0000000000064455
AUTHOR
Rami Luisto
Mappings of Finite Distortion : Compactness of the Branch Set
We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to an (n - 2)-dimensional torus and distortion arbitrarily close to the asymptotic bound. Peer reviewed
On proper branched coverings and a question of Vuorinen
We study global injectivity of proper branched coverings from the open Euclidean n$n$-ball onto an open subset of the Euclidean n$n$-space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when n=3$n=3$ or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen. Peer reviewed
A Newman property for BLD-mappings
We define a Newman property for BLD-mappings and prove that for a BLD-mapping between generalized manifolds equipped with complete path-metrics, this property is equivalent to the branch set being porous when the codomain is LLC. peerReviewed
On BLD-mappings with small distortion
We show that every $$L$$ -BLD-mapping in a domain of $$\mathbb {R}^{n}$$ is a local homeomorphism if $$L < \sqrt{2}$$ or $$K_I(f) < 2$$ . These bounds are sharp as shown by a winding map.
Stoïlow’s theorem revisited
Stoilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z -> z(k) and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps. (C) 2019 Elsevier GmbH. All rights reserved. Peer reviewed
Open and Discrete Maps with Piecewise Linear Branch Set Images are Piecewise Linear Maps
The image of the branch set of a piecewise linear (PL)‐branched cover between PL 𝑛n‐manifolds is a simplicial (𝑛−2)(n−2)‐complex. We demonstrate that the reverse implication also holds: an open and discrete map 𝑓:𝕊𝑛→𝕊𝑛f:Sn→Sn with the image of the branch set contained in a simplicial (𝑛−2)(n−2)‐complex is equivalent up to homeomorphism to a PL‐branched cover. peerReviewed