Fractal dimension of quasi-periodic orbits

✍ Scribed by Pengjian Shang; K. Widder

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 306 KB
• Volume: 14
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(01)00073-8

✦ Synopsis

In this paper, we estimate fractal dimensions of quasi-periodic orbits. Recently, Naito considered quasi-periodic orbits of Holder continuous functions and showed that if the frequency vector w satisfies certain Diophantine approximation type conditions, then in the n-frequency qussiperiodic case, the fractal dimension of its orbit is majorized by the value n/6 when it is Hiilder continuous with exponent 6, 0 < 6 5 1. We prove that the upper bound on the dimension given by Naito can be obtained rather more easily, and to our astonishment, for all frequency vectors w E IV. With the reverse Holder type condition, for a set of w, the corresponding lower bound on the dimension can be obtained by a similar argument.