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Fractal dimensions of almost periodic attractors

Published online by Cambridge University Press:  19 September 2008

Koichiro Naito
Koichiro Naito
Affiliation:
Kumamoto University, Department of Mathematics, Faculty of Engineering, Kurokami 2-39-1, Kumamoto 860Japan (email: naito@sonata.tech.kumamoto-u.ac.jp or knaito@gpo.kumamoto-u.ac.jp)
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Abstract

In this paper we estimate fractal dimensions of almost periodic orbits in terms of two kinds of exponents: the exponent in the inclusion lengths for ε-almost period and the exponent in Hölder conditions. Further, we estimate the inclusion lengths for ε-almost period of quasi-periodic functions by using Diophantine approximations. In the n-frequency quasi-periodic case we can show that the fractal dimension of its orbit is majorized by the value n/δ when it is Hölder continuous with exponent δ, 0 < δ ≤ 1.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 791 - 803
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1]Amerio, L. and Prouse, G.. Almost-Periodic Functions and Functional Equations. Van Nostrand Reinhold, 1971.CrossRefGoogle Scholar
[2]Dafermos, C. M. and Slemrod, M.. Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973), 97–106.CrossRefGoogle Scholar
[3]Feigenbaum, M. J.. Qualitative universality for a class of nonlinear transformations. J. Stat. Phys. 19 (1978), 25–52.CrossRefGoogle Scholar
[4]Frederickson, P., Kaplan, J. L., Yorke, E. D. and Yorke, J. A.. The Liapunov dimension of strange attractors. J. Differential Equations 49 (1983), 185–207.CrossRefGoogle Scholar
[5]Gill, T. L. and Zachary, W. W.. Dimensionality of invariant sets for nonautonomous processes. SIAM J. Math. Anal. 23 (1992), 1204–1229.CrossRefGoogle Scholar
[6]Grebogi, C., Ott, E. and Yorke, J. A.. Are three-frequency quasi-periodic orbits to be expected in typical nonlinear dynamical systems? Phys. Rev. Lett. 51 (1983) 339–342.CrossRefGoogle Scholar
[7]Henry, D.. Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Math. 840) Springer, 1981.CrossRefGoogle Scholar
[8]Naito, K.. On the almost periodicity of solutions of a reaction diffusion system. J. Differential Equations 44 (1982), 9–20.CrossRefGoogle Scholar
[9]Ruelle, D. and Takens, F.. On the nature of turbulence. Commun. Math. Phys. 30 (1971), 167–192.CrossRefGoogle Scholar
[10]Schmidt, W. M.. Diophantine Approximation (Lecture Notes in Math. 785). Springer, 1980.Google Scholar
[11]Temam, R.. Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Appl. Math. Sci. 68). Springer, 1988.CrossRefGoogle Scholar