Genus of a Cantor Set

Željko, Matjaž

doi:10.1216/rmjm/1181069785

Cited by 13 publications

(14 citation statements)

References 9 publications

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“…We review the definition and some facts from [Ze05] about the local genus of a point in a Cantor set in S 3 . At the end of this section, we relate the local genus of a point x in a Cantor set C ⊂ S 3 to the genus of an end x of the complementary 3-manifold S 3 \ C. We use IntX and FrX to denote the topological interior and boundary of a subset X of a space Y .…”

Section: Local Genus Of Points In a Cantor Setmentioning

confidence: 99%

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Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

Garity¹,

Repovš²

2017

Mediterr. J. Math.

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We show that for every sequence $(n_i)$, where each $n_i$ is either an integer greater than 1 or is $\infty$, there exists a simply connected open 3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every $i$, the genus of end $e_i$ is equal to $n_i$. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in $S^3$. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus

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Section: Local Genus Of Points In a Cantor Setmentioning

confidence: 99%

“…Section 2 provides details about the local genus of points in a Cantor set in S 3 (introduced in [Ze05]) and relates this to the genus of an end of the 3-manifold that is the complement of the Cantor set. Section 3 gives two replacement constructions we will need in the construction of our examples.…”

Section: Introductionmentioning

confidence: 99%

Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

Garity¹,

Repovš²

2017

Mediterr. J. Math.

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“…The genus of a Cantor set. To further classify Cantor sets embedded in R 3 , Željko [25] introduced a homeomorphic invariant called the genus of a Cantor set. Informally, this non-negative integer gives the smallest genus of handlebodies that are required by any defining sequence of the Cantor set.…”

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confidence: 99%

“…There appear to be very few constructions of Cantor sets in R 3 which are proved to be of genus at least 2 in the literature. Željko's paper [25] does construct Cantor sets of every genus, including genus infinity, but is somewhat special in that there is one point of the Cantor set where the higher genus behavior happens, and elsewhere the Cantor set looks like an Antoine's necklace. A refinement of this construction, combined with a construction of Skora [24], is given in [12], although it is not shown that the genus is at least two.…”

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confidence: 99%

Genus 2 Cantor sets

Fletcher¹,

Daniel²

2020

Preprint

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“…Every Cantor set in R 3 has a defining sequence, proved by Armentrout [2] using different terminology, but the defining sequence is far from uniquely determined. Željko [25] introduced the notion of the genus of a Cantor set X as the infimum of the genus required in the handlebodies over all possible defining sequences for X. Then the genus of Antoine's necklace is 1 and the genus of a tame Cantor set is 0.…”

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confidence: 99%

Spider's webs of doughnuts

Fletcher,

Stoertz

2018

Preprint

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If f : R 3 → R 3 is a uniformly quasiregular mapping with Julia set J(f ) a genus g Cantor set, for g ≥ 1, then for any linearizer L at any repelling periodic point of f , the fast escaping set A(L) consists of a spiders' web structure containing embedded genus g tori on any sufficiently large scale. In other words, A(L) contains a spiders' web of doughnuts. This type of structure is specific to higher dimensions, and cannot happen for the fast escaping set of a transcendental entire function in the plane. We also show that if f : R n → R n is uqr, for n ≥ 2 and J(f ) is a Cantor set, then every periodic point is in J(f ) and is repelling.

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Genus of a Cantor Set

Cited by 13 publications

References 9 publications

Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus

Genus 2 Cantor sets

Spider's webs of doughnuts

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