Numerical Values of the Hausdorff and Packing Measures for Limit
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Real Analyticity of Hausdorff Dimension of Disconnected Julia
REAL ANALYTICITY OF HAUSDORFF DIMENSION OF DISCONNECTED JULIA SETS OF CUBIC PARABOLIC POLYNOMIALS Hasina Akter Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS August 2012 APPROVED: Mariusz Urbański, Major Professor Pieter Allaart, Committee Member Lior Fishman, Committee Member Su Gao, Chair of the Department of Mathematics Mark Wardell, Dean of the Toulouse Graduate School Akter, Hasina. Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials. Doctor of Philosophy (Mathematics), August 2012, 79 pp., 36 numbered references. 2 Consider a family of cubic parabolic polynomials given by fλ (z) = z(1− z − λz ) for non-zero complex parameters λ ∈ D0 such that for each λ ∈ D0 the polynomial fλ is a parabolic polynomial, that is, the polynomial fλ has a parabolic fixed point and the Julia set of fλ , denoted by J ( fλ ) , does not contain any critical points of fλ . We also assumed that for each λ ∈ D0 , one finite critical point of the polynomial fλ escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set D0 is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff 1 dimension function D → ( ,2) defined by λ HD(J ( fλ )) is real analytic. To prove this 0 2 we have constructed a holomorphic family of holomorphic parabolic graph directed { f } Markov systems whose limit sets coincide with the Julia sets of polynomials λ λ∈D0 up to a countable set, and hence have the same Hausdorff dimension. -
Arxiv:1707.03784V6 [Math.GN] 21 Jul 2020
Complete Quasi-Metrics for Hyperspaces, Continuous Valuations, and Previsions Jean Goubault-Larrecq Universit´eParis-Saclay, CNRS, ENS Paris-Saclay, Laboratoire M´ethodes Formelles, 91190, Gif-sur-Yvette, France July 29, 2021 Abstract The Kantorovich-Rubinshte˘ınmetric is an L1-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi- metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda- complete, and that the associated topology is the weak topology. We do this not only for probability distributions, represented as normalized continuous valuations, but also for subprobability distributions, for various hyperspaces, and in general for different brands of functionals. Those functionals model probabilistic choice, angelic and demonic non- deterministic choice, and their combinations. The mathematics needed for those results are more demanding than in the simpler case of metric spaces. To obtain our results, we prove a few other results that have independent interest, notably: continuous Yoneda-complete spaces are consonant; on a continuous Yoneda-complete space, the Scott topology arXiv:1707.03784v8 [math.GN] 28 Jul 2021 on the space of R+-valued lower semicontinuous maps coincides with the compact-open and Isbell topologies, and the subspace topology on spaces of α-Lipschitz continuous maps also coincides with the topology of pointwise convergence, and is stably compact; we introduce and study the so-called Lipschitz-regular quasi-metric spaces, and we show that the formal ball functor induces a Kock-Z¨oberlein monad, of which all algebras are Lipschitz-regular; and we prove a minimax theorem where one of the spaces is not compact Hausdorff, but merely compact. -
DISERTAˇCN´I PR´ACE Multivalued Fractals and Hyperfractals
UNIVERZITA PALACKEHO´ V OLOMOUCI PRˇ´IRODOVEDECKˇ A´ FAKULTA KATEDRA MATEMATICKE´ ANALYZY´ A APLIKAC´I MATEMATIKY DISERTACNˇ ´I PRACE´ Multivalued fractals and hyperfractals Vedouc´ıdiplomov´epr´ace: Vypracoval: prof. RNDr. dr hab. Jan Andres, DSc. Mgr. Miroslav Rypka Rok odevzd´an´ı: 2012 Matematick´aanal´yza, IV. roˇcn´ık Prohl´aˇsen´ı Prohlaˇsuji, ˇze jsem disertaˇcn´ı pr´aci zpracoval samostatnˇepod veden´ım prof. RNDr. dr hab. Jana Andrese, DSc. s pouˇzit´ım uveden´eliteratury. V Olomouci dne 20. srpna 2012 Podˇekov´an´ı M˚uj d´ık patˇr´ıpˇredevˇs´ım profesoru Andresovi za rady a trpˇelivost. Chci podˇekovat sv´ym rodiˇc˚um a pˇr´atel˚um za podporu nejen pˇri psan´ıdisertaˇcn´ıpr´ace a profesoru Bandtovi za vstˇr´ıcnost a laskavost bˇehem moj´ıst´aˇze v Greifswaldu. Contents 1 Introduction 5 1.1 Currentstateoftheart........................ 5 1.2 Aimsofthethesis........................... 