Parabolic orbifolds and the dimension of the maximal measure for rational maps (Q1812872)

Let \(f : \overline{\mathbb{C}} \to \overline{\mathbb{C}}\) be a rational map of the Riemann sphere, \(\text{deg}(f) \geq 2\). A natural invariant measure \(m\) - - the measure of maximal entropy was constructed by \textit{M. Yu. Lyubich} [Ergodic Theory Dyn. Syst. 3, 351-385 (1983; Zbl 0537.58035)] and independently by \textit{A. Freire, A. Lopes} and \textit{R. Mañé} [Bol. Soc. Bras. Mat. 14, No. 1, 45-62 (1983; Zbl 0568.58027)]. The aim of this paper is to compare this measure with some Hausdorff measures. First recall the following definition. For a probability measure \(\nu\) on \(\overline{\mathbb{C}}\) (or, more generally, on a smooth manifold) the Hausdorff dimension of \(\nu\) is defined by a formula \[ HD(\nu) = \inf_{Y : \nu(Y) = 1} HD(Y) \] (where \(HD(Y)\) is the Hausdorff dimension of \(Y\)). It was conjectured by \textit{M. Yu. Lyubich} [loc.cit.] that the Hausdorff dimension of the measure \(m\) is strictly smaller than the Hausdorff dimension of the Julia set \(J(f)\) (which is a support of \(m\)) except for some very special cases, called ``critically finite with parabolic orbifold''. In the present paper we give a proof of this conjecture as well as some related results.