DOCSLIB.ORG

Hausdorff Dimension of Singular Vectors

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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). While it is easy to see that the set Sing(d) of all singular vectors in Rd has zero Lebesgue measure, it is hard to compute its Hausdorff dimension. After a few partial results by Baker and Rynne (see [2], [3] and [20]), it has been proved in [6] by the first author that the 4 Hausdorff dimension of Sing(2) is 3 . In this paper we extend this result to higher dimensions. d2 Theorem 1.1. The Hausdorff dimension of Sing(d) is d+1 . Singular vectors x 2 Rd are characterized by Mahler's criterion d+1 lim inffkwk : 0 6= w 2 gthx g = 0 t!1 Z Date: February 28, 2014. 1991 Mathematics Subject Classification. 11J13, 11K55, 37A17. Key words and phrases. singular vectors; self-similar coverings; multi-dimensional continued fractions; simultaneous Diophantine approximation; best approximations; divergent trajectories. The first author is partially supported by NSF grant DMS 0956209. 1 1 −x where g = diag(et; : : : ; et; e−dt) and h = d . Equivalently, fg h SL g is a t x 0 1 t x d+1 Z t2R divergent trajectory of the flow on SLd+1 R= SLd+1 Z induced by the left action of gt. d Corollary 1.2. Divergent trajectories of gt have Hausdorff codimension d+1 . Proof. The Hausdorff codimension of Sing(d) in Rd coincides with that for divergent trajec- tories in SLd+1 R= SLd+1 Z (see [5]). In [12] Einsiedler-Kadyrov showed that the Hausdorff dimension of divergent trajectories of gt does not increase in the case of SL3 R= SL3 Z even after enlarging it to include trajectories that diverge "on average". This result generalizes to higher dimensions; see [?]. Divergent trajectories and their Hausdorff dimension have been considered for other classes of "hyperbolic" flows, starting with Masur's Duke paper [18] which established codimension 1 ≥ 2 for Teichm¨ullerflows. Positive Hausdorff codimension for divergent trajectories in a product of real rank one spaces is also known; see [24] and also [5]. Let " be a fixed positive real number. Recall that a vector x in Rd is "-Dirichlet improvable if the system of inequalities (1.1) admits a solution for T large enough. H. Davenport and W. Schmidt have proved that the set DI"(d) of "-Dirichlet improvable vectors has zero measure for all " < 1 ([10] and [11]). This result has been generalized in several directions (see for instance [15]) but less is known about the Hausdorff dimension of DI"(d). The intersection of the sets DI"(d), " > 0, is the set of singular vectors, so it is natural to expect that the Hausdorf dimension of DI"(d) decreases to the Hausdorff dimension of Sing(d) as " goes to zero. When d = 2, the bounds on Hdim DI"(d) obtained in [5] implies this continuity at " = 0 of the Hausdorff dimension. Our second result extends these bounds to higher dimensions. Theorem 1.3. Let d ≥ 2 be an integer and let t be any real number > d. There is a constant C such that for " small enough, d2 d2 + "t ≤ Hdim DI (d) ≤ + C"d=2: d + 1 " d + 1 Actually, we give a slightly better lower bound; see corollary 6.10. In the two dimensional case, this result already improves the lower bound of [6] but there is a further improvement giving a precise rate of convergence when " goes to zero. Theorem 1.4. For all real numbers t > 1, we have 4 + "t ≤ Hdim DI (2) 3 " for " small enough. Our proof of the above Theorem doesn't work when the dimension is ≥ 3. In particular, our analysis stops short of establishing the existence of the exponent d2 log(Hdim DI"(d) − ) (1.2) lim d+1 "!0+ log " 2 for d ≥ 3. It seems that our approach could potentially be carried further to establish existence of the above limit, which we will guess to lie in the interval [d − 1; d[. At last we have to say that results about singular linear forms can be deduced using a transference Theorem ([4], chapter V, corollary section II). But a direct proof is not clear. The proofs rely on three main tools. The first is lattice theory, especially Minkowski's theorems and reduced bases. The second is the notion of self-similar covering introduced in [5] and [6]. The third is the notion of best Diophantine approximation. Section 2 is devoted to lattices theory and counting results. Section 3 is devoted to Hausdorff dimension bounds using self-similar covers and Section 4 is devoted to best Diophantine approximations. The proofs of the above Theorems actually begin in Section 5. Proving the upper bounds is the most easy parts of the proofs. While the upper bounds are proved in Section 5, the lower bounds need the Sections 6, 7 and 8. 