tailieunhanh - Đề tài " On the dimensions of conformal repellers. Randomness and parameter dependency "

Bowen’s formula relates the Hausdorff dimension of a conformal repeller to the zero of a ‘pressure’ function. We present an elementary, self-contained proof to show that Bowen’s formula holds for C 1 conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen’s formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower box dimensions and is given by a natural generalization of Bowen’s formula | Annals of Mathematics On the dimensions of conformal repellers. Randomness and parameter dependency By Hans Henrik Rugh Annals of Mathematics 168 2008 695 748 On the dimensions of conformal repellers. Randomness and parameter dependency By Hans Henrik Rugh Abstract Bowen s formula relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function. We present an elementary self-contained proof to show that Bowen s formula holds for C1 conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen s formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower box dimensions and is given by a natural generalization of Bowen s formula. For a random uniformly hyperbolic Julia set on the Riemann sphere we show that if the family of maps and the probability law depend real-analytically on parameters then so does its almost sure Hausdorff dimension. 1. Random Julia sets and their dimensions Let U du be an open connected subset of the Riemann sphere avoiding at least three points and equipped with a hyperbolic metric. Let K c U be a compact subset. We denote by E K U the space of unramified conformal covering maps f Df U with the requirement that the covering domain Df c K. Denote by Df Df R the conformal derivative of f see equation and by Df II supf- 1K Df the maximal value of this derivative over the set f-1K. Let F fn c E K U be a sequence of such maps. The intersection J F n f-1 f-1 U n 1 defines a uniformly hyperbolic Julia set for the sequence F. Let Y v be a probability space and let w G Y f G E K U be a v-measurable map. Suppose that the elements in the sequence F are picked independently each according to the law v. Then J F becomes a random variable . Our main objective is to establish the .