DIMENSION THEORY by DANIELLE WALSH a Thesis Submitted to The
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Hausdorff Dimension of Singular Vectors
HAUSDORFF DIMENSION OF SINGULAR VECTORS YITWAH CHEUNG AND NICOLAS CHEVALLIER Abstract. We prove that the set of singular vectors in Rd; d ≥ 2; has Hausdorff dimension d2 d d+1 and that the Hausdorff dimension of the set of "-Dirichlet improvable vectors in R d2 d is roughly d+1 plus a power of " between 2 and d. As a corollary, the set of divergent t t −dt trajectories of the flow by diag(e ; : : : ; e ; e ) acting on SLd+1 R= SLd+1 Z has Hausdorff d codimension d+1 . These results extend the work of the first author in [6]. 1. Introduction Singular vectors were introduced by Khintchine in the twenties (see [14]). Recall that d x = (x1; :::; xd) 2 R is singular if for every " > 0, there exists T0 such that for all T > T0 the system of inequalities " (1.1) max jqxi − pij < and 0 < q < T 1≤i≤d T 1=d admits an integer solution (p; q) 2 Zd × Z. In dimension one, only the rational numbers are singular. The existence of singular vectors that do not lie in a rational subspace was proved by Khintchine for all dimensions ≥ 2. Singular vectors exhibit phenomena that cannot occur in dimension one. For instance, when x 2 Rd is singular, the sequence 0; x; 2x; :::; nx; ::: fills the torus Td = Rd=Zd in such a way that there exists a point y whose distance in the torus to the set f0; x; :::; nxg, times n1=d, goes to infinity when n goes to infinity (see [4], chapter V). -
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. -
23. Dimension Dimension Is Intuitively Obvious but Surprisingly Hard to Define Rigorously and to Work With
58 RICHARD BORCHERDS 23. Dimension Dimension is intuitively obvious but surprisingly hard to define rigorously and to work with. There are several different concepts of dimension • It was at first assumed that the dimension was the number or parameters something depended on. This fell apart when Cantor showed that there is a bijective map from R ! R2. The Peano curve is a continuous surjective map from R ! R2. • The Lebesgue covering dimension: a space has Lebesgue covering dimension at most n if every open cover has a refinement such that each point is in at most n + 1 sets. This does not work well for the spectrums of rings. Example: dimension 2 (DIAGRAM) no point in more than 3 sets. Not trivial to prove that n-dim space has dimension n. No good for commutative algebra as A1 has infinite Lebesgue covering dimension, as any finite number of non-empty open sets intersect. • The "classical" definition. Definition 23.1. (Brouwer, Menger, Urysohn) A topological space has dimension ≤ n (n ≥ −1) if all points have arbitrarily small neighborhoods with boundary of dimension < n. The empty set is the only space of dimension −1. This definition is mostly used for separable metric spaces. Rather amazingly it also works for the spectra of Noetherian rings, which are about as far as one can get from separable metric spaces. • Definition 23.2. The Krull dimension of a topological space is the supre- mum of the numbers n for which there is a chain Z0 ⊂ Z1 ⊂ ::: ⊂ Zn of n + 1 irreducible subsets. DIAGRAM pt ⊂ curve ⊂ A2 For Noetherian topological spaces the Krull dimension is the same as the Menger definition, but for non-Noetherian spaces it behaves badly. -
Dimension and Dimensional Reduction in Quantum Gravity
