Jordan Arcs Research Articles

Bergman kernel functions associated to measures supported on totally real submanifolds

Abstract We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in C n \mathbb{C}^{n} is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.

Dipolar Komatu–Loewner differential equation

By constructing a suitable holomorphic vector field for a finitely connected domain we derive that any Jordan arc in a strip standard slit domain D starting on the real axis can be described through dipolar Komatu–Loewner equation. The Komatu–Loewner equations for the slits of strip standard domain are established. We prove that conversely, the dipolar Komatu–Loewner equation generates a family of conformal maps from sub-domains of D onto another strip standard slit domains.

Delta‐convex structure of the singular set of distance functions

AbstractFor the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains Omega (n^{frac{1}{2}}) pairwise disjoint edges and a plane cycle (and hence path) of length Omega (frac{log n }{log log n}). Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.

Julia Sets, Jordan Curves and Quasi-circles

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and parabolic basins into a single critical point of highest possible multiplicity.

On the structure of self-affine Jordan arcs in ℝ2

Abstract We prove that if a self-affine arc γ ∈ R 2 \gamma \in {{\mathbb{R}}}^{2} does not satisfy weak separation condition, then it is a segment of a parabola or a straight line. If a self-affine arc γ \gamma is not a segment of a parabola or a line, then it is a component of the attractor of a Jordan multizipper with the same set of generators.

On a planar six-neighbor theorem and its application

This study focuses on the normal tiling and proves a theorem that a planar normal tiling must contain one tile that admits at least six neighbors if each tile is enclosed by a boundary consisting of finitely many Jordan arcs, where the tile can be multiply connected and share non-connected boundaries with its neighbors. It provides an optimal lower bound in the general case and can be applied, e.g., to a complexity assessment of nonlinear optimization involving the gradient flow.

Empty Triangles in Generalized Twisted Drawings of $K_n$

Simple drawings are drawings of graphs in the plane such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once, and there is a ray emanating from $O$ which crosses every edge exactly once. We show that all generalized twisted drawings of $K_n$ contain exactly $2n-4$ empty triangles, by this making a substantial step towards proving the conjecture that any simple drawing of $K_n$ contains at least $2n-4$ empty triangles.

Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential V(M)=−13M3+tM, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, t∈C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.

The traveling salesman theorem for Jordan curves

If Γ⊂Rn, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfiesℓ(Γ)−|z−w|≃∑QβΓ2(Q)diam(Q), where the sum is over all dyadic cubes in Rn and βΓ(Q) is Peter Jones's β-number that measures the deviation of Γ from a straight line inside 3Q. This estimate sharpens previously known results by replacing an O(diam(Γ)) term by |z−w|. Applications of this improvement to the study of Weil-Petersson curves are described, and a new proof of Jones's traveling salesman theorem is given.

Feedback equivalence and uniform ensemble reachability

This paper considers feedback methods for ensemble reachability of parameter-dependent linear systems (A(θ),B(θ)), where the parameter θ is varying over a compact Jordan arc in the complex plane. Recently, pointwise testable sufficient conditions for uniform ensemble reachability have been developed. Beside the necessity of pointwise reachability these conditions put restrictions on the spectra of the matrices A(θ) and the Hermite indices of the pair (A(θ),B(θ)). In this paper we show that these conditions can be ensured by applying a suitable feedback transformation if the pair (A(θ),B(θ)) is pointwise reachable and its Kronecker indices are independent of the parameter.

Extremal polynomials on a Jordan arc

Let Γ be a C2+ Jordan arc and let Γ0 be the open arc which consists of interior points of Γ. We find concrete upper and lower bounds for the limit of Widom factors for L2(μ) extremal polynomials on Γ which was given in Widom (1969). In addition, we show that the upper bound for the limit supremum of Widom factors for the weighted Chebyshev polynomials which was obtained in Widom (1969) can be improved once two normal derivatives of the Green function do not agree at one point z∈Γ0. We also show that if Γ0 is not analytic then we have improved upper bounds.

Weak quasicircles have Lipschitz dimension 1

We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.

Bi-Lipschitz isomorphisms of self-similar Jordan arcs

Bi-Lipschitz isomorphisms of self-similar Jordan arcs

On Multipoint Padé Approximants whose Poles Accumulate on Contours that Separate the Plane

In this note we consider asymptotics of the multipoint Pade approximants to Cauchy integrals of analytic non-vanishing densities defined on a Jordan arc connecting $ -1 $ and $ 1 $. We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.

A Sharp Bound for the Growth of Minimal Graphs

We consider minimal graphs \(u = u(x,y) > 0\) over unbounded domains \(D \subset R^2\) bounded by a Jordan arc \(\gamma \) on which \(u = 0\). We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector $$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$then the rate of growth is at most \(r^{\pi /\lambda }\).

Multiplicity of zeros of polynomials

Sharp bounds are given for the highest multiplicity of zeros of polynomials in terms of their norm on Jordan curves and arcs. The results extend a theorem of Erdős and Turán and solve a problem of them from 1940.

Torsions and integrations

ABSTRACT The solutions of a number of well-known boundary value problems of complex analysis (for example, the Riemann boundary value problem) can be found in the form of curvilinear integrals over the boundaries of domains under consideration. In this connection, the classical results on that problems concern domains with rectifiable boundaries only. On the other hand, the boundary value problems themselves make sense for non-rectifiable boundaries, too. This is the reason for recent development of theory of generalized integration over plane non-rectifiable Jordan curves and arcs. The existence of such generalized integrations over non-rectifiable arcs depends on certain geometry properties of the arcs in neighborhoods of their ends. Here we consider the connections of generalized integration and so-called torsions of the path of integration at its end points.

On Hilbert and Riemann problems for generalized analytic functions and applications

The research of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincare problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. First of all, here it is proved theorems on the existence of solutions to the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the natural parameter and angular (nontangential) limits, moreover, in arbitrary Jordan domains in terms of harmonic measure and principal asymptotic values. Moreover, it is established the existence theorems on solutions for the appropriate boundary value problems of Hilbert and Riemann with arbitrary measurable data along the so-called Bagemihl–Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincare boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.

On the distribution of Julia sets of holomorphic maps

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.