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Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations
Author(s):
Koichiro
Naito
Journal:
Trans. Amer. Math. Soc.
354
(2002),
1137-1151.
MSC (2000):
Primary 11K60, 28A80, 35B15
Posted:
September 21, 2001
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Abstract:
In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE.
References:
-
- 1.
- V.Barbu and Th.Precupanu, Convexity and Optimization in Banach Spaces, Sijthoff & Noordhoff, 1978. MR 87k:49045
- 2.
- P. Bergé, Y. Pomeau, C. Vidal, Order within chaos, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York; Hermann, Paris, 1986. MR 88f:58099
- 3.
- H.Brezis, Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. MR 50:1060
- 4.
- M.G.Crandall and A.Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57-94. MR 45:9214
- 5.
- C.M.Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, in Dynamical Systems, Proc. Univ. Florida Inter. Symp." (A.R.Bednarek and L.Cesari, Eds.), 43-57, Academic Press, New York, 1977. MR 57:3545
- 6.
- T.Kato, Nonlinear semi-groups and evolution equations J. Math. Soc. Japan 19 (1967), 508-520. MR 37:1820
- 7.
- K.Naito, On the almost periodicity of solutions of a reaction diffusion system, J. Differential Equations 44 (1982), 9-20. MR 83f:35009
- 8.
- -, Fractal dimensions of almost periodic attractors, Ergodic Theory and Dynamical Systems 16 (1996), 791-803. MR 97e:58156
- 9.
- -, Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation, J. Differential Equations, 141 (1997), 179-200. MR 98k:35215
- 10.
- -, Correlation dimensions of quasi-periodic orbits with frequencies given by Roth numbers, Differential equations and applications (Chinju, 1998), 119-129, Nova Sci. Publ., Huntington, NY, 2000. MR 2001e:37038
- 11.
- -, Correlation dimensions of quasi-periodic orbits with frequencies given by quasi Roth numbers, J. Korean Math. Soc. 37 (2000), 857-870. MR 2001h:37051
- 12.
- -, Fractal dimensions of quasi-periodic orbits given by Weierstrass type functions, to appear in J. Dyn. Diff. Eq.
- 13.
- Y. B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997. MR 99b:58003
- 14.
- R.T.Rockafellar, Integrals which are convex functionals, I, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
- 15.
- -, Integrals which are convex functionals, II, Pacific J. Math. 39 (1971), 439-469. MR 46:9710
- 16.
- W.M.Schmidt, Diophantine Approximation, Springer Lecture Notes in Math. 785, 1980. MR 81j:10038
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Additional Information:
Koichiro
Naito
Affiliation:
Faculty of Engineering, Kumamoto University, Kurokami 2-39-1, Kumamoto, 860-8555, Japan
Email:
naito@cs.kumamoto-u.ac.jp
DOI:
10.1090/S0002-9947-01-02901-4
PII:
S 0002-9947(01)02901-4
Keywords:
Diophantine approximation,
quasi-periodic solutions,
fractal dimension
Received by editor(s):
October 29, 2000
Received by editor(s) in revised form:
May 9, 2001
Posted:
September 21, 2001
Copyright of article:
Copyright
2001,
American Mathematical Society
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