2. Counting lattice points 2.1. Notation. We shall use the standard notations ; ; in number theory: f(a; b; :::) g(a; b; :::) means that f(a; b; :::) ≤ Cg(a; b; :::) where C is a constant that does not depend on the parameters a; b; ::: etc. Most of the time, C depends only the dimension or on the norm. The symbol is used for the reverse inequality and when both inequalities hold. Suppose that Rn is endowed with a norm k · k. For a lattice L in Rn and 1 ≤ k ≤ n, we denote by λk(L) = inffr ≥: B(0; r) contains at least k linearly independent vector of Lg the kth minimum of L. The successive minima of a lattice L in Rn are the real numbers λ1(L); : : : ; λn(L). This notation suppresses the dependence on the norm. 2.2. Successive minima and reduced basis. Assume that Rn is equipped with a norm. By the (second) Minkowski theorem, the successive minima of a lattice L in Rn satisfy (see [4]) 2n (M) vol(L) ≤ λ (L) ··· λ (L) vol(B(0; 1)) ≤ 2n vol(L): n! 1 n We shall use this fact many times and simply write λ1(L) ··· λn(L) vol(L) where the constants involved in depend only on the dimension and on the volume of the unit ball. 3 We shall need reduced bases in a lattice. Minkowski reduced bases are well-suited to our needs. Other kinds of reduced bases could work as well. We shall use the following properties of a Minkowski reduced basis v1; : : : ; vn. For all i, i (M') λi(L) ≤ kvik ≤ 2 λi(L); A proof of this inequality can be found in [23]. Actually, Minkowski reduced bases are associated to Euclidean norms and the above inequality holds only in these cases. When dealing with a non Euclidean norm, we consider a reduced basis associated to the standard Euclidean norm and by the norm equivalence, one has kvik λi where the constants involved depend not only on the dimension but also on the norm. An important property of a reduced basis is that the angles between a vector of the basis and the subspace spanned by other vectors cannot be too small. More precisely, there is positive constant cn depending only on the dimension such that ? (M") vi ≥ cnλi(L) ? where vi is the orthogonal projection of vi on the line orthogonal to the subspace spanned by the other basis vectors. Once again, the above inequality holds for Euclidean norms and for other norms, one has ? vi λi(L) where the implicit constants depend on the norm. To prove (M"), note that by (M') one has ? ? λ1(L) ··· λi−1(L) vi λi+1(L) ··· λn(L) j det(v1; : : : ; vi−1; vi ; vi+1; : : : ; vn)j = vol(L) and then use the Minkowski theorem. For sake of simplicity, the reader can suppose that all the results are stated for the standard Euclidean norm. However, it is clear that the results of Section 2 hold with any norm with constants possibly depending on norms. The results in Section 4 hold for any norm. 2.3. Codimension one sublattice of minimal volume. Fix a norm k · k on Rn.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • CHARACTERIZATIONS of SPACES of FINITE HAUSDORFF MEASURE: ONE-DIMENSIONAL CASE Supervisor
    CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE: ONE-DIMENSIONAL CASE JORDAN DOUCETTE Supervisor: Dr. Murat Tuncali MAJOR RESEARCH PAPER Submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Computer Science and Mathematics Nipissing University North Bay, Ontario c Jordan Doucette December 2012 ! Abstract. An intriguing problem that was proposed by Samuel Eilenberg and O.G. Harrold in 1941 involves the characterization of separable metric spaces, whose topological dimension is n, for which there is a metrization so that the n- dimensional Hausdorff measure is finite. The focus of this paper will be to examine several conditions in order to characterize metric continua of finite one-dimensional Hausdorff measure. Our main theorem will consist of showing that for a metric continuum being of finite one-dimensional Hausdorff measure is equivalent to being totally regular, as well as being an inverse limit of graphs with monotone bonding maps. iv ACKNOWLEDGEMENTS First and foremost, I am grateful to my supervisor Dr. Murat Tuncali, for his invaluable support, instruction and encouragement throughout the preparation of this paper. Without Murat’s continual guidance this paper may never have come to fruition. Thanks are also due to my second reader, Dr. Alexandre Karassev, external examiner, Dr. Ed Tymchatyn and examination committee chair, Dr. Vesko Valov for their helpful feedback. I would also like to extend thanks the entire mathematics department at Nipissing University and visiting professors throughout the years, whose teaching has lead me to this point. Lastly, thank you to my family and friends for their reassurance and unwavering confidence in me.
