12 citations · 32 across the 20 of their papers we have counts for
23 papers
Quasisymmetric rectifiability of uniformly disconnected sets
Jacob Honeycutt, Vyron Vellis
We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological…
Hölder spiral arcs
Efstathios Konstantinos Chrontsios Garitsis, Vyron Vellis
We establish a quantitative necessary and sufficient condition for a spiral arc to be a Hölder arc. The class of spiral arcs contains the polynomial spirals studied by Fraser, and…
Universal quasiconformal trees
Efstathios Konstantinos Chrontsios Garitsis, Fotis Ioannidis, Vyron Vellis
A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer ,…
Hölder curves with exotic tangent spaces
Eve Shaw, Vyron Vellis
An important implication of Rademacher's Differentiation Theorem is that every Lipschitz curve infinitesimally looks like a line at almost all of its points in the sense that a…
Bi-Lipschitz arcs in metric spaces with controlled geometry
Jacob Honeycutt, Vyron Vellis, Scott Zimmerman
We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supportin…
Parametrizability of infinitely generated attractors
Eve Shaw, Vyron Vellis
An infinite iterated function system (IIFS) is a countable collection of contraction maps on a compact metric space. In this paper we study the conditions under which the attractor…
Genus Cantor sets and germane Julia sets
Alastair N. Fletcher, Daniel Stoertz, Vyron Vellis
The primary aim of this paper is to give topological obstructions to Cantor sets in being Julia sets of uniformly quasiregular mappings. Our main tool is the genus o…
Time complexity of the Analyst's Traveling Salesman algorithm
Anthony Ramirez, Vyron Vellis
The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of is contained on a curve of finite length. We show that for…
Bi-Lipschitz embeddings of quasiconformal trees
Guy C. David, Sylvester Eriksson-Bique, Vyron Vellis
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show…
Decomposing Multitwists
Alastair N. Fletcher, Vyron Vellis
The Decomposition Problem in the class is to decompose any bi-Lipschitz map as a composition of finitely many maps of arbitrar…
Bi-Lipschitz geometry of quasiconformal trees
Guy C. David, Vyron Vellis
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry…
On uniformly disconnected Julia sets
Alastair N. Fletcher, Vyron Vellis
It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes,…
Hölder parameterization of iterated function systems and a self-affine phenomenon
Matthew Badger, Vyron Vellis
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 attra…
Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces
Vasileios Chousionis, Sean Li, Vyron Vellis +1
The Heisenberg group equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedd…
Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem
Matthew Badger, Lisa Naples, Vyron Vellis
We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image o…
Geometry of measures in real dimensions via Hölder parameterizations
Matthew Badger, Vyron Vellis
We investigate the influence that -dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in when is a real number between $0…
Uniformization of Cantor sets with bounded geometry
Vyron Vellis
In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particula…
Quasisymmetric extension on the real line
Vyron Vellis
We give a geometric characterization of the sets that satisfy the following property: every quasisymmetric embedding extends to a quas…
Quasiconformal non-parametrizability of almost smooth spheres
Pekka Pankka, Vyron Vellis
We show that, for each , there exists a smooth Riemannian metric on a punctured sphere for which the associated length metric extends to…
Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces
Matthew Romney, Vyron Vellis
We show that, for all , the generalized Grushin plane is bi-Lipschitz homeomorphic to a -dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2…
Weak chord-arc curves and double-dome quasisymmetric spheres
Vyron Vellis
Let be a planar Jordan domain and . We consider double-dome-like surfaces over where the height of the surface over any point equa…
Quasisymmetric spheres over Jordan domains
Vyron Vellis, Jang-Mei Wu
Let be a planar Jordan domain. We consider double-dome-like surfaces defined by graphs of functions of over . The goal is to find the right co…
Sets of constant distance from a Jordan curve
Vyron Vellis, Jang-Mei Wu
We study the -level sets of the signed distance function to a planar Jordan curve , and ask what properties of ensure that the -level sets are Jordan curves, or unifor…