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Quasiconformal groups and a theorem of Bishop and Jones. (English) Zbl 1079.30017
This paper is a sequel to a series of papers by the authors (with other co-authors) on conformal dynamics, in particular the theory developed by Patterson, Sullivan and Tukia. A basic result in that theory, whose general version is due to Bishop and Jones, states that the Hausdorff dimension of the conical limit set of a non-elementary Kleinian group acting on the hyperbolic space \(\mathbb{H}^n\) can be estimated in terms of the exponent of convergence of the Poincaré series of that action [see C. Bishop and P. Jones, Acta Math. 179, 1–39 (1997; Zbl 0921.30032)]. In the paper under review, the authors obtain such a result in a general setting where the group action is not necessarily Kleinian, but quasiconformal. They obtain new bounds on the exponent of convergence of planar discrete quasiconformal groups in terms of the associated dilatation and the Hausdorff dimension of its conical limit set. They show that if \(G\) is a discrete non-elementary \(K\)-quasiconformal group acting on \(\mathbb{S}^2\), then its exponent of convergence is uniformly bounded, from above by a function of the dilatation \(K\) and of the Hausdorff dimension of their conical limit set. More precisely, they prove the following bound: \[ \dim L_C(G)\leq \delta(G)\leq {2K \dim L_C(G)\over 2+(K-1)\dim L_C(G)} \] where \(\dim L_C(G)\) is the Hausdorff dimension of the conical limit set and \(\delta(G)\) is the exponent of convergence. The theorem of Bishop and Jones is then obtained as an asymptotic limit in the dilatation.
MSC:
30C62 Quasiconformal mappings in the complex plane
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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