Losing Hausdorff dimension while generating pseudogroups

Volume 149 / 1996

Paweł Walczak Fundamenta Mathematicae 149 (1996), 211-237 DOI: 10.4064/fm-149-3-211-237

Abstract

Considering different finite sets of maps generating a pseudogroup G of locally Lipschitz homeomorphisms between open subsets of a compact metric space X we arrive at a notion of a Hausdorff dimension $dim_H G$ of G. Since $dim_H G ≤ dim_H X$, the dimension loss $dl_HG = dim_HX - dim_H G$ can be considered as a "topological price" one has to pay to generate G. We collect some properties of $dim_H$ and $dl_H$ (for example, both of them are invariant under Lipschitz isomorphisms of pseudogroups) and we either estimate or calculate $dim_HG$ for pseudogroups arising from classical dynamical systems, group actions, foliations, etc.

Authors

  • Paweł Walczak

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