7 1.3 Theoreticalframework . 9 1.4 Appliedmethods ........................... 9 1.5 Mainresults.............................. 9 2 Spacesandhyperspaces,mapsandhypermaps 11 3 Iteratedfunctionsystemsandinvariantmeasures 20 3.1 Iteratedfunctionsystems. 20 3.2 Measure ................................ 25 3.3 Dimensionandself-similarity. 31 3.4 LiftedIFSandsuperfractals . 34 4 Fixedpointtheoryinhyperspaces 38 5 Multivaluedfractalsandhyperfractals 42 5.1 Existenceresults ........................... 42 5.2 Address structure of multivalued fractals . .. 51 5.3 Ergodicapproach ........................... 57 5.4 Chaosgame .............................. 62 6 -
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales. -
Intermediate Dimensions
Mathematische Zeitschrift (2020) 296:813–830 https://doi.org/10.1007/s00209-019-02452-0 Mathematische Zeitschrift Intermediate dimensions Kenneth J. Falconer1 · Jonathan M. Fraser1 · Tom Kempton2 Received: 21 March 2019 / Accepted: 6 November 2019 / Published online: 26 December 2019 © The Author(s) 2019 Abstract We introduce a continuum of dimensions which are ‘intermediate’ between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that |U|≤|V |θ for all sets U, V used in a particular cover, where θ ∈[0, 1] is a parameter. Thus, when θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when θ = 0there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of θ), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman’s lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford–McMullen carpets. Keywords Hausdorff dimension · Box dimension · Self-affine carpet Mathematics Subject Classification Primary 28A80; Secondary 37C45 1 Intermediate dimensions: definitions and background We work with subsets of Rn throughout, although much of what we establish also holds in more general metric spaces. We denote thediameter of a set F by |F|, and when we refer to a { } ⊆ { } cover Ui of a set F we mean that F i Ui where Ui is a finite or countable collection of sets. -
SMALL FURSTENBERG SETS 3 One Wants to Measure How Fast the Quotient (Or Gap) H Between G, H H G ∈ Grows, Whenever G H Goes to Zero
SMALL FURSTENBERG SETS URSULA MOLTER AND EZEQUIEL RELA Abstract. For α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E ∩ ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of −γ 1 zero-dimensional Furstenberg type sets. Namely, for hγ (x) = log ( x ), ∈ γ > 0, we construct a set Eγ Fhγ of Hausdorff dimension not greater 1 1 than 2 . Since in a previous work we showed that 2 is a lower bound for ∈ the Hausdorff dimension of any E Fhγ , with the present construction, 1 the value 2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions hγ . 1. Introduction We study dimension properties of sets of Furstenberg type. In particular we are interested in being able to construct very small Furstenberg sets in a given class. We begin with the definition of classical Furstenberg sets. Definition 1.1. For α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension (dimH ) of the set E ℓe is greater than or equal to α. -
Finding Subsets of Positive Measure∗
Finding Subsets of Positive Measure∗ Bjørn Kjos-Hanssen† Jan Reimann‡ October 10, 2018 Abstract An important theorem of geometric measure theory (first proved by Besicov- itch and Davies for Euclidean space) says that every analytic set of non-zero s- dimensional Hausdorff measure Hs contains a closed subset of non-zero (and in- deed finite) Hs-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that 1 0 given a (lightface) S1 set of reals in Cantor space, there is always a P1(O) sub- set on non-zero Hs-measure definable from Kleene’s O. On the other hand, there 0 are P2 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure. 1 Introduction A most useful property of Lebesgue measure l is its (inner) regularity: For any mea- surable set E, we can find an Fs set F ⊆ E with l(E) = l(F). In other words, any measurable set can be represented as an Fs set plus a nullset. This means that, for mea- sure theoretic considerations, E can be replaced by an Fs , simplifying the complicated topological structure of arbitrary measurable sets. It is a basic result in geometric measure theory that regularity holds for s-dimensional Hausdorff measure Hs (s > 0), too, with one important restriction. Theorem 1 (Besicovitch and Moran [4]). If E is Hs-measurable and of finite Hs- s s measure, then there exists an Fs set F ⊆ E so that H (F) = H (E). -
