On Hausdorff dimension of Julia sets of hyperbolic rational semigroups
Let \(G\) be a finitely generated semigroup of rational maps of the Riemann sphere to itself. The purpose of this paper is first to investigate various notions of hyperbolicity of the semigroup. The author first shows that hyperbolicity is equivalent to being expanding and he introduces the stronger notion of ``the strong open set condition''. At the end of the paper, he constructs a sequence of examples satisfying this condition. Assuming it, the author shows first that the Lebesgue measure of the Julia set of \(G\) is 0, and, under a minor additional condition, shows that not only does the Hausdorff measure lie between 0 and 2 but that the Hausdorff measure at this dimension is finite. Under the assumption that \(G\) is expanding, the author gives an upper bound for the Hausdorff dimension of the Julia set.
scientific article; zbMATH DE number 1208192