Jordan Arcs Research Articles

Carleman approximation by conformal minimal immersions and directed holomorphic curves

Let R be an open Riemann surface. In this paper we prove that every continuous function M→Rn, n≥3, defined on a divergent Jordan arc M⊂R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalisation of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map.As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.

Recognizing Weak Embeddings of Graphs

We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding φ : G → M of a graph G into a 2-manifold M maps the vertices in V ( G ) to distinct points and the edges in E ( G ) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs due to data compression or low resolution. This raises the computational problem of deciding whether a given map φ : G → M comes from an embedding. A map φ : G → M is a weak embedding if it can be perturbed into an embedding ψ ε : G → M with ‖ φ − ψ ε ‖ < ε for every ε > 0, where ‖.‖ is the unform norm. A polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over Z 2 . It runs in O ( n 2ω )≤ O ( n 4.75 ) time, where ω ∈ [2,2.373) is the matrix multiplication exponent and n is the number of vertices and edges of G . We improve the running time to O ( n log n ). Our algorithm is also conceptually simpler: We perform a sequence of local operations that gradually “untangles” the image φ( G ) into an embedding ψ( G ) or reports that φ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.

Fractal curves and rugs of prescribed conformal dimension

We construct Jordan arcs of prescribed conformal dimension which are “minimal for conformal dimension,” meaning the Hausdorff and conformal dimensions are equal. These curves are used to design fractal rugs, similar to Rickman's rug, that are also minimal for conformal dimension. These fractal rugs could potentially settle a standing conjecture regarding the existence of metric spaces of prescribed topological conformal dimension.

Christoffel functions with power type weights

Precise asymptotics for Christoffel functions are established for power type weights on unions of Jordan curves and arcs. The asymptotics involve the equilibrium measure of the support of the measure. The result at the endpoints of arc components is obtained from the corresponding asymptotics for internal points with respect to a different power weight. On curve components the asymptotic formula is proved via a sharp form of Hilbert's lemniscate theorem while taking polynomial inverse images. The situation is completely different on the arc components, where the local asymptotics is obtained via a discretization of the equilibrium measure with respect to the zeros of an associated Bessel function. The proofs are potential theoretical, and fast decreasing polynomials play an essential role in them.

Critical Points on Convex Surfaces

The cut locus C(x) of some point x on an open convex surface is a forest and has measure zero. However, we show here that topologically it can be quite large, namely residual. All critical points with respect to x belong to C(x). We also show that, irrespective of how large C(x) might be, there is a Jordan arc in C(x) containing all critical points.

On the number of touching pairs in a set of planar curves

Given a set of planar curves (Jordan arcs), each pair of which meets — either crosses or touches — exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has O(t2n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

Moduli space of cubic Newton maps

In this article, we study the topology and bifurcations of the moduli space M3 of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure (Cˆ,(2,3,∞)). We prove two results:• The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves;• The Head's angle map is surjective and monotone. The fibers of this map are characterized completely.The first result is a moduli space analogue of the first author's dynamical regularity theorem [37]. The second result confirms a conjecture of Tan Lei.

On the accumulation sets of exponential rays

We show that there exist non-landing exponential rays with bounded accumulation sets. By introducing folding models of certain rays, we prove that each of the corresponding accumulation sets is an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc.

Stochastic Komatu–Loewner evolutions and BMD domain constant

For Komatu–Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of Schramm (2000) to find the SDEs satisfied by the induced motion ξ(t) on ∂H and the slit motion s(t). The diffusion coefficient α and drift coefficient b of such SDEs are homogeneous functions.Next with solutions of such SDEs, we study the corresponding stochastic Komatu–Loewner evolution, denoted as SKLEα,b. We introduce a function bBMD measuring the discrepancy of a standard slit domain from H relative to BMD. We show that SKLE6,−bBMD enjoys a locality property.

Quasiconformal extension of quasimöbius mappings of Jordan domains

We introduce the new class of Jordan arcs (curves) of bounded rotation which includes all arcs (curves) of bounded turning. We prove that if the boundary of a Jordan domain has bounded rotation everywhere but possibly one singular point then every quasimobius embedding of this domain extends to a quasiconformal automorphism of the entire plane.

