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By B. O. (-stan) Urbański and Mariusz (-ntxs


Jarník and Julia: a Diophantine analysis for parabolic rational maps for geometrically finite Kleinian groups with parabolic elements. (English summary) Math. Scand. 91 (2002), no. 1, 27–54. In this paper, the authors generalise two results from number theory, a theorem by Dirichlet and a theorem by Jarník, to the context of iterated parabolic rational maps. They then use these results to give a “weak multifractal analysis ” of the unique conformal measure of dimension h (where h is the Hausdorff dimension of the Julia set of the rational map) which is supported on the Julia set. Corresponding results already exist for Kleinian groups, so the paper contributes to the famous “dictionary ” of Sullivan, which lists parallel results in the theories of rational maps and Kleinian groups. The two results from number theory concern the approximability of real numbers by rationals. Dirichlet’s theorem states that there exists a universal constant κ> 0 such that given any sufficiently small positive number α and any positive real number x, there exists a reduced rational p/q with 1/q2> α such that p/q lies within κ √ α/q2 of x. Jarník’s theorem gives the Hausdorff dimension of the set of points in R+ for which the approximatio

Year: 2011
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