Coincidence of various dimensions associated with metrics and measures on metric spaces

Moisey Guysinsky; Serge Yaskolko

doi:10.3934/dcds.1997.3.591

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1997, Volume 3, Issue 4: 591-603. Doi: 10.3934/dcds.1997.3.591

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Coincidence of various dimensions associated with metrics and measures on metric spaces

Moisey Guysinsky^1, and
Serge Yaskolko^1,

1.
Department of Mathematics, The Pennsylvania State University, State College, PA 16802

Received: January 31, 1997

Published: September 1997

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Abstract

We establish coincidence of major types of dimensions for a broad class of separable metric spaces with finite borel measures. To do this we introduce a new type of separable metric spaces, so called tight spaces, for which these dimensions coincide naturally. This class includes, for example, all manifolds of the curvature bounded from below and any their subsets with induced metric. In particular, we prove that Hentshel-Procaccia and Renyi spectra for dimensions are equal in tight spaces for any measure. We also give the examples that demonstrate that all known dimensions can differ for bad enough metric spaces.

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Mathematics Subject Classification: 58F15, 28D20.

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