New topological object connects complex dynamics and topology.
problem Understanding dynamics of complex maps.
method Axiomatic characterisation and ambient homeomorphism proof.
result Hairy Cantor sets are ambiently homeomorphic.
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…
We found a new simple family of Cantor sets whose projections are one-dimensional.
problem Finding simple Cantor sets with specific projection properties.
method Developed a new series of self-similar Cantor sets in R3. result All projections of these new Cantor sets are connected and one-dimensional.
First constructed genus 2 Cantor set in 3D space.
problem Constructing a geometrically self-similar Cantor set of genus 2.
method Geometrically self-similar construction in R3. result First uniformly quasiregular mapping with a genus 2 Cantor set Julia set.
In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the cont…
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…
New proof shows abelian Cantor groups can act on spaces.
problem Understanding actions of Cantor groups on metric spaces.
method Examined actions of abelian Cantor groups on metric spaces.
result Cantor groups can be abelian for n>1 in space actions.
All projections of typical Cantor sets in high dimensions are Cantor sets.
problem Whether all projections of a typical Cantor set in high dimensions are Cantor sets.
method Proving that for a dense Gδ subset of Cantor sets, all projections into non-zero linear subspaces are Cantor sets.
result There exists a dense Gδ subset of Cantor sets such that all projections into non-zero linear subspaces are Cantor sets.
New Cantor sets with high-dimensional projections discovered.
problem Understanding projections of Cantor sets in high dimensions.
method Construction and analysis of Cantor sets in Rn. result Cantor sets can be moved to have (n−2)-dimensional projections in (n−1)-planes. Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
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.
problem Approximating hyperbolic 3-manifolds using Cantor set complements in the 3-sphere.
method Using exhaustion by π1-injective sub-manifolds and removing Cantor sets. result Hyperbolic 3-manifolds can be geometrically approximated by removing Cantor sets from the 3-sphere.
The group of C1-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…
For each Cantor set C in R3, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in R3 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…
We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.
Study of limits of Cantor and \sier sets, showing homeomorphic spaces.
problem Understanding topological properties of certain geometric structures.
method Analyzing direct limits of embedded Cantor sets and \sier curves, showing homeomorphism.
result Morse boundaries of specific groups are homeomorphic to limits of Cantor and \sier sets.
The study confirms most Cantor sets are in general position for all projections.
problem Understanding the general position of Cantor sets under various projections.
method Proof of the theorem stated in the title.
result Most Cantor sets are in general position with respect to all projections.
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=…
We construct a large class of pathological n-dimensional topological spheres in Rn+1 by showing that for any Cantor set C⊂Rn+1 there is a topological embedding f:Sn→Rn+1 of the Sobolev class W1,n whose image contains the Cantor set C.
For every finitely generated abelian group G, we construct an irreducible open 3-manifold MG whose end set is homeomorphic to a Cantor set and with end homogeneity group of MG 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…
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 studies the moduli space of generalized Cantor sets and their properties.
problem Understanding the moduli space of generalized Cantor sets and their equivalence.
method Constructing generalized Cantor sets and studying their moduli space properties.
result There are uncountably many moduli spaces and most have vanishing volume.
New criteria for Cantor set tameness and wildness via projections.
problem Characterize dimensions of projections of Cantor sets.
method Geometric measure theory and Baire category theory.
result New criteria for Cantor set tameness and wildness.
The study examines distortion in specific homeomorphisms of Cantor sets.
problem Distortion in homeomorphisms of Cantor sets.
method Analyzes equivalence of conditions related to discontinuities and conjugacy.
result Elements are distorted if they satisfy certain conditions.
New minimal surfaces found with Cantor ends in convex domains.
problem Finding complex structures for minimal surfaces with Cantor ends.
method Proving existence of complete minimal surfaces with Cantor ends in minimally convex domains.
result Existence of a Cantor set whose complement forms a complete minimal surface.
The paper proves identities for hyperconvex Anosov representations and their applications to Cantor sets.
problem Establishing identities for hyperconvex Anosov representations.
method Analyzing holomorphic families of Cantor non-conformal repellers and studying series identities.
result The series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is less than 1.
CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.
problem Finding CMC-1 surfaces on compact Riemann surfaces.
method Uniform approximation theorems for holomorphic null curves in C2imesC∗. result Cantor set removal allows for CMCext−1 immersions in hyperbolic and de Sitter spaces. Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
A subset of Rd is called "sticky" if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in Rd for each d≥4.
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.
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
The paper constructs wild Cantor sets in high dimensions.
problem Embedding Cantor sets in high-dimensional spaces.
method Constructing embeddings of Cantor sets in \(\mathbb{R}^n\).
result Embeddings create pairwise wild Cantor sets that are ambiently incomparable.
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence…
Infinite clique of rays in plane minus Cantor set.
problem Understanding the mapping class group of plane minus Cantor set.
method Using a graph of loops and cliques of high-filling rays.
result Construction of an infinite clique of high-filling rays.
Every countable compact subset of sphere is tame.
problem Characterizing compact subsets of spheres.
method Proving homeomorphic complements imply homeomorphic subsets.
result Wild subspaces like Antoine contain Cantor sets.
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
A little complement concerning the dynamics of non-metric manifolds is provided, by showing that any flow on an ω-bounded surface with non-zero Euler character has a fixed point.
Every normal subgroup of Cantor tree's mapping class group is geometric.
problem Characterizing normal subgroups of mapping class groups.
method Generalized curve graph study and adaptation of Brendle-Margalit strategy.
result All normal subgroups of Cantor tree's mapping class group are geometric.
Two graph homologies help compute embedding space.
problem Computing the rational homotopy group of long embeddings.
method Invented two graph homologies and constructed a map between them.
result A monomorphism from top hairy graph homology to top BCR graph homology.
The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.
problem Understanding the structure and properties of homeomorphism groups of self-similar 2-manifolds.
method Survey of recent results, exposition of classical results, treatment of stable sets, and proof of new theorems.
result Characterization of homeomorphisms of perfectly self-similar 2-manifolds and extensions of existing results.
New actions of mapping class groups on circles are classified.
problem Understanding actions of mapping class groups on circles.
method Minimal actions on the simple circle.
result Every action of the mapping class group on the circle is semi-conjugate to a unique minimal action on the simple circle.
We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendix…
The study bounds the complexity of meromorphic differentials' directions.
problem Understanding the descriptive complexity of meromorphic differentials.
method Geometric lemma and topological analysis of saddle connections.
result Sharp upper bound on the Cantor-Bendixson rank of meromorphic differentials.
Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.
problem Characterizing SL(2,R) representations on a once-punctured torus.
method Introduction of spectrum as a subset of projective measured laminations, analysis of dynamics of cocycles.
result Spectrum of a generic representation on a once-punctured torus is a Cantor set.
Groups acting on product trees are boundary rigid.
problem Understanding boundary rigidity of groups acting on product trees.
method Analyzing geometric actions and visual boundaries of groups.
result Visual boundaries of CAT(0) spaces are homeomorphic to a join of two Cantor sets.
The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray g…
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.