May 2017 Dimension and Dimensional Reduction in Quantum Gravity S. Carlip∗ Department of Physics University of California Davis, CA 95616 USA Abstract , A number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of “dimension” and concluding with some speculative ideas of what dimensional reduction might mean for physics. arXiv:1705.05417v2 [gr-qc] 29 May 2017 ∗email: [email protected] 1 Why Dimensional Reduction? What is the dimension of spacetime? For most of physics, the answer is straightforward and uncontroversial: we know from everyday experience that we live in a universe with three dimensions of space and one of time. For a condensed matter physicist, say, or an astronomer, this is simply a given. There are a few exceptions—surface states in condensed matter that act two-dimensional, string theory in ten dimensions—but for the most part dimension is simply a fixed, and known, external parameter. Over the past few years, though, hints have emerged from quantum gravity suggesting that the dimension of spacetime is dynamical and scale-dependent, and shrinks to d 2 at very small ∼ distances or high energies. The purpose of this review is to summarize this evidence and to discuss some possible implications for physics. 1.1 Dimensional reduction and quantum gravity As early as 1916, Einstein pointed out that it would probably be necessary to combine the newly formulated general theory of relativity with the emerging ideas of quantum mechanics [1]. -
The Small Inductive Dimension Can Be Raised by the Adjunction of a Single Point
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector MATHEMATICS THE SMALL INDUCTIVE DIMENSION CAN BE RAISED BY THE ADJUNCTION OF A SINGLE POINT BY ERIC K. VAN DOUWEN (Communicated by Prof. R. TIBXMAN at the meeting of June 30, 1973) Unless otherwise indicated, all spaces are metrizable. 1. INTRODUCTION In this note we construct a metrizable space P containing a point p such that ind P\@} = 0 and yet ind P = 1, thus showing that the small inductive dimension of a metrizable space can be raised by the adjunction of a single point. (Without the metrizsbility condition such on example has already been constructed by Dowker [3, p. 258, exercise Cl). So the behavior of the small inductive dimension is in contrast with that of the dimension functions Ind and dim, which, as is well known, cannot be raised by the edjunction of a single point [3, p. 274, exercises B and C]. The construction of our example stems from the proof of the equality of ind and Ind on the class of separable metrizable spaces, as given in [lo, p. 1781. We make use of the fact that there exists a metrizable space D with ind D = 0 and Ind D = 1 [ 141. This space is completely metrizable. Hence our main example, and the two related examples, also are com- pletely metrizable. Following [7, p, 208, example 3.3.31 the example is used to show that there is no finite closed sum theorem for the small inductive dimension, even on the class of metrizable spaces. -
Metrizable Spaces Where the Inductive Dimensions Disagree
TRANSACTIONS OF THE AMERICAN MATHEMATICALSOCIETY Volume 318, Number 2, April 1990 METRIZABLE SPACES WHERE THE INDUCTIVE DIMENSIONS DISAGREE JOHN KULESZA Abstract. A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy's technique, these spaces can often be shown to have positive large inductive dimension. Examples of N-compact, complete metrizable spaces with ind = 0 and Ind = 1 are provided, answering questions of Mrowka and Roy. An example with weight c and positive Ind such that subspaces with smaller weight have Ind = 0 is produced in ZFC. Assuming an additional axiom, for each cardinal X a space of positive Ind with all subspaces with weight less than A strongly zero-dimensional is constructed. I. Introduction There are three classical notions of dimension for a normal topological space X. The small inductive dimension, denoted ind(^T), is defined by ind(X) = -1 if X is empty, ind(X) < n if each p G X has a neighborhood base of open sets whose boundaries have ind < n - 1, and ind(X) = n if ind(X) < n and n is as small as possible. The large inductive dimension, denoted Ind(Jf), is defined by Ind(X) = -1 if X is empty, \t\c\(X) < n if each closed C c X has a neighborhood base of open sets whose boundaries have Ind < n — 1, and Ind(X) = n if Ind(X) < n and n is as small as possible. The covering dimension, denoted by dim(X), is defined by dim(A') = -1 if X is empty, dim(A') < n if for each finite open cover of X there is a finite open refining cover such that no point is in more than n + 1 of the sets in the refining cover and, dim(A") = n if dim(X) < n and n is as small as possible. -