Fractal Curvature Measures and Image Analysis
Diplomarbeit zur Erlangung des akademischen Grades Diplom-Mathematiker an der Fakult¨atf¨urMathematik und Wirtschaftswissenschaften der Universit¨atUlm Fractal Curvature Measures and Image Analysis vorgelegt von Peter Straka arXiv:1501.00530v1 [math.MG] 3 Jan 2015 Gutachter: Prof. Dr. E. Spodarev , Prof. Dr. W. Arendt Universit¨atUlm, Abteilung Stochastik Oktober 2007 Abstract Since the recent dissertation by Steffen Winter, for certain self-similar sets F the growth behaviour of the Minkowski functionals of the parallel sets F" := fx 2 Rd : d(x; F ) ≤ "g as " # 0 is known, leading to the notion of fractal curvatures f Ck (F ), k 2 f0; : : : ; dg. The dependence of the growth behaviour on the fractal f dimension s = dim F is exploited, and estimators for s and Ck (F ) are derived. The performance of these estimators is tested on binary images of self-similar sets. ii Danksagung Meinen tiefsten Dank m¨ochte ich meinen Eltern aussprechen, die mich w¨ahrend meiner langen Studienzeit immer bedingungslos un- terstutzt¨ haben. Ohne sie w¨are diese Arbeit nicht zustandegekom- men. Prof. Dr. Evgeny Spodarev m¨ochte ich fur¨ dieses sch¨one Thema danken, das es auch einem Neuling in der fraktalen Geometrie erm¨oglicht hat innovative Arbeit zu leisten. Auch danke ich ihm fur¨ seine stete Hilfsbereitschaft w¨ahrend meiner gesamten Zeit an der Universit¨at Ulm. Obwohl Dr. Steffen Winter keine offizielle Betreuerfunktion inne hatte, war er uber¨ Email und Telefon eine allgegenw¨artige und prompte Hilfe bei meinen zahlreichen geometrischen Fragen. Ihm danke ich auch insbesondere fur¨ die gewissenhafte Durchsicht die- ser Arbeit. Danke an Daniel Meschenmoser und Johannes Mayer fur¨ ihre Hil- fe bei Programmierfragen und allgemeinen Fragen zum GeoStoch- Projekt. -
DIMENSION FUNCTIONS of CANTOR SETS 1. Introduction A
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 10, October 2007, Pages 3151–3161 S 0002-9939(07)09019-3 Article electronically published on June 21, 2007 DIMENSION FUNCTIONS OF CANTOR SETS IGNACIO GARCIA, URSULA MOLTER, AND ROBERTO SCOTTO (Communicated by Michael T. Lacey) Abstract. We estimate the packing measure of Cantor sets associated to non- increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets. 1. Introduction A Cantor set is a compact perfect and totally disconnected subset of the real line. In this article we consider Cantor sets of Lebesgue measure zero. Different kinds of these sets appear in many areas of mathematics, such as number theory and dynamical systems. They are also interesting in themselves as theoretical examples and counterexamples. A classical way to understand them quantitatively is through the Hausdorff measure and dimension. A function h :(0,λh] → (0, ∞], where λh > 0, is said to be a dimension function if it is continuous, nondecreasing and h(x) → 0asx → 0. We denote by D the set of dimension functions. Given E ⊂ R and h ∈ D,weseth(E)=h(|E|), where |E| is the diameter of the set E. Recall that a δ-covering of a given set E is a countable family of subsets of R covering E whose diameters are less than δ.Theh-Hausdorff measure of E is defined as ∞ h (1.1) H (E) = lim h(Ui):{Ui} is a δ-covering of E . -
Refined Size Estimates for Furstenberg Sets Via Hausdorff Measures: a Survey of Some Recent Results
REFINED SIZE ESTIMATES FOR FURSTENBERG SETS VIA HAUSDORFF MEASURES: A SURVEY OF SOME RECENT RESULTS EZEQUIEL RELA Abstract. In this survey we collect and discuss some recent results on the so called \Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension (dimH ) of the sets in the Fα-class: for a 2 given α 2 (0; 1], a set E ⊆ R is in the Fα-class if for each e 2 S there exists a unit line segment `e in the direction of e such that dimH (` \ E) ≥ α. For α = 1, this problem is essentially equivalent to the \Kakeya needle problem". Define γ(α) = inf fdimH (E): E 2 Fαg. The best known results on γ(α) are the following inequalities: max f1=2 + α; 2αg ≤ γ(α) ≤ (1 + 3α)=2: In this work we approach this problem from a more general point of view, in terms of a generalized Hausdorff