# American Institute of Mathematical Sciences

October  1997, 3(4): 591-603. doi: 10.3934/dcds.1997.3.591

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

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

Received  February 1997 Published  July 1997

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.
Citation: Moisey Guysinsky, Serge Yaskolko. Coincidence of various dimensions associated with metrics and measures on metric spaces. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 591-603. doi: 10.3934/dcds.1997.3.591
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