NDLI: Dimensions of Points in Self-Similar Fractals

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lutz, Jack H. Mayordomo, Elvira
Copyright Year	2008
Abstract	Self-similar fractals arise as the unique attractors of iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying an open set condition. Each point x in such a fractal F arising from an IFS S is naturally regarded as the outcome of an infinite coding sequence T (which need not be unique) over the alphabet $\Sigma_k = \{0, \ldots, k-1\}$, where k is the number of contracting similarities in S. A classical theorem of Moran (1946) and Falconer (1989) states that the Hausdorff and packing dimensions of a self-similar fractal coincide with its similarity dimension, which depends only on the contraction ratios of the similarities. The theory of computing has recently been used to provide a meaningful notion of the dimensions of individual points in Euclidean space. In this paper, we use (and extend) this theory to analyze the dimensions of individual points in fractals that are computably self-similar, meaning that they are unique attractors of IFSs that are computable and satisfy the open set condition. Our main theorem states that, if $F \subseteq \mathbb{R}^n$ is any computably self-similar fractal and S is any IFS testifying to this fact, then the dimension identities $\operatorname{dim}(x) = \operatorname{sdim}(F) \operatorname{dim}^{\pi_S}(T)$ and $\operatorname{Dim}(x) = \operatorname{sdim}(F) \operatorname{Dim}^{\pi_S}(T)$ hold for all $x \in F$ and all coding sequences T for x. In these equations, $\operatorname{sdim}(F)$ denotes the similarity dimension of the fractal F; $\operatorname{dim}(x)$ and $\operatorname{Dim}(x)$ denote the dimension and strong dimension, respectively, of the point x in Euclidean space; and $\operatorname{dim}^{\pi_S}(T)$ and $\operatorname{Dim}^{\pi_S}(T)$ denote the dimension and strong dimension, respectively, of the coding sequence T relative to a probability measure $\pi_S$ that the IFS S induces on the alphabet $\Sigma_k$. The above-mentioned theorem of Moran and Falconer follows easily from our main theorem by relativization. Along the way to our main theorem, we develop the elements of the theory of constructive dimensions relative to general probability measures. The proof of our main theorem uses Kolmogorov complexity characterizations of these dimensions.
Starting Page	1080
Ending Page	1112
Page Count	33
File Format	PDF
ISSN	00975397
DOI	10.1137/070684689
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-07-02
Access Restriction	Subscribed
Subject Keyword	fractal geometry iterated function system constructive dimension geometric measure theory computability self-similar fractal Metric theory of other algorithms and expansions; measure and Hausdorff dimension packing dimension Complexity classes Billingsley dimension Hausdorff dimension Hausdorff and packing measures Algorithmic information theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Effective Strong Dimension in Algorithmic Information and Computational Complexity

Revisiting multifractal analysis

Cantor series expansions and packing dimension faithfulness

Exact Hausdorff Measure of Certain Non-Self-Similar Cantor Sets

Complexity and fractal dimensions for infinite sequences with positive entropy

Boundaries of Disk-like Self-affine Tiles

Packing Dimension of Measures Associated with $${\widetilde{Q}}$$ -Representation

Curvature-direction measures of self-similar sets

Curvature Bounds for Neighborhoods of Self-Similar Sets

Dimensions of Points in Self-Similar Fractals

Similar Documents

Effective Strong Dimension in Algorithmic Information and Computational Complexity

Revisiting multifractal analysis

Cantor series expansions and packing dimension faithfulness

Exact Hausdorff Measure of Certain Non-Self-Similar Cantor Sets

Complexity and fractal dimensions for infinite sequences with positive entropy

Boundaries of Disk-like Self-affine Tiles

Packing Dimension of Measures Associated with $${\widetilde{Q}}$$ -Representation

Curvature-direction measures of self-similar sets

Curvature Bounds for Neighborhoods of Self-Similar Sets

Dimensions of Points in Self-Similar Fractals