Correspondence Principles for Effective Dimensions

Hitchcock, John M.

doi:10.1007/s00224-004-1122-1

Cited by 47 publications

(45 citation statements)

References 4 publications

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“…We show in Section 3.3.2 that Staiger's computable entropy rate coincides with computable dimension and also that a polynomial-space entropy rate characterizes polynomial-space dimension. The material on entropy rates is based on [18].…”

Section: Entropy Rates and Kolmogorov Complexitymentioning

confidence: 99%

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Effective fractal dimension: foundations and applications

Hitchcock

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Section: Entropy Rates and Kolmogorov Complexitymentioning

confidence: 99%

“…We show that these results are optimal in the arithmetical hierarchy. This section is based on [18].…”

Section: Correspondence Principlesmentioning

confidence: 99%

Effective fractal dimension: foundations and applications

Hitchcock

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“…This notion of dimension has been shown to be geometrically meaningful. For example, if X ⊆ R n is a "reasonably simple" set, in the sense that X is a union of Π 0 1 (i.e., computably closed) sets, then dim H (X) = sup x∈X dim(x), (1.1) which is a nonclassical, pointwise characterization of the classical Hausdorff dimensions of such sets [16,12]. The self-similar fractals form the best known and best understood class of fractals.…”

Section: Introductionmentioning

confidence: 99%

Dimension spectra of random subfractals of self-similar fractals

Lutz

Mayordomo

et al. 2014

Annals of Pure and Applied Logic

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The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a general-purpose computer with arbitrarily high precisions 2 −r . The dimension spectrum of a set X in Euclidean space is the subset of [0, n] consisting of the dimensions of all points in X.The dimensions of points have been shown to be geometrically meaningful (Lutz 2003, Hitchcock 2003, and the dimensions of points in self-similar fractals have been completely analyzed (Lutz and Mayordomo 2008). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given self-similar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite two-player game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selector-coder games are, from the coder's point of view, games against nature.We determine the dimension spectra of a wide class of such randomly selected subfractals. We show that each such fractal has a dimension spectrum that is a closed interval whose endpoints can be computed or approximated from the parameters of the fractal. In general, the maximum of the spectrum is determined by the degree to which the coder can reinforce the randomness in the selector, while the minimum is determined by the degree to which the coder can cancel randomness in the selector. This constructive and destructive interference between the players' randomnesses is somewhat subtle, even in the simplest cases. Our proof techniques include van Lambalgen's theorem on independent random sequences, measure preserving transformations, an application of network flow theory, a Kolmogorov complexity lower bound argument, and a nonconstructive proof that this bound is tight.

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“…Although it may at first seem counter-intuitive to assign dimensions, which may be positive, to individual points, there are now several indications that these dimensions are geometrically meaningful in Euclidean space. For example, results of Hitchcock [7] and Lutz [11] imply that, if X ⊆ R n is a union (not necessarily effective) of Π 0 1 (i.e., computably closed) sets, then dim H (X) = sup x∈X dim(x), (1.1) where dim H (X) is the classical Hausdorff dimension of X. We thus have a "pointwise" characterization of Hausdorff dimension, which is the most important dimension in fractal geometry, on unions of Π 0 1 sets.…”

Section: Introductionmentioning

confidence: 99%

Connectivity properties of dimension level sets

Lutz¹,

Weihrauch

2008

Mathematical Logic Qtrly

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This paper initiates the study of sets in Euclidean space R n (n ≥ 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, 1], we are interested in the connectivity properties of the set DIM I consisting of all points in R n whose (constructive Hausdorff) dimensions lie in the interval I. It is easy to see that the sets DIM [0,1) and DIM (n−1,n] are totally disconnected. In contrast, we show that the sets DIM [0,1] and DIM [n−1,n] are path-connected. Our proof of this fact uses geometric properties of Kolmogorov complexity in Euclidean space.

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Correspondence Principles for Effective Dimensions

Cited by 47 publications

References 4 publications

Effective fractal dimension: foundations and applications

Effective fractal dimension: foundations and applications

Dimension spectra of random subfractals of self-similar fractals

Connectivity properties of dimension level sets

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