| On Hausdorff Measures And SBR Measures For Parabolic Rational Maps | |||||||||||||||
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| If J is the Julia set of a parabolic rational map having Hausdorff dimension h 0 or 0 for some explicitly computable 0 > 0. x1 A dichotomy for SBR measures and Hausdor measures An analytic endomorphism T of the Riemannian sphere C and of degree 2 is called parabolic if its Julia set does not contain any critical point but a rationally indifferent periodic point (i.e. its multiplier is a root of unity). It is known that the Hausdorff dimension h of the Julia set of such a transformation satisfies 1 2 < h < 2 (see [1]) and that there exists a unique non-atomic probability measure m determined by the Jacobian dm Æ T dm = jT 0 j h a.s. Such measures are called h-conformal in the sense of Sullivan ([11]). In the hyperbolic case this measure is a multiple of the h... | |||||||||||||||
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