Self-conformal multifractal measures
Let \((S,p)\) be an iterated function system generated by selfmappings \(S_1,\dots, S_n\) of an open and connected subset of a smooth Riemannian manifold and a probability vector \((p_1,\dots, p_n)\). The mappings \(S_1,\dots, S_n\) are assumed to be conformal \(C^{1+\gamma}\) diffeomorphisms with \(\gamma\in (0,1)\) and to satisfy \(0<| S_i'|< 1\) for \(i= 1,\dots, n\) and the following Morans open set condition: there is an open (and bounded) set \(V\) with pairwise disjoint images \(S_1(V),\dots, S_n(V)\subset V\). It is well known that there is a unique compact \(\Xi\) with \(\Xi= \bigcup^n_{i= 1} S_i(\Xi)\) (attractor of \((S,p)\)) and a unique probability measure \(\varphi\) on \(\text{cl }V\) satisfying \(\varphi= \sum^n_{i=1} p_i\varphi\circ S^{-1}_i\) (fractal measure of \((S,p)\) with the support \(\Xi\)). The main result of the paper (Theorem 4.7) describes the local structure of the attractor \(\Xi\) by computing the Hausdorff and packing dimensions of the decomposition parts \(\Xi_a\) of \(\Xi\) defined for \(a\in\mathbb{R}\) by \(\Xi_a= \{x\in \Xi: \lim_{r\to 0}\log\varphi(B(x, r))/\log r=a\}\). The formulae obtained for these dimensions have been proved under the assumption \(V\cap\Xi\neq \emptyset\) and make use of some functions defined via the so-called equilibrium measures.
conformal selfmapping of a Riemannian manifold
scientific article; zbMATH DE number 1139872