Weak Tangent Research Articles

Quasisymmetrically rigid self-similar carpets

Abstract We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

WEAK TANGENTS ON CLOSED POPCORN GRAPHS

Given [Formula: see text], we study the Assouad dimension and weak tangents of closed [Formula: see text]-popcorn graphs. For all [Formula: see text], we prove that [Formula: see text] is a weak tangent of the closed [Formula: see text]-popcorn graphs by using some arguments on prime numbers. For all [Formula: see text], we first show that the Assouad dimension of the closed [Formula: see text]-popcorn graphs is 1, and then prove that [Formula: see text] is a weak tangent of them. We also discuss some specific weak tangents of closed [Formula: see text]-popcorn graphs when [Formula: see text] and [Formula: see text].

Quasisymmetric embeddability of weak tangents

In this paper, we study the quasisymmetric embeddability of weak tangents of metric spaces. We first show that quasisymmetric embeddability is hereditary, i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak tangent of $X$ can be quasisymmetrically embedded into some weak tangent of $Y$, given that $X$ is proper and doubling. However, the converse is not true in general; we will illustrate this with several counterexamples. In special situations, we are able to show that the embeddability of weak tangents implies global or local embeddability of the ambient space. Finally, we apply our results to Gromov hyperbolic groups and visual spheres of expanding Thurston maps.

Assouad dimension and local structure of self-similar sets with overlaps in [formula omitted

For a self-similar set in Rd that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least 1. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of overlapping directions and the Assouad dimension of the orthogonal projection of the self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than 2, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.

Weak tangent and level sets of Takagi functions

In this paper, we study some properties of Takagi functions and their level sets. We show that for Takagi functions T_{a,b} with parameters a, b such that ab is a root of a Littlewood polynomial, there exist large level sets. As a consequence, we show that for some parameters a, b, the Assouad dimension of graphs of T_{a,b} is strictly larger than their upper box dimension. In particular, we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.

On the upper regularity dimensions of measures

We study the \emph{upper regularity dimension} which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of \emph{doubling}. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the $L^q$-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.

Distance sets, orthogonal projections and passing to weak tangents

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.

Arithmetic patches, weak tangents, and dimension

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including: the presence (or lack of) arithmetic progressions (or patches in dimensions $\geq 2$); the structure of tangent sets; and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to `asymptotically' containing arbitrarily large arithmetic patches. We present some applications of our results concerning sets of integers, which include a weak solution to the Erd\"os-Tur\'an conjecture on arithmetic progressions.

Some Properties of the Derivatives on Sierpinski Gasket Type Fractals

In this paper, we focus on Strichartz’s derivatives, a family of derivatives including the normal derivative, on post critically finite fractals, which are defined at vertices in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives, of the neighboring vertices will decay to zero. The rates of approximations are described, and several nontrivial examples are provided to illustrate that our estimates are optimal. We also study the boundedness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex is provided. Furthermore, we give a counterexample of a conjecture of Strichartz on the existence of higher-order weak tangents.

Conformal dimension and boundaries of planar domains

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in C ^ \hat {\mathbb {C}} have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco’s “uniformly well-spread cut point” condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.

Assouad type dimensions for self-affine sponges

We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity properties of doubling measures in the context studied by Bylund and Gudayol.

Nematic Equilibria on a Two‐Dimensional Annulus

We study planar nematic equilibria on a two‐dimensional annulus with strong and weak tangent anchoring, in the Oseen–Frank theoretical framework. We analyze a radially invariant defect‐free state and compute analytic stability criteria for this state in terms of the elastic anisotropy, annular aspect ratio, and anchoring strength. In the strong anchoring case, we define and characterize a new spiral‐like equilibrium which emerges as the defect‐free state loses stability. In the weak anchoring case, we compute stability diagrams that quantify the response of the defect‐free state to radial and azimuthal perturbations. We study sector equilibria on sectors of an annulus, including the effects of weak anchoring and elastic anisotropy, giving novel insights into the correlation between preferred numbers of boundary defects and the geometry. We numerically demonstrate that these sector configurations can approximate experimentally observed equilibria with boundary defects.

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below

Abstract We show that in any infinitesimally Hilbertian 𝖢𝖣 * ( K , N ) $\mathsf {CD}^*(K,N)$ -space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov–Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian 𝖢𝖣 * ( 0 , N ) $\mathsf {CD}^*(0,N)$ -spaces.

Electrical Properties of Na1 - xKxNbO3 (0.28 ≤ x ≤ 0.40)

Sodium potassium niobate (Na 1 − x K x NbO 3 ) pellet samples, with x = 0.28, 0.30, 0.32, 0.325 and 0.4, were prepared by solid state reaction method. In the XRD patterns a break in the peak shifting tendency was observed for the composition with x = 0.32. Dielectric constant, loss tangent and dielectric conductivity of the prepared samples were measured in the frequency range from 10 to 1000 KHz, and in the temperature range from 30 to 500°C. Transition temperatures determined from the dielectric constant measurements were found independent of the measured frequencies, and no relaxator type behavior was observed for the presently prepared compositions. Dielectric constant and loss tangent were observed decreasing, and dielectric conductivity increasing with increasing frequency. Weak loss tangent anomalies were observed, near the transition temperatures. Transition anomalies observed from the dielectric conductivity measurements were consistent with those from the dielectric constant measurements. Among the prepared compositions dielectric constant, loss tangent and dielectric conductivity were observed minimum for Na 0.68 K 0.32 NbO 3 , at different frequencies and temperatures, which indicate a strong composition dependence of the electrical properties. Among the prepared n-type Na 1 − x K x NbO 3 compositions, an increase in the Na site vacancies and structural anomaly may be held responsible for the observed minimum dielectric conductivity of the composition with x = 0.32.

Conformal assouad dimension and modulus

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.

Weak Tangent Cones and Optimization in a Banach Space

A general notion of a $\tau $-tangent cone is introduced and developed for optimization purposes. This includes as special cases both the weak and strong tangent cones that appear in the literature. First order conditions with and without constraint qualification are examined and particular examples are provided to demonstrate that these conditions properly subsume those previously in the literature. Emphasis is placed on weak Kuhn–Tucker sufficiency conditions.