    [Show full text]
  • On the Hausdorff Volume in Sub-Riemannian Geometry
    On the Hausdorff volume in sub-Riemannian geometry Andrei Agrachev SISSA, Trieste, Italy and MIAN, Moscow, Russia - [email protected] Davide Barilari SISSA, Trieste, Italy - [email protected] Ugo Boscain CNRS, CMAP Ecole Polytechnique, Paris, France - [email protected] October 29, 2018 Abstract For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spher- ical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is 3 (and 4 on every smooth curve) but in general not 5. These results answer to a question addressedC byC Montgomery about the relation between twoC intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general. 1 Introduction In this paper, by a sub-Riemannian manifold we mean a triple S = (M, ∆, g), where M is a connected orientable smooth manifold of dimension n, ∆ is a smooth vector distribution of constant rank k<n, satisfying the H¨ormander condition and g is an Euclidean structure on ∆. A sub-Riemannian manifold has a natural structure of metric space, where the distance is the so called Carnot-Caratheodory distance T d(q0,q1) = inf gγ(t)(γ ˙ (t), γ˙ (t)) dt γ : [0, T ] M is a Lipschitz curve, { 0 | → arXiv:1005.0540v3 [math.DG] 15 Apr 2011 Z q γ(0) = q , γ(T )= q , γ˙ (t) ∆ a.e.
    [Show full text]
  • Convex Bodies with Almost All K-Dimensional Sections Polytopes by LEONI DALLA ASH D
    Math. Proc. Camb. Phil. Soc. (1980), 88, 395 395 Printed in Great Britain Convex bodies with almost all k-dimensional sections polytopes BY LEONI DALLA ASH D. G. LARMAN University College, London (Received 12 September 1979) 1. It is a well-known result of V. L. Klee (2) that if a convex body K in En has all its fc-dimensional sections as polytopes (k > 2) then K is a polytope. In (6,7) E. Schneider has asked if any properties of a polytope are retained by K if the above assumptions are weakened to hold only for almost all ^-dimensional sections of K. In particular, for 2-dimensional sections in E3 he asked whether ext K is countable, or, if not, is the 1-dimensional Hausdorff measure of ext K zero? Here we shall give an example (Theorem 1) of a convex body K in E3, almost all of whose 2-sections are polygons but ext K has Hausdorff dimension 1. Our example does not, however, yield a convex body K with ext K of positive 1-dimensional Hausdorff measure although we believe that such a body exists. Using deep measure-theoretic results of J. M. MarstrandO), as extended by P. Mattila(5), we shall prove a very general result (Theorem 2) showing that a convex body K in En has ext (K n L) of dimension at most s for almost all of its ^-dimensional sections L, if, and only if, the dimension of the (n — &)-skeleton of K is at most n — k + s. 3 THEOREM 1.
    [Show full text]
  • A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals
    mathematics Article A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals Mohsen Soltanifar 1,2 1 Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, Canada; [email protected] 2 Real World Analytics, Cytel Canada Health Inc., 802-777 West Broadway, Vancouver, BC V5Z 1J5, Canada Abstract: How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions. Keywords: cantor set; fractals; Hausdorff dimension; continuum; aleph-two MSC: 28A80; 03E10 1. Introduction Benoit Mandelbrot (1924–2010) coined the term fractal and its dimension in his 1975 essay on the quest to present a mathematical model for self-similar, fractured ge- ometric shapes in nature [1]. Both nature and the virtual world have prominent examples Citation: Soltanifar, M. A of fractals: some natural examples include snowflakes, clouds and mountain ranges, while Generalization of the Hausdorff some virtual instances are the middle third Cantor set, the Sierpinski triangle and the Dimension Theorem for Deterministic Vicsek fractal. A key element in the definition of the term fractal is its dimension usually Fractals. Mathematics 2021, 9, 1546. indexed by the Hausdorff dimension. This dimension has played vital roles in develop- https://doi.org/10.3390/math9131546 ing the concept of fractal dimension and the definition of the term fractal and has had extensive applications [2–5].
