Talk:Attractor dimensions

Reviewer A

The review is, of course, scientifically sound, my remarks point in the direction of easier readability, and more illustrations only.

At the very beginning, the presentation of a plot of a factal chaotic attractor would be enlightening.

I would recommend to start the section 'Box counting dimension' with a more qualitative picture, by just recalling the relation \(N(\epsilon) \sim \epsilon^{-D_0}\) and giving an example of the \(\ln N(\epsilon)\) vs \(\ln(1/\epsilon)\) plot. It might be worth mentioning that this \(D_0\) is also called the fractal dimension.

Section 'The natural measure' I would recommend to start with a qualititve description of the fact that on any attractor a natural distribution develops which is the probability that a certain part of the attractor is visited by the long term dynamics. Here a spatial view of a natural distributon would be very useful, whch could also be used to motivate the need for a fractal characterization of such disribution. (If needed, it would be a pleasure for me to provide such a figure.)

In 'The Renyi Dimension D_q' section the alternative term of generalized dimensions could be mentioned.

In the section on 'Lyapunov dimension ..' I recommend first to give the result for two-dimensional maps (or three-dimensional flows): \[D_L=1-\lambda_1/\lambda_2\], which is much easier, and this can then be followed by (6).

As for references, the author might wish to add Beck, Schlogl: Thermodynamics of Chaotic Systems, CUP, 1993 in relation to Renyi dimensions, and Tel, Gruiz: Chaotic Dynamics, CUP 2006 as a recent book on chaos and transient chaos.

Thank you for your help.