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

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JournalInventiones Mathematicae
DateAccepted/In press - 25 Mar 2017
DateE-pub ahead of print - 26 Apr 2017
DatePublished (current) - Oct 2017
Issue number1
Volume210
Number of pages50
Pages (from-to)85-134
Early online date26/04/17
Original languageEnglish

Abstract

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.

    Research areas

  • 37C40, 37D20, Primary 37C45, Secondary 37D35

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