Mendes France and thermodynamical spectra: a comparative study of contractive and expansive fractal processes

This paper presents a comparative study of two families of curves in R(n). The first ones comprise self-similar bounded fractals obtained by contractive processes, and have a non-integer Hausdorff dimension. The second ones are unbounded, locally rectifiable, locally smooth, obtained by expansive processes, and characterized by a fractional dimension defined by M. Mendes France. We present a way to relate the two types of curves and their respective non-integer dimensions. Thus, to one fractal bounded curve we associate, at first, a finite range of Mendes France dimensions, identifying the minimal and the maximal ones. Later, we show that this discrete spectrum can be made continuous, allowing it to be compared with some other multifractal spectra encountered in the literature. We discuss the corresponding physical interpretations.