I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical property of a complex system, is there any interpretation for that kind of series of slopes? Any literature around?
And, generally - if we do that kind of fractal analysis, what does the overall shape of logN vs log(e) tell us (if anything)?



Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?
The canonical reference for this material (at least with the people I hang out with) is Ken Falconer's Fractal Geometry Mathematical Foundations and Applications. 


For most sets that are fully self-similar with infinite levels of geometry such as the Cantor set the function $\log(N)/\log(e)$ will be a similar polygonal chain far away from 0. It will get smoother as you approach zero and there will be a non-zero slope there. What you have however is a finite set of points so for an $e$ below a certain threshold $N$ stays constant. That is the function will have slope zero and you have a set of fractal dimension zero. In the book there is some discussion of how to interpret situations like there where you are approximating something that will have a non-zero dimension. But that is not something I can say much about. 


