Abstract :
We analyse the multifractal decomposition of a measure defined on a general Sierpinski carpet. We compute the dimension spectrum f(α) and we show that this function is real-analytic on a certain domain when it is not degenerated. Actually, we prove that f is the Legendre-Fenchel transformation of a free energy function F which is also real-analytic. These two functions have the typical behaviours. In particular, F is strictly increasing and is in general strictly concave (respectively linear in the degenerate case), and f, on its domain, has a typical shape of a strictly concave function (respectively defined in one point in the degenerate case). We associate also to the singularity sets Cα measures which are singular with respect to each other, and we see that these measures are very well fitted to these singularity sets. This work completes with (D. Simpelaere, Chaos, Solitons and Fractals 4(12), 2223–2235 (1994)) the study of the multifractal analysis of the Sierpinski carpets.
