S-dimensional Hausdorff Measure Research Articles

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

Pointwise densities of homogeneous Cantor measure and critical values

Let N ⩾ 2 and ρ ∈ (0, 1/N 2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system Let s = dim H E be the Hausdorff dimension of E, and let be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density and upper s-density Θ∗s (μ, x) of μ at x. This extends some early results of Feng et al (2000 J. Math. Anal. Appl. 250 692–705). Furthermore, we determine two critical values a c and b c for the sets respectively, such that dim H E *(a) > 0 if and only if a < a c, and that dim H E*(b) > 0 if and only if b > b c. We emphasize that both values a c and b c are related to the Thue–Morse type sequences, and our strategy to find them relies on ideas from open dynamical systems and techniques from combinatorics on words.

Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete 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 &gt; 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 Hausdorff 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-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

Peano-type curves, Liouville numbers, and microscopic sets

Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is given.

Peano-Type Curves, Liouville Numbers, and Microscopic Sets

Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is given.

Partial regularity of minimizers of [formula omitted]-growth functionals with [formula omitted

We prove partial regularity of minimizers u for functionals of the following type A(u)=∫Ω[(Aijαβ(x,u)DαuiDβuj)p(x)/2+g(x,u,Du)]dx, assuming that Aijαβ(x,u) and p(x) are sufficiently smooth and that p(x)>1.We prove that u∈C0,α(Ω0) for some α∈(0,1) and an open set Ω0⊂Ω with Hm−γ1(Ω−Ω0)=0, where Hs denotes the s-dimensional Hausdorff measure and γ1=infΩp(x).

Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.

Partial regularity of minimizers of p (x )-growth functionals with 1 < p (x ) < 2

We prove partial regularity of minimizers u for p ( x ) -energy functionals of the following type: E ( u ) = ∫ Ω ( A i j α β ( x , u ) D α u i D β u j ) p ( x ) / 2 d x , assuming that A i j α β ( x , u ) and p ( x ) are sufficiently smooth and that p ( x ) is subquadratic. We prove that u ∈ C 0 , α ( Ω 0 ) for some α ∈ ( 0 , 1 ) and an open set Ω 0 ⊂ Ω with H m - γ 1 ( Ω - Ω 0 ) = 0 , where H s denotes the s-dimensional Hausdorff measure and γ 1 = inf Ω p ( x ) .

Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces

We give a new estimate for the ratio of s-dimensional Hausdorff measure \({\mathcal{H}^s}\) and (radius-based) packing measure \({\mathcal{P}^s}\) of a set in any metric space. This estimate is$$ \inf_{E}\frac{\mathcal{P}^s(E)}{\mathcal{H}^s(E)}\ge 1+\left(2-\frac{3}{2^{1/s}}\right)^s, $$where 0 < s < 1/2 and the infimum is taken over all metric spaces X and sets \({E\subset X}\) with \({0 < \mathcal{H}^s(E) < \infty}\). As an immediate consequence we improve the upper bound for the lower s-density of such sets in \({\mathbb{R}^n}\).

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

Density theorems for Hausdorff and packing measures of self-similar sets

We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff dimension and packing dimension equal s, a special case of our main results says that if K satisfies the Open Set Condition, then there exists a number r 0 such that 1 $${\mathcal{H}}^{s}(K \cap B(x, r)) \leq (2r)^{s}$$ and 2 $$(2r)^{s} \leq {\mathcal{P}}^{s}(K \cap B(x, r))$$ for all x ∈ K and all 0 < r < r 0, where $${\mathcal{H}}^{s}$$ denotes the s-dimensional Hausdorff measure and $${\mathcal{P}}^{s}$$ denotes the s-dimensional packing measure. Inequality (1) and inequality (2) are used to obtain a number of very precise density theorems for Hausdorff and packing measures of self-similar sets. These density theorems can be applied to compute the exact value of the s-dimensional Hausdorff measure $${\mathcal{H}}^{s}(K)$$ and the exact value of the s-dimensional packing measure $${\mathcal{P}}^{s}(K)$$ of self-similar sets K.

A new lower bound of the Hausdorff measure of the Sierpinski gasket

For the Sierpinski gasket, by using a sort of cover consisting of special regular hexagons, we define a new measure that is equivalent to the Hausdorff measure and obtain a lower bound of this measure. Moreover, the following lower bound of the Hausdroff measure of the Sierpinski gasket has been achieved $$H^s (S) \geqslant 0.670432,$$ where S denotes the Sierpinski gasket, s =dimH(S) =log 23, and Hs(S) denotes the s-dimensional Hausdorff measure of S. The above result improves that developed in ba][2].

