A complete realization of the orbits of generalized derivatives of quasiregular mappings (Q2024622)

Quasiregular mappings are differentiable only almost everywhere. This paper mainly studies the behavior of quasiregular mappings at points of non-differentiability. Let \(f: U\rightarrow \mathbb R^n\) be a quasiregular mapping (with \(U\subset \mathbb R^n\) a domain) and \(x_0\in U\). There is a notion of generalized derivative of \(f\) at \(x_0\), which is obtained by modifying the denominator in the definition of derivative and then considering subsequential limits. The generalized derivatives of \(f\) at \(x_0\) is not unique. The infinitesimal space of \(f\) at \(x_0\) is the set of generalized derivatives of \(f\) at \(x_0\). So the infinitesimal space is a collection of mappings from \(\mathbb R^n\) to itself. For any \(x\) in \(\mathbb R^n\), the orbit space of \(x\) (with respect to \(f\) and \(x_0\)) is the set of images of \(x\) under the mappings in the infinitesimal space. It is known that the orbit space is always a compact connected subset of \(\mathbb R^n\backslash\{0\}\). One of the main results in this paper is a construction of quasiregular mappings (actually quasiconformal) with any given compact connected subset of \(\mathbb R^n \backslash\{0\}\) as an orbit space. To construct the examples, the authors introduce the so-called Zorich transform (another main contribution of the paper). Roughly the Zorich transform of a mapping \(f\) is the conjugation of \(f\) by a Zorich map. Since a Zorich map is not a bijection, one needs to restrict the domain of the Zorich map. While Zorich maps are analogues in higher-dimensional Euclidean spaces of the exponential map on the complex plane, the Zorich transform is a generalization of the logarithmic transform.