Minkowski dimension and the Poincaré exponent
The lower and upper Minkowski dimensions \(\underline{\text{Mdim}}(K)\) and \(\overline{\text{Mdim}}(K)\) are defined for a compact set \(K\) of the two-sphere. If both quantities are equal, then it is called the Minkowski dimension and denoted by \(\text{Mdim}(K)\). If \(G\) is a Kleinian group, then we may use \(\Lambda(G)\) (the limit set of \(G\)) for \(K\). Another quantity associated to \(G\) is the Poincaré exponent defined by \(\delta(G)=\inf\{s:\sum_{g\in G}\text{dist}(g(z),\Lambda(G))<\infty\}\). The main result of this paper relates both the Poincaré exponent of a group \(G\) and the upper Minkowski dimension of its limit. More precisely: Theorem. Let \(G\) be an analytically finite, non-elementary Kleinian group. If \(\text{area}(\Lambda(G))=0\), then \(\delta(G)=\overline{\text{Mdim}}(\Lambda(G))\). The facts that (i) the Hausdorff dimension \(\dim(K)\) is less or equal to the lower Minkowski dimension of \(K\), and (ii) \(\delta(G)\leq\dim(\Lambda(G))\) for a Kleinian group, can be used together with the above theorem to obtain the following: Corollary. If \(G\) is an analytically finite, non-elementary Kleinian group with \(\text{area}(\Lambda(G))=0\), then \(\delta(G)=\dim(\Lambda(G))\). In particular, the above holds for geometrically finite Kleinian groups \(G\).
scientific article; zbMATH DE number 934195