Quasiorthogonal Dimension
Quasiorthogonal Dimension Paul C. Kainen1 and VˇeraK˚urkov´a2 1 Department of Mathematics and Statistics, Georgetown University Washington, DC, USA 20057 [email protected] 2 Institute of Computer Science, Czech Academy of Sciences Pod Vod´arenskou vˇeˇz´ı 2, 18207 Prague, Czech Republic [email protected] Abstract. An interval approach to the concept of dimension is pre- sented. The concept of quasiorthogonal dimension is obtained by relax- ing exact orthogonality so that angular distances between unit vectors are constrained to a fixed closed symmetric interval about π=2. An ex- ponential number of such quasiorthogonal vectors exist as the Euclidean dimension increases. Lower bounds on quasiorthogonal dimension are proven using geometry of high-dimensional spaces and a separate argu- ment is given utilizing graph theory. Related notions are reviewed. 1 Introduction The intuitive concept of dimension has many mathematical formalizations. One version, based on geometry, uses \right angles" (in Greek, \ortho gonia"), known since Pythagoras. The minimal number of orthogonal vectors needed to specify an object in a Euclidean space defines its orthogonal dimension. Other formalizations of dimension are based on different aspects of space. For example, a topological definition (inductive dimension) emphasizing the recur- sive character of d-dimensional objects having d−1-dimensional boundaries, was proposed by Poincar´e,while another topological notion, covering dimension, is associated with Lebesgue. A metric-space version of dimension was developed -
A Metric on the Moduli Space of Bodies
A METRIC ON THE MODULI SPACE OF BODIES HAJIME FUJITA, KAHO OHASHI Abstract. We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of sym- plectic toric manifolds. We also discuss related problems. 1. Introduction An n-dimensional Delzant polytope is a convex polytope in Rn which is simple, rational and smooth1 at each vertex. There exists a natu- ral bijective correspondence between the set of n-dimensional Delzant polytopes and the equivariant isomorphism classes of 2n-dimensional symplectic toric manifolds, which is called the Delzant construction [4]. Motivated by this fact Pelayo-Pires-Ratiu-Sabatini studied in [14] 2 the set of n-dimensional Delzant polytopes Dn from the view point of metric geometry. They constructed a metric on Dn by using the n-dimensional volume of the symmetric difference. They also studied the moduli space of Delzant polytopes Den, which is constructed as a quotient space with respect to natural action of integral affine trans- formations. It is known that Den corresponds to the set of equivalence classes of symplectic toric manifolds with respect to weak isomorphisms [10], and they call the moduli space the moduli space of toric manifolds. In [14] they showed that, the metric space D2 is path connected, the moduli space Den is neither complete nor locally compact. Here we use the metric topology on Dn and the quotient topology on Den. -
Lectures on Fractal Geometry and Dynamics
Lectures on fractal geometry and dynamics Michael Hochman∗ June 27, 2012 Contents 1 Introduction 2 2 Preliminaries 3 3 Dimension 4 3.1 A family of examples: Middle-α Cantor sets . 4 3.2 Minkowski dimension . 5 3.3 Hausdorff dimension . 9 4 Using measures to compute dimension 14 4.1 The mass distribution principle . 14 4.2 Billingsley's lemma . 15 4.3 Frostman's lemma . 18 4.4 Product sets . 22 5 Iterated function systems 25 5.1 The Hausdorff metric . 25 5.2 Iterated function systems . 27 5.3 Self-similar sets . 32 5.4 Self-affine sets . 36 6 Geometry of measures 40 6.1 The Besicovitch covering theorem . 40 6.2 Density and differentiation theorems . 