Fractal Geometry
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SUBFRACTALS INDUCED by SUBSHIFTS a Dissertation
SUBFRACTALS INDUCED BY SUBSHIFTS A Dissertation Submitted to the Graduate Faculty of the North Dakota State University of Agriculture and Applied Science By Elizabeth Sattler In Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Department: Mathematics May 2016 Fargo, North Dakota NORTH DAKOTA STATE UNIVERSITY Graduate School Title SUBFRACTALS INDUCED BY SUBSHIFTS By Elizabeth Sattler The supervisory committee certifies that this dissertation complies with North Dakota State Uni- versity's regulations and meets the accepted standards for the degree of DOCTOR OF PHILOSOPHY SUPERVISORY COMMITTEE: Dr. Do˘ganC¸¨omez Chair Dr. Azer Akhmedov Dr. Mariangel Alfonseca Dr. Simone Ludwig Approved: 05/24/2016 Dr. Benton Duncan Date Department Chair ABSTRACT In this thesis, a subfractal is the subset of points in the attractor of an iterated function system in which every point in the subfractal is associated with an allowable word from a subshift on the underlying symbolic space. In the case in which (1) the subshift is a subshift of finite type with an irreducible adjacency matrix, (2) the iterated function system satisfies the open set condition, and (3) contractive bounds exist for each map in the iterated function system, we find bounds for both the Hausdorff and box dimensions of the subfractal, where the bounds depend both on the adjacency matrix and the contractive bounds on the maps. We extend this result to sofic subshifts, a more general subshift than a subshift of finite type, and to allow the adjacency matrix n to be reducible. The structure of a subfractal naturally defines a measure on R . -
March 12, 2011 DIAGONALLY NON-RECURSIVE FUNCTIONS and EFFECTIVE HAUSDORFF DIMENSION 1. Introduction Reimann and Terwijn Asked Th
March 12, 2011 DIAGONALLY NON-RECURSIVE FUNCTIONS AND EFFECTIVE HAUSDORFF DIMENSION NOAM GREENBERG AND JOSEPH S. MILLER Abstract. We prove that every sufficiently slow growing DNR function com- putes a real with effective Hausdorff dimension one. We then show that for any recursive unbounded and nondecreasing function j, there is a DNR function bounded by j that does not compute a Martin-L¨ofrandom real. Hence there is a real of effective Hausdorff dimension 1 that does not compute a Martin-L¨of random real. This answers a question of Reimann and Terwijn. 1. Introduction Reimann and Terwijn asked the dimension extraction problem: can one effec- tively increase the information density of a sequence with positive information den- sity? For a formal definition of information density, they used the notion of effective Hausdorff dimension. This effective version of the classical Hausdorff dimension of geometric measure theory was first defined by Lutz [10], using a martingale defini- tion of Hausdorff dimension. Unlike classical dimension, it is possible for singletons to have positive dimension, and so Lutz defined the dimension dim(A) of a binary sequence A 2 2! to be the effective dimension of the singleton fAg. Later, Mayor- domo [12] (but implicit in Ryabko [15]), gave a characterisation using Kolmogorov complexity: for all A 2 2!, K(A n) C(A n) dim(A) = lim inf = lim inf ; n!1 n n!1 n where C is plain Kolmogorov complexity and K is the prefix-free version.1 Given this formal notion, the dimension extraction problem is the following: if 2 dim(A) > 0, is there necessarily a B 6T A such that dim(B) > dim(A)? The problem was recently solved by the second author [13], who showed that there is ! an A 2 2 such that dim(A) = 1=2 and if B 6T A, then dim(B) 6 1=2. -
A Note on Pointwise Dimensions
A Note on Pointwise Dimensions Neil Lutz∗ Department of Computer Science, Rutgers University Piscataway, NJ 08854, USA [email protected] April 6, 2017 Abstract This short note describes a connection between algorithmic dimensions of individual points and classical pointwise dimensions of measures. 1 Introduction Effective dimensions are pointwise notions of dimension that quantify the den- sity of algorithmic information in individual points in continuous domains. This note aims to clarify their relationship to the classical notion of pointwise dimen- sions of measures, which is central to the study of fractals and dynamics. This connection is then used to compare algorithmic and classical characterizations of Hausdorff and packing dimensions. See [15, 17] for surveys of the strong ties between algorithmic information and fractal dimensions. 