There are many definitions/methods of calculating the fractal dimension (Hausdorff, Box-counting, Higuchi, Katz, Correlation, etc. dimensions) and I think the main one is the Hausdorff dimension $D$ defined as:
$$D=-\frac{\log{N}}{\log{\epsilon}}$$
where $N$ is the number of measured units and $\epsilon$ is the scaling factor [wikipedia]. Are all these different definitions the estimates of the same fractal dimension (Hausdorff) or are they independent measures of complexity?
Wikipedia suggests the latter but in this paper for example the estimation of methods are compared with a theoretical fractal dimension (Hausdorff) of a synthetic time series.