arXiv research

For each Cantor set C in $R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $R^{3}$ with complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple c…

New criteria for Cantor set tameness and wildness via projections.

The paper studies the moduli space of generalized Cantor sets and their properties.

New proof shows abelian Cantor groups can act on spaces.

The paper studies Lipschitz equivalence of Cantor sets via polynomial irreducibility.

New topological object connects complex dynamics and topology.

New embeddings show answer to Baker-Laidacker question can be yes or no.

We found a new simple family of Cantor sets whose projections are one-dimensional.

The study bounds the complexity of meromorphic differentials' directions.

Sticky Cantor sets in higher dimensions can't be easily moved away.

Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…

First constructed genus 2 Cantor set in 3D space.

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=…

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise…

All projections of typical Cantor sets in high dimensions are Cantor sets.

Cantor Riemannium is a new type of space from holomorphic germs.

Non-ergodic geodesic flow on Cantor tree surfaces found.

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

New Cantor sets with high-dimensional projections discovered.

Study of limits of Cantor and \sier sets, showing homeomorphic spaces.

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…

Hyperbolic 3-manifolds can be approximated by removing Cantor sets from the 3-sphere.

Researchers create a Fredholm module on fractal shapes like the Cantor set.

The paper studies identities for Cantor sets and their Hausdorff dimension.

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds,…

We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.

The study confirms most Cantor sets are in general position for all projections.

The study examines distortion in specific homeomorphisms of Cantor sets.

New minimal surfaces found with Cantor ends in convex domains.

New proof shows set of saddle connections has finite Cantor-Bendixson rank.

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.

The study allocates income based on journal impact and coauthorship contributions.

We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements …

The paper proves identities for hyperconvex Anosov representations and their applications to Cantor sets.

This paper introduces Hausdorff measure and its applications in fractal geometry.

Curious examples of lifting spaces not as inverse limits of covering spaces.

For every finitely generated abelian group G, we construct an irreducible open 3-manifold $M_{G}$ whose end set is homeomorphic to a Cantor set and with end homogeneity group of $M_{G}$ isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-m…

Uncountably many fibrations found on genus 2 handlebody.

CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the…

Study shows similar result to Margulis for Cantor set homeomorphisms.

Every normal subgroup of Cantor tree's mapping class group is geometric.

The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.

The paper constructs wild Cantor sets in high dimensions.

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

Gromov boundary described for ray graph action.

In an earlier paper, we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz. the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundamental conjecture in nonc…

New method uses Cantor embeddings and Wasserstein distances to analyze predictive states in time series data.