. Math. (to appear).
 [8]
V. Millionshchikov, Metric theory of linear systems of differential equations, Math. USSRSb. 6 (1968), 149158.
 [9]
, Proof of the existence of nonirreducible systems of linear differential equations with almost periodic coefficients, J. Differential Equations 4 (1968), 203205.
 [10]
V.
I. Oseledec, A multiplicative ergodic theorem. Characteristic
Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat.
Obšč. 19 (1968), 179–210 (Russian). MR 0240280
(39 #1629)
 [11]
L. Pontryagin, Topologische Gruppen, Deutsche Ubersetzung, Teubner Verlag, Leipzig, 1957.
 [12]
Robert
J. Sacker and George
R. Sell, Existence of dichotomies and invariant splittings for
linear differential systems. I, J. Differential Equations
15 (1974), 429–458. MR 0341458
(49 #6209)
 [13]
Robert
J. Sacker and George
R. Sell, A spectral theory for linear differential systems, J.
Differential Equations 27 (1978), no. 3,
320–358. MR 0501182
(58 #18604)
 [14]
W.
A. Veech, Almost automorphic functions on groups, Amer. J.
Math. 87 (1965), 719–751. MR 0187014
(32 #4469)
 [1]
 A. Coppel, Dichotomies and stabilitity theory, Lecture Notes in Math., vol. 629, SpringerVerlag, Berlin and New York, 1978.
 [2]
 J. Daletskii and M. Krein, Stability of solutions of differential equations in Banach space, Transl. Math. Monos., vol. 43, Amer. Math. Soc., Providence, R.I., 1974. MR 0352639 (50:5126)
 [3]
 R. Ellis, Lectures on topological dynamics, Benjamin, Reading, Mass., 1969. MR 0267561 (42:2463)
 [4]
 R. Ellis and R. Johnson, Topological dynamics and linear differential systems, J. Differential Equations 44 (1982), 2139. MR 651685 (83c:54058)
 [5]
 H. Furstenberg, H. Keynes and L. Shapiro, Prime flows in topological dynamics, Israel J. Math. 14 (1973), 2638. MR 0321055 (47:9588)
 [6]
 R. Johnson, The recurrent Hills equation, J. Differential Equations 46 (1982), 165194. MR 675906 (84m:34037)
 [7]
 R. Johnson, K. Palmer and G. Sell, Ergodic theory of linear dynamical systems, SIAM J. Appl. Math. (to appear).
 [8]
 V. Millionshchikov, Metric theory of linear systems of differential equations, Math. USSRSb. 6 (1968), 149158.
 [9]
 , Proof of the existence of nonirreducible systems of linear differential equations with almost periodic coefficients, J. Differential Equations 4 (1968), 203205.
 [10]
 V. Oseledec, A multiplicative ergodic theorem, Trans. Moscow Math. Soc. 19 (1968), 197231. MR 0240280 (39:1629)
 [11]
 L. Pontryagin, Topologische Gruppen, Deutsche Ubersetzung, Teubner Verlag, Leipzig, 1957.
 [12]
 R. Sacker and G. Sell, Dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429458. MR 0341458 (49:6209)
 [13]
 , A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320358. MR 0501182 (58:18604)
 [14]
 W. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719751. MR 0187014 (32:4469)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
34C35,
54H20,
58F19,
58F27
Retrieve articles in all journals
with MSC:
34C35,
54H20,
58F19,
58F27
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708707827
PII:
S 00029939(1987)08707827
Keywords:
Almost periodic,
minimal set,
Lyapunov number
Article copyright:
© Copyright 1987
American Mathematical Society
