MULTIFRACTAL ANALYSIS OF NEARLY CIRCULAR JULIA SET AND THERMODYNAMICAL FORMALISM

The multifractal description of measures supported by strange sets has been introduced in order to analyse actual experimental data in chaotic systems. A justification of this approach on the basis of the thermodynamical formalism has previously been given for Cantor sets invariant under Markov maps. Here we give the analogous derivation for connected sets invariant under polynomial maps. More precisely, for polynomials close to z(q), we consider the uniform Z(U) and the dynamical Z(D) partition functions associated to coverings of the Julia set and we show that the corresponding thermodynamical limits F(U) and F(D) exist, when the size of the pieces of the coverings goes to zero. We then show an explicit relation between F(U) and F(D) and explain how to expand F(D) close to the circle case, corresponding to z(q). Results from the large deviation probability theory are used to show the relation between F(U) and the dimension spectrum, which includes Hausdorff dimension as a particular case. The method used here provides a complete description of the multifractal properties of nearly circular Julia sets and an explicit perturbation procedure for the Hausdorff dimension and for the multifractal spectrum.