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

TL;DR

This paper compares contractive and expansive fractal processes, relating their non-integer dimensions and spectra, and explores their physical interpretations and potential connections to multifractal spectra.

Contribution

It introduces a method to relate bounded and unbounded fractal curves and their dimensions, bridging different types of fractal processes and spectra.

Findings

Established a relation between contractive and expansive fractal dimensions.

Converted discrete Mendes France spectra into a continuous form for comparison.

Discussed physical interpretations of the fractal spectra.

Abstract

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.