On BLD-mappings with small distortion

Abstract

We show that every L-BLD-mapping in a domain of R n is a local homeomorphism if L < √ 2 or KI (f ) < 2. These bounds are sharp as shown by a winding map.

Remarks on Martio's conjecture

“…Especially, these mappings form a subclass of quasiregular mappings. Thus, Theorem 1.2 improves the result in [KLT21] by relaxing the boundedness conditions of the Jacobian determinant. In this article we also obtain several alternative proofs for the local homeomorphism property of BLD-mappings with a small inner dilatation, see Remark 2.2 and 4.1.…”

“…A simple construction shows that the integrability condition of the reciprocal of the Jacobian determinant in Theorem 1.2 is actually sharp in the planar case, see Example 3.2. Furthermore, Theorem 1.2 can be considered as a continuation of the author's earlier joint work [KLT21] with Kauranen and Luisto where the strong Martio's conjecture was verified for mappings of bounded length distortion (abbr. BLD-mappings).…”

“…(1) One can use the upper estimate in (2.1) and radiality properties of BLD-mappings to show that f is a local homemorphism, see [KLT21]. (2) Proposition 2.1 implies the local homeomorphism property of f .…”

A self-contained proof to Martio's conjecture in the class of BLD-maps

“…The proof of this estimate requires several layers of preliminary results which makes it technical and rather lengthy. In this article we prove the strong Martio's conjecture for BLD-mappings without any use of the estimate (1.4) by providing a rather short and self-contained proof for the following result from [KLT21] which is valid also in the planar case: Theorem 1.2 (Kauranen, Luisto, and Tengvall, 2021). Every non-constant BLD-mapping…”