Rigidity of Schottky sets
By M. Bonk, Bruce Kleiner, S. Merenkov. 2009
64 citations recorded by the source corpus.
We call the complement of a union of at least three disjoint (round) open balls in the unit sphere ${\Bbb S}^n$ a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a M\"obius transformation on ${\Bbb S}^n$. In the other direction we show that every Schottky set in ${\Bbb S}^2$ of positive measure admits nontrivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1dc574d47b94594acf6a23cce7a79621dd19b80c
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In Applied Mathematics
In Geometry and Topology
- S. Merenkov
- M. Bonk
- M. Lyubich
- Bruce Kleiner
- Kai Rajala
- Zoltán M. Balogh
- M. Bourdon
- Matias Carrasco Piaggio
Citations in the corpus
- Uniformization of Sierpiński carpets in the plane (2010, background): “where {Bi : i∈ I} is a collection of pairwise disjoint round open disks. One can define the notion of a Schottky set similarly for subsets of higher-dimensional spheres. This concept was introduced in [BKM] (with the additional requirement #I ≥ 3). By [BKM, Prop. 2.2] every Schottky set is 1-LLC(the ”
- Quasisymmetric rigidity of square Sierpinski carpets (2011, background): “t sends one family to the other family. The proof of Corollary 5.2 relies on some previous work. In Section 4 we collect certain uniformization and rigidity results that were established in [Bo1] and [BKM], and derive some consequences. Among these results is Proposition 4.9 which gives an explicit ”
- Quasisymmetries of Sierpi\'nski carpet Julia sets (2014, background): “ discussion is applicable, e.g., to the previously mentioned hyperbolic rational map f 0(z) = z2 1=(16z2) whose Julia set is Sierpinski carpet. 1.2. Previous rigidity results for quasisymmetries. In [BKM] the authors considered quasisymmetric rigidity questions for Schottky sets, i.e., subsets of th”