The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

• 成果类型: Article
• 署名作者: Das, Tushar; Simmons, David
• 署名单位: University of Wisconsin System; University of Wisconsin La Crosse; University of York - UK
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0725-5
• 发表日期: 2017
• 页码: 85-134
• 关键词: multifractal sierpinski sponges; full dimension; limit-sets; invariant-measures; ergodic-theory; analyticity; exponents; geometry; entropy; MAPS

摘要

We construct a self-affine sponge in R-3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.