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- Title
Effective Hausdorff Dimension in General Metric Spaces.
- Authors
Mayordomo, Elvira
- Abstract
We introduce the concept of effective dimension for a wide class of metric spaces whose metric is not necessarily based on a measure. Effective dimension was defined by Lutz (Inf. Comput., 187(1), 49-79, <xref>2003</xref>) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses gambling, in particular the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated.
- Subjects
MATHEMATICAL models; FRACTAL dimensions; METRIC spaces; ALGORITHMIC randomness; KOLMOGOROV complexity
- Publication
Theory of Computing Systems, 2018, Vol 62, Issue 7, p1620
- ISSN
1432-4350
- Publication type
Article
- DOI
10.1007/s00224-018-9848-3