45 6.3 Dimension of a measure at a point . 50 6.4 Upper and lower dimension of measures . 52 ∗Send comments to [email protected] 1 6.5 Hausdorff measures and their densities . 55 7 Projections 59 7.1 Marstrand's projection theorem . 60 7.2 Absolute continuity of projections . 63 7.3 Bernoulli convolutions . 65 7.4 Kenyon's theorem . 71 8 Intersections 74 8.1 Marstrand's slice theorem . 74 8.2 The Kakeya problem . 77 9 Local theory of fractals 79 9.1 Microsets and galleries . 80 9.2 Symbolic setup . 81 9.3 Measures, distributions and measure-valued integration . 82 9.4 Markov chains . 83 9.5 CP-processes . 85 9.6 Dimension and CP-distributions . 88 9.7 Constructing CP-distributions supported on galleries . 90 9.8 Spectrum of an Markov chain . -
On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes
Munich Personal RePEc Archive On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes Dominique, C-Rene Independent Research 9 August 2009 Online at https://mpra.ub.uni-muenchen.de/16723/ MPRA Paper No. 16723, posted 10 Aug 2009 10:42 UTC ON THE COMPUTATION OF THE HAUSDORFF DIMENSION OF THE WALRASIAN ECONOMY: FURTHER NOTES C-René Dominique* ABSTRACT: In a recent paper, Dominique (2009) argues that for a Walrasian economy with m consumers and n goods, the equilibrium set of prices becomes a fractal attractor due to continuous destructions and creations of excess demands. The paper also posits that the Hausdorff dimension of the attractor is d = ln (n) / ln (n-1) if there are n copies of sizes (1/(n-1)), but that assumption does not hold. This note revisits the problem, demonstrates that the Walrasian economy is indeed self-similar and recomputes the Hausdorff dimensions of both the attractor and that of a time series of a given market. KEYWORDS: Fractal Attractor, Contractive Mappings, Self-similarity, Hausdorff Dimensions of the Walrasian Economy, the Hausdorff dimension of a time series of a given market. INTRODUCTION Dominique (2009) has shown that the equilibrium price vector of a Walrasian pure exchange economy is a closed invariant set E* Rn-1 (where R is the set of real numbers and n-1 are the number of independent prices) rather than the conventionally assumed stationary fixed point. And that the Hausdorff dimension of the attractor lies between one and two if n self-similar copies of the economy can be made. -
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. -
Typical Behavior of Lower Scaled Oscillation
TYPICAL BEHAVIOR OF LOWER SCALED OSCILLATION ONDREJˇ ZINDULKA Abstract. For a mapping f : X → Y between metric spaces the function lipf : X → [0, ∞] defined by diam f(B(x,r)) lipf(x) = liminf r→0 r is termed the lower scaled oscillation or little lip function. We prove that, given any positive integer d and a locally compact set Ω ⊆ Rd with a nonempty interior, for a typical continuous function f : Ω → R the set {x ∈ Ω : lipf(x) > 0} has both Hausdorff and lower packing dimensions exactly d − 1, while the set {x ∈ Ω : lipf(x) = ∞} has non-σ-finite (d−1)- dimensional Hausdorff measure. This sharp result roofs previous results of Balogh and Cs¨ornyei [3], Hanson [11] and Buczolich, Hanson, Rmoutil and Z¨urcher [5]. It follows, e.g., that a graph of a typical function f ∈ C(Ω) is microscopic, and for a typical function f : [0, 1] → [0, 1] there are sets A, B ⊆ [0, 1] of lower packing and Hausdorff dimension zero, respectively, such that the graph of f is contained in the set A × [0, 1] ∪ [0, 1] × B. 1. Introduction For a mapping f : X Y between metric spaces the function lipf : X [0, ] defined by → → ∞ diam f(B(x, r)) (1) lipf(x) = lim inf r→0 r is termed the lower scaled oscillation or little lip function. (B(x, r) denotes the closed ball of radius r centered at x.) Let us mention that some authors, cf. [3], define the scaled oscillations from the version of ωf (cf.(6)) given by ωf (x, r) = sup f(y) f(x) that may be more suitable for, e.g., differentiation con- y∈B(x,r)| − | siderations.