Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-25T17:53:55.873Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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