Cutting algebraic curves into pseudo-segments and applications

We show that a set of n algebraic plane curves of constant maximum degree can be cut into O(n3/2polylogn) Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then apply a variant of a technique of Aronov and Sharir to eliminate these depth cycles by making a small number of cuts, which corresponds to a small number of cuts to the original planar arrangement of curves. After these cuts have been performed, the resulting curves form a collection of pseudo-segments.Our cutting bound leads to new incidence bounds between points and constant-degree algebraic curves. The conditions for these incidence bounds are slightly stricter than those for the current best-known bound of Pach and Sharir; for our result to hold, the curves must be algebraic and of bounded maximum degree, while Pach and Sharir's bound only imposes weaker, purely topological constraints on the curves. However, when our conditions hold, the new bounds are superior for almost all ranges of parameters. We also obtain new bounds on the complexity of a single level in an arrangement of constant-degree algebraic curves, and a new bound on the complexity of many marked faces in an arrangement of such curves.

Asymptotic Markov inequality on Jordan arcs

Markov's inequality for the derivative of algebraic polynomials is considered on -smooth Jordan arcs. The asymptotically best estimate is given for the th derivative for all . The best constant is related to the behaviour around the endpoints of the arc of the normal derivative of the Green's function of the complementary domain. The result is deduced from the asymptotically sharp Bernstein inequality for the th derivative at inner points of a Jordan arc, which is derived from a recent result of Kalmykov and Nagy on the Bernstein inequality on analytic arcs. In the course of the proof we shall also need to reduce the analyticity condition in this last result to -smoothness. Bibliography: 21 titles.

Асимптотическое неравенство Маркова на жордановых дугах

Изучаются неравенства Маркова для производной алгебраического многочлена на $C^2$-гладкой жордановой дуге. Получена асимптотически точная оценка для производной порядка $k$ при всех $k=1,2,…$ . Наилучшая константа в этой оценке связана с поведением в окрестности концов дуги нормальной производной функции Грина дополнения к дуге. Данный результат выводится из асимптотически точного неравенства Бернштейна для $k$-й производной во внутренних точках жордановой дуги. Последний результат в свою очередь получен с использованием недавнего результата С. И. Калмыкова и Б. Надя о неравенстве Бернштейна на аналитических дугах. В ходе доказательства будет ослаблено условие аналитичности дуг из результата Калмыкова и Надя до условия $C^2$-гладкости. Библиография: 21 название.

The jump problem for certain Beltrami equation on arcs

We consider the jump boundary problem on Jordan arc for solutions of certain Beltrami equations.

Möbius bilipschitz homogeneous arcs on the plane

A mobius bilipschitz mapping is an η-quasimobius mapping with the linear distortion function η(t) = Kt. We show that if an open Jordan arc γ ⊂ C with distinct endpoints a and b is homogeneous with respect to the family F K of mobius bilipschitz automorphisms of the sphere C with K specified then γ has bounded turning RT(γ) in the sense of Rickman and, consequently, γ is a quasiconformal image of a rectilinear segment. The homogeneity of γ with respect to F K means that for all x, y ∈ γ {a, b} there exists f ∈ F K with f(γ) = γ and f(x) = y. In order to estimate RT(γ) from above, we introduce the condition BR(δ) of bounded rotation of γ, and then the explicit bound depends only on K and δ.

Polynomial and rational inequalities on analytic Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein–Erdélyi inequality for derivative of rational functions on the unit circle, Gonchar–Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.

Empty Triangles in Good Drawings of the Complete Graph

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph \(K_n\) with \(n\) vertices is at least \(n\).

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.

On a conjecture of Widom

In 1969 Harold Widom published his seminal paper (Widom, 1969) which gave a complete description of orthogonal and Chebyshev polynomials on a system of smooth Jordan curves. When there were Jordan arcs present the theory of orthogonal polynomials turned out to be just the same, but for Chebyshev polynomials Widom’s approach proved only an upper estimate, which he conjectured to be the correct asymptotic behavior. In this note we make some clarifications which will show that the situation is more complicated.

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.