Fractal dimension of quasi-periodic orbits Original Research Article

In this paper, we estimate fractal dimensions of quasi-periodic orbits. Recently, Naito considered quasi-periodic orbits of Hölder continuous functions and showed that if the frequency vector ω satisfies certain Diophantine approximation type conditions, then in the n-frequency quasi-periodic case, the fractal dimension of its orbit is majorized by the value n/δ when it is Hölder continuous with exponent δ, 0 < δ ≤ 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 ω ϵ View the MathML sourcen. With the reverse Hölder type condition, for a set of ω, the corresponding lower bound on the dimension can be obtained by a similar argument.