| Porosity Of Collet-Eckmann Julia Sets (1998) | |||||||||||||||
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| . We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2. 1. Introduction Let f : b C ! b C be a rational map. Then f is said to satisfy the Collet-Eckmann condition if there are constants C ? 0 and ? 1 such that (CE) j(f n ) 0 (f(c))j C n for all n and all critical points c 2 J(f) of f whose forward orbit does not meet another critical point (J(f) stands for the Julia set of f ). Here and in what follows derivatives and distances are always with respect to the spherical metric of b C ; unless stated otherwise. A set E ae b C is called mean porous if there are constants p 1 ! 1 and p 2 ? 0 such that for each z 2 E the following holds: There is an increasing sequence n j of integers and points z j with dist(z; z j ) 2 \Gamman j such that n j ! p 1 j and dist(z j ; E) ? ... | |||||||||||||||
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