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Metric diophantine approximation in Julia sets of expanding rational maps. (English) Zbl 0885.11051

In classical metric number theory one investigates various sets of real numbers defined by their approximability by rational numbers. In particular one is often interested in the Hausdorff dimension of such sets and, when this has been determined, the measure with respect to the appropriate Hausdorff measure. Many such results can be extended to the context of Kleinian groups, where the role of the rationals is taken over by the orbit of a parabolic or hyperbolic fixed point. Around 1980, D. Sullivan emphasized the analogy between the theory of Kleinian groups and the theory of the iteration of rational maps of the Riemann sphere.
Here, the authors consider the theory of an expanding rational map of the Riemann sphere. In this context, they define the notion of a well-approximable point of the Julia set with respect to the backward orbit of any fixed point of the sphere (and to a Hölder function). They determine the Hausdorff dimension of this set in terms of the vanishing of pressure function in the sense of Ruelle’s thermodynamic formalism. The main tool used in the proof is M. Denker and M. Urbański’s theory of conformal measures carried on the Julia set [Nonlinearity 4, 365-384 (1991; Zbl 0722.58028)]. In the course of the proof they also prove counting results describing the development of a backward orbit.

MSC:

11K60 Diophantine approximation in probabilistic number theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
26A16 Lipschitz (Hölder) classes
28A78 Hausdorff and packing measures
30C20 Conformal mappings of special domains
37E99 Low-dimensional dynamical systems

Citations:

Zbl 0722.58028
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References:

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