    [Show full text]
  • The Hausdorff Dimension Distribution of Finite Measures in Euclidean Space
    Can. J. Math., Vol. XXXVIII, No. 6, 1986, pp. 1459-1484 THE HAUSDORFF DIMENSION DISTRIBUTION OF FINITE MEASURES IN EUCLIDEAN SPACE COLLEEN D. CUTLER 1. Introduction. Let E be a Borel set of R^. The a-outer Hausdorff measure of E has been defined to be H"(E) = lirn H%{E) where H%E)= inf 2 (</(*,•))" UB^E and each Bt is a closed ball. d(Bt) denotes the diameter of Bt. It is easily seen that the same value Ha(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed. By dim(£') we will mean the usual Hausdorff-Besicovitch dimension of E, where dim(E) = sup{a\Ha(E) = oo} = inf{a\Ha(E) = 0}. The following (see [7] ) are well known elementary properties of dim(E): (1) 0 S dim(E) ^ N. (2) if E is countable then dim(E) = 0 while if X(E) > 0 then dim(E) = N (where X denotes iV-dimensional Lebesgue measure, a notation to be maintained throughout this paper). (3) dim^S En) = sup dim(£J. These properties will be used freely without comment in the following. Let ju be a probability measure on R^. In this paper we introduce the notion of the dimension distribution jx of fi. ji is a probability measure on [0, N] and the quantity jl(E) can be interpreted as the proportion of mass of JU supported strictly on sets with dimension lying in E. Associated with JU and JU is a real-valued random variable à (which we will call the dimension concentration map determined by n) and a family [\p(-, a) }, 0 ^ a ^ N, of probability measures on R^ to be referred to as the Received February 19, 1985 and in revised form July 21, 1985.
    [Show full text]
  • What Is Dimension?
    What is dimension? Somendra M Bhattacharjee1;2 1 Institute of Physics, Bhubaneswar 751005, India 2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India email:[email protected] This chapter explores the notion of \dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG flow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices. Contents 5 Dimensions related to physical problems 12 5.1 Spectral dimension . 12 1 Introduction 2 5.1.1 Diffusion, random walk . 12 5.1.2 Density of states . 13 2 Does \dimension" matter? 2 6 Which d? 13 3 Euclidean and topological dimensions 3 6.1 Thermodynamic equation of state . 13 3.1 Euclidean dimension . 3 6.2 Phase transitions . 14 3.1.1 d via analytic continuation . 4 6.2.1 Discrete case: Ising model . 14 3.2 Topological dimension . 4 6.2.2 Continuous case: crystal, xy, Heisen- berg . 15 4 Fractal dimension: Hausdorff, Minkowski (box) dimensions 6 6.3 Bound states in quantum mechanics .
    [Show full text]
  • FRACTAL DIMENSIONS and MEASURES 1. Historical Context
    AN INTRODUCTION TO DIMENSION THEORY AND FRACTAL GEOMETRY: FRACTAL DIMENSIONS AND MEASURES ERIN PEARSE 1. Historical context and motivation Poincar´e's topological reinterpretation of Euclid's initial concept of dimension: \When we say that space has the dimensions three, what do we mean? If to divide a continuum C it suffices to consider as cuts a certain number of distinguishable points, we say that this continuum is of dimension one. If, on the contrary, to divide a continuum it suffices to use cuts which form one or several continua of dimension one, then we say that C is a continuum of dimension two. If cuts which form one or several continua of at most dimension two suffice, we say that C is a continuum of dimension three; and so on." This idea of dimension can be rephrased (`a la Brouwer) inductively in more modern language: (1) We take a single point to have dimension 0. (2) If a set A contains points for which the boundaries of arbitrarily small neigh- bourhoods all have dimension n, then A is said to have dimension n + 1. These basically justify thinking of dimension as the number of parameters required to identify a point in a given space. This idea was turned on its head by Peano at the beginning of this century, when he constructed a continuous function of the unit interval with the unit square. This construction of a space-filling curve showed how the dimension of an object could be changed by a continuous transformation and thus contradicted the idea of dimension as \minimum number of parameters".
    [Show full text]