Multifractal Spectrum and Thermodynamical Formalism of the Farey Tree

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. [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 and Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et al., 1986]. The task of comparing fH , fC , and fL for different measures μ was tackled by several authors ([Cawley and Mauldin, 1992], [Mandelbrot and Riedi, 1997], [Riedi and 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 μIH , fundamentally different from any self-similar one. In this paper we compare fH , fC , and fL for μ IH .