Multifractal Spectrum and Thermodynamical Formalism of the Farey Tree

TL;DR

This paper compares different multifractal spectra of the Farey Tree's hyperbolic measure, highlighting differences from self-similar measures and advancing understanding of fractal measure spectra.

Contribution

It provides a detailed comparison of the Hausdorff, computational, and Legendre spectra for the Farey Tree's hyperbolic measure, which is not self-similar.

Findings

Spectra differ significantly for the hyperbolic measure.

Self-similar measures have coinciding spectra.

New insights into multifractal analysis of non-self-similar measures.

Abstract

The task of comparing the Hausdorff spectrum, the computational spectrum, and the Legendre spectrum of a fractal set endowed with a probability measure, was tackled by several authors - Cawley and Mauldin, Riedi and Mandelbrot, among others. For self-similar measures all three spectra coincide. We compare these spectra for the hyperbolic measure (inducing the Farey Tree partition), fundamentally different from the self-similar one.