Multifractal spectrum and thermodynamical formalism of the Farey tree

Let (Omega, mu) be a set of real numbers to which we associate a measure mu. Let alpha greater than or equal to 0, let Omega (alpha) = {chi is an element of Omega/alpha(chi) = alpha}, where alpha is the concentration index defined by Halsey et al. [1986]. Let f(H)(alpha) be the Hausdorff dimension of Omega (alpha). Let f(L)(alpha) be the Legendre spectrum of Omega, as defined in [Riedi & Mandelbrot, 1998]; and f(C)(alpha) the classical computational spectrum of Omega, defined in [Halsey et al., 1986]. The task of comparing fH, fe and fL for different measures mu was tackled by several authors [Cawley & Mauldin, 1992; Mandelbrot & Riedi, 1997; Riedi & Mandelbrot, 1998] working, mainly, on self-similar measures mu. The Farey tree partition in the unit segment induces a probability measure mu on an universal class of fractal sets Omega that occur in physics and other disciplines. This measure mu is the Hyperbolic measure mu (H), fundamentally different from any self-similar one. In this paper we compare f(H), f(C) and f(L) for mu (H).