Multifractal spectra and hyperbolic geometry

In recent years Procaccia, Kohmoto, and others studied non-regular fractal sets appearing in physics -such as the strange attractors- using the multifractal spectral decomposition technique introduced by Procaccia. These studies allowed the discovery of the so-called universal functions — and constants. Thus, a number of apparently unconnected physical phenomena have associated fractal sets with the same spectral decomposition function. Tél proposed a qualitative, tentative classification of the universal functions known so far. We model these using hyperbolic geometry, by decomposing the limit sets of finitely generated groups of motions in the Poincarécircle. In connection with the metrics of the spectral decomposition curves, we dérive the same constants obtained in physics.