2 Pointwise Notions of Dimension arXiv:1612.05849v2 [cs.CC] 5 Apr 2017 We begin by defining the two most well-studied formulations of algorithmic dimension [5], the effective Hausdorff and packing dimensions. Given a point x ∈ Rn, a precision parameter r ∈ N, and an oracle set A ⊆ N, let A A n Kr (x) = min{K (q): q ∈ Q ∩ B2−r (x)} , where KA(q) is the prefix-free Kolmogorov complexity of q relative to the oracle −r A, as defined in [10], and B2−r (x) is the closed ball of radius 2 around x. ∗Research supported in part by National Science Foundation Grant 1445755. 1 Definition. The effective Hausdorff dimension and effective packing dimension of x ∈ Rn relative to A are KA(x) dimA(x) = lim inf r r→∞ r KA(x) DimA(x) = lim sup r , r→∞ r respectively [11, 14, 1]. -
Assouad Type Dimensions and Homogeneity of Fractals
Assouad type dimensions and homogeneity of fractals Jonathan M. Fraser The University of St Andrews Jonathan M. Fraser Assouad type dimensions In particular, it usually studies how much space the set takes up on small scales. Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. Jonathan M. Fraser Assouad type dimensions Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. In particular, it usually studies how much space the set takes up on small scales. Jonathan M. Fraser Assouad type dimensions Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. -
Conformal Assouad Dimension and Modulus S
GAFA, Geom. funct. anal. Vol. 14 (2004) 1278 – 1321 c Birkh¨auser Verlag, Basel 2004 1016-443X/04/061278-44 DOI 10.1007/s00039-004-0492-5 GAFA Geometric And Functional Analysis CONFORMAL ASSOUAD DIMENSION AND MODULUS S. Keith and T. Laakso Abstract. Let α ≥ 1andlet(X, d, µ)beanα-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, µ) with uniformly big 1-modulus. 1 Introduction The existence of families of curves with non-vanishing modulus in a metric measure space is a strong and useful property, perhaps most appreciated in the study of abstract quasiconformal and quasisymmetric maps, Sobolev spaces, Poincar´e inequalities, and geometric rigidity questions. See for ex- ample [BoK3], [BouP2], [Ch], [HK], [HKST], [K2,3], [KZ], [P1], [Sh], [T2] and the many references contained therein. In this paper we give an exis- tence theorem for families of curves with non-vanishing modulus in weak tangents of certain metric measure spaces, and show that this existence is fundamentally related to the conformal Assouad dimension of the underly- ing metric space. Theorem 1.0.1. Let α ≥ 1 and let (X, d, µ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, µ) with uniformly big 1-modulus. The terminology of Theorem 1.0.1 will be explained in section 2, and several applications will be given momentarily. For the moment we note that the notion of uniformly big 1-modulus is new, and describes those metric measure spaces that are rich in curves at every location and scale, as measured in a scale-invariant way by modulus. -
Arxiv:1511.03461V2 [Math.MG] 19 May 2017 Osadfrhrasmtoso Hs Asaesiuae.I on If Stipulated
ON THE DIMENSIONS OF ATTRACTORS OF RANDOM SELF-SIMILAR GRAPH DIRECTED ITERATED FUNCTION SYSTEMS SASCHA TROSCHEIT Abstract. In this paper we propose a new model of random graph directed fractals that encompasses the current well-known model of random graph di- rected iterated function systems, V -variable attractors, and fractal and Man- delbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of 1- variable and ∞-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespec- tive of overlap. Under the additional assumption of the uniform strong separa- tion condition we give an expression for the almost sure Hausdorff and Assouad dimension. 1. Introduction The study of deterministic and random fractal geometry has seen a lot of interest over the past 30 years. While we assume the reader is familiar with standard works on the subject (e.g. [7], [12], [13]) we repeat some of the material here for completeness, enabling us to set the scene for how our model fits in with and also differs from previously considered models. In the study of strange attractors in dynamical systems and in fractal geometry, one of the most commonly encountered families of attractors is the invariant set under a finite family of contractions. This is the family of Iterated Function System (IFS) attractors. An IFS is a set of mappings I = {fi}i∈I , with associated attractor F that satisfies (1.1) F = fi(F ). i[∈I d d If I is a finite index set and each fi : R → R is a contraction, then there exists a unique compact and non-empty set F in the family of compact subsets K(Rd) that satisfies this invariance (see Hutchinson [21]). -
