scholarly journals From curves to currents

2021 ◽  
Vol 9 ◽  
Author(s):  
Dídac Martínez-Granado ◽  
Dylan P. Thurston
Keyword(s):  
Continuous Extension ◽  
Extremal Length ◽  
Continuous Functions ◽  
Simple Criterion ◽  
Main Condition ◽  
Closed Curves ◽  
Smoothing Property ◽  
Geodesic Currents ◽  
Nonpositively Curved ◽  
Almost All

Abstract Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

On the Extension of Uniformly Continuous Functions(1)

1975 ◽  
Vol 18 (1) ◽  
pp. 143-145 ◽  
Author(s):  
L. T. Gardner ◽  
P. Milnes
Keyword(s):  
Continuous Function ◽  
Uniform Space ◽  
Continuous Extension ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Totally Bounded ◽  
Special Cases ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

AbstractA theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.

The packing measure of the graphs and level sets of certain continuous functions

1988 ◽  
Vol 104 (2) ◽  
pp. 347-360 ◽  
Author(s):  
Fraydoun Rezakhanlou
Keyword(s):  
Continuous Function ◽  
Level Sets ◽  
Continuous Functions ◽  
Packing Dimension ◽  
Packing Measure ◽  
Occupation Measure ◽  
Weierstrass Function ◽  
Local Growth ◽  
Almost All ◽  
The Relationship

AbstractThe relationship between the local growth of a continuous function and the packing measure of its level sets and of its graph is studied. For the Weierstrass function with b an integer such that b ≥ 2 and with 0 < α < 1, and for x ∈ Range (W) outside a set of first category, the level set W−1(x) has packing dimension at least 1 − α. Furthermore, for almost all x ∈ Range (W), the packing dimension of f is at most 1 − α. Finer results on the occupation measure and the size of the graph of a continuous function satisfying the Zygmund Λ-condition are obtained.

scholarly journals Vassiliev measures of complexity of open and closed curves in 3-space

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Eleni Panagiotou ◽  
Louis H. Kauffman
Keyword(s):  
Continuous Function ◽  
Jones Polynomial ◽  
Continuous Functions ◽  
Topological Invariant ◽  
Integral Formulation ◽  
Closed Curves ◽  
Vassiliev Invariants ◽  
Polygonal Curves ◽  
Gauss Code

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

A non-constant invariant function for certain ergodic flows

2008 ◽  
Vol 28 (3) ◽  
pp. 1031-1035
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Differentiable Manifold ◽  
Invariant Function ◽  
Continuous Functions ◽  
The Real ◽  
Almost All ◽  
Uniformly Continuous

AbstractLet U be the vector space of uniformly continuous real-valued functions on the real line $\mathbb {R}$ and let U0 denote the subspace of U consisting of all bounded uniformly continuous functions. If X is a compact differentiable manifold and we are given a flow on X, then we associate with the flow a function F:X→H1(X,U/U0) that is invariant under the flow. We give examples for which the flow on X is ergodic but there is no λ∈H1(X,U/U0) such that F(p)=λ for almost all points p.

scholarly journals Asymptotic Behaviour of the Iterates of Positive Linear Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Ioan Gavrea ◽  
Mircea Ivan
Keyword(s):  
Asymptotic Behaviour ◽  
General Result ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Compact Set ◽  
Almost All

We present a general result concerning the limit of the iterates of positive linear operators acting on continuous functions defined on a compact set. As applications, we deduce the asymptotic behaviour of the iterates of almost all classic and new positive linear operators.

scholarly journals Module of Annulus

1961 ◽  
Vol 18 ◽  
pp. 37-41 ◽  
Author(s):  
Tohru Akaza ◽  
Tadashi Kuroda
Keyword(s):  
Continuous Functions ◽  
Closed Curves ◽  
Lower Semi ◽  
The Family ◽  
Rectifiable Curves

Let C and C′ be two simple closed curves in the complex z-plane which have no point in common and surround the origin. Denote by D the annulus bounded by C and C′. Consider a family {γ} of rectifiable curves γ in D and the family P of all non-negative lower semi-continuous functions ρ = ρ(z) in D. Put

scholarly journals On continuous extension of uniformly continuous functions and metrics

2009 ◽  
Vol 116 (2) ◽  
pp. 191-202 ◽  
Author(s):  
T. Banakh ◽  
N. Brodskiy ◽  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Continuous Functions ◽  
Uniformly Continuous

Degree and holomorphic extendibility

2013 ◽  
Vol 143 (1) ◽  
pp. 129-139
Author(s):  
Darja Govekar Leban
Keyword(s):  
Continuous Function ◽  
Finite Number ◽  
Bounded Domain ◽  
Pairwise Disjoint ◽  
Continuous Functions ◽  
Closed Curves ◽  
Preceding Theorem ◽  
Holomorphic Extendibility ◽  
Class Of Functions

Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

ON THE PRESENCE OF NORMALLY ATTRACTING MANIFOLDS CONTAINING PERIODIC OR QUASIPERIODIC ORBITS IN COUPLED MAP LATTICES

1993 ◽  
Vol 03 (06) ◽  
pp. 1503-1514 ◽  
Author(s):  
CLAUDIO GIBERTI ◽  
CECILIA VERNIA
Keyword(s):  
Invariant Manifolds ◽  
Chaotic Maps ◽  
Period Doubling ◽  
Coupled Map Lattices ◽  
Saddle Node Bifurcation ◽  
Closed Curves ◽  
One Dimensional ◽  
Weak Hyperbolicity ◽  
Long Time ◽  
Almost All

The significant presence of normally attracting invariant manifolds, formed by closed curves or two-tori, is investigated in two-dimensional lattices of coupled chaotic maps. In the case of a manifold formed by closed curves, it contains symmetrically placed periodic orbits, with the property of a very weak hyperbolicity along the manifold itself. The resulting dynamics is an extremely slow relaxation to periodic behavior. Analogously, a manifold consisting of two-tori includes very weakly hyperbolic periodic (or quasiperiodic) orbits, which in this case also implies quite a long time before any solution approaches periodicity or quasiperiodicity. The normally attracting manifolds and the contained weak attractors can undergo several global bifurcations. Some of them, including saddle-node bifurcation, period-doubling and Hopf bifurcation, are illustrated. Almost all the asymptotic solutions that we discuss have flat rows or flat columns, which means that they can occur also in one-dimensional lattices.

scholarly journals On the structure of the set of solutions of the Darboux problem for hyperbolic equations

1986 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
F. S. de Blasi ◽  
J. Myjak
Keyword(s):  
Integrable Function ◽  
Hyperbolic Equations ◽  
Continuous Functions ◽  
Darboux Problem ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Image Position ◽  
Almost All

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

Sign in / Sign up

Export Citation Format

Share Document