GEOMETRICAL VERSUS TOPOLOGICAL PROPERTIES OF MANIFOLDS

AbstractGiven a compact n-dimensional immersed Riemannian manifold M n in some Euclidean space we prove that if the Hausdorff dimension of thesingular set of the Gauss map is small, then M n is homeomorphic to thesphere S n .Also, we define a concept of finite geometrical type and prove that fi-nite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite numberof points. A characterization of the 2n-catenoid is obtained. 1 Introduction Let f: M n → N m be a C 1 map. We denote byrank(f) := min p∈M rank(D p f).If n= dimM = dimN = m, let C := {p∈ M: detD p f= 0} the set of critical points of fand S:= f(C) the set of critical values of f.Now, let M n a compact, connected, boundary less, n-dimensional manifold.Denote by H s the s-dimensional Hausdorff measure and dim H (A) the Hausdorffdimension of A⊂ M n . For definitions see section 2 below. Let xbe an immer-sion x: M n → R n+1 . In this case, let G: M → S n the Gauss map associatedto x, C the critical points of Gand S the critical values of G. We denoteby dim

On the parametrization of self-similar and other fractal sets

We prove for many self-similar, and some more general, sets E C in that if s is the Hausdorff dimension of E and f :]RI -* Rn is H6lder continuous with exponent m/s, then the s-dimensional Hausdorff measure of Enff(Rm) is o.

An integral expression of Hausdorff measures in terms of s-densities

In a previous Note [7], we gave alternative definitions of Hausdorff measures. In this Note, we give some applications. Namely, we obtain an expression of the s-dimensional Hausdorff measure of a subset E of R d as an integral of the lower inverse s-density of a suitable measure supported on E.

Equivalent definitions of Hausdorff measures

This paper defines, on ℝ d, the s-dimensional Hausdorff measure, the s-dimensional full covering measure and the s-dimensional fine variational measure which are based on Thomson's uniform, fine and full covering relations, respectively, and proves that they are equivalent to each other for 0 < s ≤ d.

Spherical functions and conformal densities on spherically symmetric 𝐶𝐴𝑇(-1)-spaces

Let X X be a C A T ( − 1 ) CAT(-1) -space which is spherically symmetric around some point x 0 ∈ X x_{0}\in X and whose boundary has finite positive s − s- dimensional Hausdorff measure. Let μ = ( μ x ) x ∈ X \mu =(\mu _{x})_{x\in X} be a conformal density of dimension d &gt; s / 2 d&gt;s/2 on ∂ X \partial X . We prove that μ x 0 \mu _{x_{0}} is a weak limit of measures supported on spheres centered at x 0 x_{0} . These measures are expressed in terms of the total mass function of μ \mu and of the d − d- dimensional spherical function on X X . In particular, this result proves that μ \mu is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

Hausdorff Dimension of Symmetric Perfect Sets

We consider nowhere dense perfect subsets of [0, 1] that are symmetric but have no additional nice properties. We prove that if E = ∩En is a symmetric perfect set and the length of the basic intervals in En is denoted by ln then the Hausdorff dimension of E is $$s = \mathop {\lim inf}\limits_{n \to \infty } \{ s_n : 2^n l_n^{9_n } = 1\} = \mathop {\lim inf}\limits_{n \to \infty } \frac{{\log 2^n }}{{ - \log l_n }}$$ . The argument we use also shows that using natural covers of E; i.e., covers consisting of the 2n closed, equal length intervals of the nth stage, yield an estimate for the s-dimensional Hausdorff measure within a factor of four.

Local Dimensions, Average Densities and Self-Conformal Measures

We consider subsets F of \(\mathbb{R}^n\) generated by iterated function systems with contracting conformal C1+γ-diffeomorphisms whose Hausdorff dimension is s. The unique s-self-conformal probability measure μ agrees with the normalised s-dimensional Hausdorff measure on F. Using the associated dynamical system and ergodic theory we develop a potential theoretic representation of the average densities of μ at almost all points of F. Before we formulate some general relationships between average densities and local dimensions of measures.

Rectifiability of the singular set of energy minimizing maps

The regularity theory of Schoen and Uhlenbeck [SU] for energy minimizing maps u from a domain ~2 C R n (equipped with any smooth Riemannian metric) into a compact smooth Riemannian target manifold N , established that the singular set s ingu always has Hausdorff dimension _ 0, where ~ s denotes s-dimensional Hausdorff measure. Similar results were obtained independently by Giaquinta and Giusti [GG] in the case when the image is contained in a coordinate chart. In case the target manifold N is real analytic, the main theorems of this paper (Theorems 1-3 below) establish rectifiability properties for such singular sets in the dimension n 3, and in other dimensions m < n 3 in case N happens to be such that all energy minimizing maps u into N have dim sing u < m. Recall that a subset A C R n is said to be m-rectifiable if ,~flm(A) < oo, and if A has an approximate tangent space a.e. in the sense that for ~ m a . e . z E A there is an m-dimensional subspace Lz such that