As ‘Fractals are Everywhere’ in geo-ref data, which GIS other than ArcGIS can calculate fractal dimensions?
QGIS could too via a plugin it seems :
SAGA GIS seems a good candidate : Library Fractals
- Bifurcation - Fractal Dimension of Grid Surface - Gaussian Landscapes - Mandelbrot Set (interactive) - Newton-Raphson (interactive) - Pythagoras' Tree
As @mkennedy mentions there's GRASS too :
I know you exclude Arcgis, but just for generic info, it seems it has some add-on tools that could help :