measure Hh associated with the dimen- sion function h. We define the class Fh of Furstenberg sets associated to a given dimension function h. The natural requirement for a set E to belong h to Fh, is that H (`e \ E) > 0 for each direction. We generalize the known results in terms of \logarithmic gaps" and obtain analogues to the estimates given above. Moreover, these analogues allow us to extend our results to the endpoint α = 0. For the upper bounds we exhibit an explicit construction of Fh-sets which are small enough. To that end we adapt and prove some re- sults on Diophantine Approximation about the the dimension of a set of \well approximable numbers". -
The Dimensionality of Discourse
The dimensionality of discourse Isidoros Doxasa,1, Simon Dennisb,2, and William L. Oliverc aCenter for Integrated Plasma Studies, University of Colorado, Boulder, CO 80309; bDepartment of Psychology, Ohio State University, Columbus, OH 43210; and cInstitute of Cognitive Science, University of Colorado, Boulder, CO 80309 Edited* by Richard M. Shiffrin, Indiana University, Bloomington, IN, and approved February 2, 2010 (received for review July 26, 2009) The paragraph spaces of five text corpora, of different genres and accurately estimate passage coherence. In addition, LSA has found intended audiences, in four different languages, all show the same application in many areas, including selecting educational materials two-scale structure, with the dimension at short distances being for individual students, guiding on-line discussion groups, providing lower than at long distances. In all five cases the short-distance feedback to pilots on landing technique, diagnosing mental disor- dimension is approximately eight. Control simulations with ran- ders from prose, matching jobs with candidates, and facilitating domly permuted word instances do not exhibit a low dimensional automated tutors (3). structure. The observed topology places important constraints on By far the most surprising application of LSA is its ability to grade the way in which authors construct prose, which may be universal. student essay scripts. Foltz et al. (9) summarize the remarkable reliability with which it is able to do this, especially when compared correlation dimension | language | latent semantic analysis against the benchmark of expert human graders. In a set of 188 essays written on the functioning of the human heart, the average s we transition from paragraph to paragraph in written dis- correlation between two graders was 0.83, whereas the correlation course, one can think of the path through which one passes as a A of LSA’s scores with the graders was 0.80. -
On the Computation of the Kantorovich Distance for Images
DN THE COMPUTATION OF THE KANTOROV ICH DISTANCE FOR IMAGES Th omas Ka·i j ser June 1996 Report nr LiTH-ISY-R-1858 ,.. f ON THE COMPUTATION OF THE KANTOROVICH DISTANCE FOR IMAGES Thomas Kaijser June 1996 Report nr LiTH - ISY -R-1858 ,. l .. l Introduction. The Kantorovich distance for images can be defined in two ways. Either as the infimum of all "costs" for "transporting" one image into the other, where the cost-function is induced by the distance-function ehosen for measuring distances beetween pixels, - or as a supremum of all integrals for which the integration is with respect to the measure obtained by taking the difference of the two images considered as mass distributions, and for which the integrand is any function whose "slope" is bounded. (For the precise definitions see the next sections). We shall call the first of these two definitions the primal version and the seeond definition the dual version. It was Kantarovich [10], (see also [11]), who in the beginning of the 40:ies introduced this "transportation-metric" for prohability measures and proved the equality between the two definitions. This result is a special case of what in optimization theory is called the duality theorem. The first to use the primal version of the Kantarovich distance as a distance-function between 2-dimensional grey valued images with equal to tal grey value was probably Werman et al [19]. (See also Werman [20]). The conclusions in their paper was that the metric is applicable in many domains such as co-occurance matrices, shape matching, and picture half toning.