arXiv subjects
Publications and source records attributed to Ville Tengvall.
We provide a self-contained proof to so-called Martio's conjecture in the class of mappings of bounded length distortion. Unlike the earlier proofs, our proof is not based on the modulus of continuity estimate of Martio from 1970.
We introduce a certain integrability condition for the reciprocal of the Jacobian determinant which guarantees the local homeomorphism property of quasiregular mappings with a small inner dilatation. This condition turns out to be sharp in the planar case. We also show that every branch point of a quasiregular mapping with a small inner dilatation is a Lebesgue point of the differential matrix of the mapping.
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.
We study global injectivity of proper branched coverings defined on the Euclidean $n$-ball in the case when the branch set is compact. In particular we show that such mappings are homeomorphisms when $n=3$ or when the branch set is empty. This proves the corresponding cases of a question of Vuorinen from [Vuo79].
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 $(n-2)$-dimensional torus and distortion arbitrarily close to the asymptotic bound.