Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets (Q2716113)

Let \(K\subset\mathbb{R}^n\) compact be defined from a set of contractions \(\{\varphi_i\}_{i=1}^{m}\) via the self-reflexive equation \(K=\bigcup_{i=1}^{m}\varphi_i(K)\). In the case that \(K\) is self-similar (\(\varphi_i\) is a similitude) the Hausdorff measure of \(K\) satisfies (i) \(H^{\dim K}(K)>0\) if (ii) the open set condition (OSC) holds, i.e., there exists \(U\not =\emptyset\) open such that \(\varphi_i(U)\subset U\) for all \(i\) and \(\varphi_i(U)\cap\varphi_j(U)=\emptyset\) for \(i\not = j\) [\textit{J. E. Hutchinson}, Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)]. The remarkable reverse implication was established by \textit{A. Schief} [Proc. Am. Math. Soc. 122, No. 1, 111-115 (1994; Zbl 0807.28005)] who in fact proved that (i) \(\Rightarrow\) (iii) the strong OSC, i.e., the open set \(U\) in (ii) can be chosen so that \(U\cap K\not =\emptyset\), by making use of (iv) an alternative algebraic characterization of (i) given by \textit{C. Bandt} and \textit{S. Graf} [Proc. Am. Math. Soc. 114, No. 4, 995-1001 (1992; Zbl 0823.28003)]. NEWLINENEWLINENEWLINEWhether the equivalence (i)\(\iff\)(ii)\(\iff\)(iii) holds when the \(\varphi_i\)'s are conformal (\(K\) is called self-conformal) has been asked by R. D. Mauldin. This paper clears the questions: the full equivalence (i)\(\iff\)(ii)\(\iff\)(iii)\(\iff\)(iv) extends mutatis mutandis to the self-conformal case. The proofs, however, of the linear case do not translate to the nonlinear setting, that requires new ideas to take care of distorsion.