On the dimensions of conformal repellers. Randomness and parameter dependency

• 成果类型: Article
• 署名作者: Rugh, Hans Henrik
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.168.695
• 发表日期: 2008
• 页码: 695-748
• 关键词: hausdorff dimension; DYNAMICAL-SYSTEMS; expanding maps; sets; PRODUCTS

摘要

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. For a random uniformly hyperbolic Julia set on the Reimann 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.