At the moment I want to estimate the fractal dimension of the Rössler attractor. I have written a program, which is counting the boxes hitted N(ɛ) (with ɛ := side length of boxes) by a trajectory on the attractor. The trajectory is calculated numerically. So for a rising number of iterations, N(ɛ) is rising, like it is shown in the following graph (for ɛ = 1/128, horizontal axis is for iterations, vertical axis for N(ɛ)).

Graph for N(ɛ) against iterations:

The thing is, that the Graph is not converging, even if I run it for 50*10^6 iterations. There is a similar work for the Lorentz attractor, written by Mark J.McGuinness called "The Fractal Dimension of the Lorentz attractor" published in the Physics Letters Volume 99A number 1 on the 14 November 1983. Since the Lorentz attractor is similar to the Rössler attractor, you could get a good orientation from this work. In the paper they have the same problem for the Lorentz attractor and they guess the convergence to be algebraic and not logarithmic.

**Question**
What is known about the (algebraic ?) convergence of N(ɛ) (boxes bitted by a trajectory on the attractor) for strange attractors ?

researchand he has encountered a difficulty. The given problem does not seem to be trivial, and is definitely of research level. It's not Grothendieck-like algebraic geometry, but it is still research. $\endgroup$1more comment