Hölder Parameterization of Iterated Function Systems and a Self-A Ne
Anal. Geom. Metr. Spaces 2021; 9:90–119 Research Article Open Access Matthew Badger* and Vyron Vellis Hölder Parameterization of Iterated Function Systems and a Self-Ane Phenomenon https://doi.org/10.1515/agms-2020-0125 Received November 1, 2020; accepted May 25, 2021 Abstract: We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a com- plete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdor measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-ane Bedford-McMullen carpets and build parameterizations of self-ane sponges. An inter- esting phenomenon emerges in the self-ane setting. While the optimal parameter s for a self-similar curve in Rn is always at most the ambient dimension n, the optimal parameter s for a self-ane curve in Rn may be strictly greater than n. -
Divergence Points of Self-Similar Measures and Packing Dimension
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Mathematics 214 (2007) 267–287 www.elsevier.com/locate/aim Divergence points of self-similar measures and packing dimension I.S. Baek a,1,L.Olsenb,∗, N. Snigireva b a Department of Mathematics, Pusan University of Foreign Studies, Pusan 608-738, Republic of Korea b Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, UK Received 14 December 2005; accepted 7 February 2007 Available online 1 March 2007 Communicated by R. Daniel Mauldin Abstract Rd ∈ Rd log μB(x,r) Let μ be a self-similar measure in . A point x for which the limit limr0 log r does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions. © 2007 Elsevier Inc. All rights reserved. MSC: 28A80 Keywords: Fractals; Multifractals; Hausdorff measure; Packing measure; Divergence points; Local dimension; Normal numbers 1. -
Fractal-Bits
fractal-bits Claude Heiland-Allen 2012{2019 Contents 1 buddhabrot/bb.c . 3 2 buddhabrot/bbcolourizelayers.c . 10 3 buddhabrot/bbrender.c . 18 4 buddhabrot/bbrenderlayers.c . 26 5 buddhabrot/bound-2.c . 33 6 buddhabrot/bound-2.gnuplot . 34 7 buddhabrot/bound.c . 35 8 buddhabrot/bs.c . 36 9 buddhabrot/bscolourizelayers.c . 37 10 buddhabrot/bsrenderlayers.c . 45 11 buddhabrot/cusp.c . 50 12 buddhabrot/expectmu.c . 51 13 buddhabrot/histogram.c . 57 14 buddhabrot/Makefile . 58 15 buddhabrot/spectrum.ppm . 59 16 buddhabrot/tip.c . 59 17 burning-ship-series-approximation/BossaNova2.cxx . 60 18 burning-ship-series-approximation/BossaNova.hs . 81 19 burning-ship-series-approximation/.gitignore . 90 20 burning-ship-series-approximation/Makefile . 90 21 .gitignore . 90 22 julia/.gitignore . 91 23 julia-iim/julia-de.c . 91 24 julia-iim/julia-iim.c . 94 25 julia-iim/julia-iim-rainbow.c . 94 26 julia-iim/julia-lsm.c . 98 27 julia-iim/Makefile . 100 28 julia/j-render.c . 100 29 mandelbrot-delta-cl/cl-extra.cc . 110 30 mandelbrot-delta-cl/delta.cl . 111 31 mandelbrot-delta-cl/Makefile . 116 32 mandelbrot-delta-cl/mandelbrot-delta-cl.cc . 117 33 mandelbrot-delta-cl/mandelbrot-delta-cl-exp.cc . 134 34 mandelbrot-delta-cl/README . 139 35 mandelbrot-delta-cl/render-library.sh . 142 36 mandelbrot-delta-cl/sft-library.txt . 142 37 mandelbrot-laurent/DM.gnuplot . 146 38 mandelbrot-laurent/Laurent.hs . 146 39 mandelbrot-laurent/Main.hs . 147 40 mandelbrot-series-approximation/args.h . 148 41 mandelbrot-series-approximation/emscripten.cpp . 150 42 mandelbrot-series-approximation/index.html . -
Assouad Dimension and the Open Set Condition
Abstract: The Assouad dimension is a measure of the complexity of a fractal set similar to the box counting dimension, but with an additional scaling requirement. We generalize Moran's open set condition and introduce a notion called grid like which allows us to compute upper bounds and exact values for the Assouad dimension of certain fractal sets that arise as the attractors of self-similar iterated function systems. Then for an arbitrary fractal set A, we explore the question of whether the Assouad dimension of the set of differences A − A obeys any bound related to the Assouad dimension of A. This question is of interest, as infinite dimensional dynamical systems with attractors possessing sets of differences of finite Assouad dimension allow embeddings into finite dimensional spaces without losing the original dynamics. We find that even in very simple, natural examples, such a bound does not generally hold. This result demonstrates how a natural phenomenon with a simple underlying structure can be difficult to measure. Assouad Dimension and the Open Set Condition Alexander M. Henderson Department of Mathematics and Statistics University of Nevada, Reno 19 April 2013 Selected References I Jouni Luukkainen. Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures. Journal of the Korean Mathematical Society, 35(1):23{76, 1998. I Eric J. Olson and James C. Robinson. Almost bi-Lipschitz embeddings and almost homogeneous sets. Transactions of the American Mathematical Society, 362:145{168, 2010. I K. J. Falconer. The Geometry of Fractal Sets. Cambridge University Press, New York, 1985. I John M. Mackay. Assouad dimension of self-affine carpets. -
Dimension Theory of Random Self-Similar and Self-Affine Constructions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository DIMENSION THEORY OF RANDOM SELF-SIMILAR AND SELF-AFFINE CONSTRUCTIONS Sascha Troscheit A Thesis Submitted for the Degree of PhD at the University of St Andrews 2017 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/11033 This item is protected by original copyright This item is licensed under a Creative Commons Licence Dimension Theory of Random Self-similar and Self-affine Constructions Sascha Troscheit This thesis is submitted in partial fulfilment for the degree of Doctor of Philosophy at the University of St Andrews April 21, 2017 to the North Sea Acknowledgements First and foremost I thank my supervisors Kenneth Falconer and Mike Todd for their countless hours of support: mathematical, academical, and otherwise; for reading through many drafts and providing valuable feedback; for the regular Analysis group outings that were more often than not the highlight of my week. I thank my father, Siegfried Troscheit; his wife, Christiane Miethe; and my mother, Krystyna Troscheit, for their support throughout all the years of university that culminated in this thesis. I would not be where I am today if it was not for my teachers Richard St¨ove and Klaus Bovermann, who supported my early adventures into physics and mathematics, and enabled me to attend university lectures while at school. I thank my friends, both in St Andrews and the rest of the world. -
Sustainability As "Psyclically" Defined -- /
Alternative view of segmented documents via Kairos 22nd June 2007 | Draft Emergence of Cyclical Psycho-social Identity Sustainability as "psyclically" defined -- / -- Introduction Identity as expression of interlocking cycles Viability and sustainability: recycling Transforming "patterns of consumption" From "static" to "dynamic" to "cyclic" Emergence of new forms of identity and organization Embodiment of rhythm Generic understanding of "union of international associations" Three-dimensional "cycles"? Interlocking cycles as the key to identity Identity as a strange attractor Periodic table of cycles -- and of psyclic identity? Complementarity of four strategic initiatives Development of psyclic awareness Space-centric vs Time-centric: an unfruitful confrontation? Metaphorical vehicles: temples, cherubim and the Mandelbrot set Kairos -- the opportune moment for self-referential re-identification Governance as the management of strategic cycles Possible pointers to further reflection on psyclicity References Introduction The identity of individuals and collectivities (groups, organizations, etc) is typically associated with an entity bounded in physical space or virtual space. The boundary may be defined geographically (even with "virtual real estate") or by convention -- notably when a process of recognition is involved, as with a legal entity. Geopolitical boundaries may, for example, define nation states. The focus here is on the extent to which many such entities are also to some degree, if not in large measure, defined by cycles in time. For example many organizations are defined by the periodicity of the statutory meetings by which they are governed, or by their budget or production cycles. Communities may be uniquely defined by conference cycles, religious cycles or festival cycles (eg Oberammergau). Biologically at least, the health and viability of individuals is defined by a multiplicity of cycles of which respiration is the most obvious -- death may indeed be defined by absence of any respiratory cycle.