MULTIFRACTAL SPECTRUM AND THERMODYNAMICAL FORMALISM OF THE FAREY TREE

Abstract

Let (Ω, μ) be a set of real numbers to which we associate a measure μ. Let α≥0, let Ωα={x∈Ω/α(x)=α}, where α is the concentration index defined by Halsey et al. [1986]. Let fH(α) be the Hausdorff dimension of Ωα. Let fL(α) be the Legendre spectrum of Ω, as defined in [Riedi & Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et al., 1986]. The task of comparing fH, fCand fLfor different measures μ was tackled by several authors [Cawley & Mauldin, 1992; Mandelbrot & Riedi, 1997; Riedi & Mandelbrot, 1998] working, mainly, on self-similar measures μ. The Farey tree partition in the unit segment induces a probability measure μ on an universal class of fractal sets Ω that occur in physics and other disciplines. This measure μ is the Hyperbolic measure μℍ, fundamentally different from any self-similar one. In this paper we compare fH, fCand fLfor μℍ.