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)?

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 selfsimilar 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 nonzero 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 nonzero dimension. But that is not something I can say much about. 

