Effective Hausdorff Dimension in General Metric Spaces

Abstract

We introduce the concept of effective dimension for a wide class of metric spaces that are not required to have a computable measure. Effective dimension was defined by Lutz in for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses 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.

Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization

“…If the entropy grows linearly, its asymptotic slope coincides with the Minkowski-Bouligand or box-counting dimension of X; otherwise the latter is infinite. Compare also [KSZ16], [Mayo16]. .…”

Algorithmic Fractal Dimensions in Geometric Measure Theory

“…While Euclidean space has a very well-behaved metric based on a Borel measure µ, where for instance s-Hausdorff measure coincides with µ for s = 1, this is not the case for other metric spaces. Since both Hausdorff and packing dimension can be defined in any metric space, the second author has considered in [70] the extension of algorithmic dimension to a large class of separable metric spaces, the class of spaces with a computable nice cover. This extension includes an algorithmic information characterization of constructive dimension, based on the concept of Kolmogorov complexity of a point at a certain precision, which is an extension of the concept presented in section 2 for Euclidean space.…”

Extending the Reach of the Point-to-Set Principle

“…The classical Hausdorff and packing dimensions work not only in Euclidean spaces, but in arbitrary metric spaces. In contrast, nearly all work on algorithmic dimensions to date (the exception being [29]) has been in Euclidean spaces or in spaces of infinite sequences over finite alphabets. Our objective here is to significantly reduce this gap by extending the theory of algorithmic dimensions, along with the point-to-set principle, to arbitrary separable metric spaces.…”