The orbits of generalized derivatives

Abstract

The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al [5]. Quasiregular mappings are only differentiable almost everywhere, and so the infinitesimal space generalizes the notion of derivative to this class of mappings. In this paper, we show that the infinitesimal space is either simple, that is, it consists of only one mapping, or it contains uncountable many. To achieve this, we define the orbit of a given point as its image under all elements of the infinitesimal space. We prove that this orbit is a compact and connected subset of R n \ {0} and moreover, every such set can be realized as an orbit space in dimension two. We conclude with some examples exhibiting features of orbits.

On the mean radius of quasiconformal mappings

“…In this paper, we will study the regularity properties of ρ f itself. It is clear from the definition that ρ f is increasing and, moreover, it was shown in the proof of [4,Lemma 3.1] that ρ f is continuous as a consequence of the Lusin (N) condition. It follows from a standard result in analysis that, as a continuous monotonic real function, ρ f is differentiable almost everywhere.…”

“…The mean radius was used by Gutlyanskii et al [5] to construct generalized derivatives of quasiregular mappings, a topic which has been studied more recently in papers of the first named author and Wallis [4] and the authors [3]. In this paper, we will study the regularity properties of ρ f itself.…”

A Complete Realization of the orbits of generalized derivatives of Quasiregular Mappings

“…This property was shown by the first author and Wallis [6] to lead to a dichotomy: either the infinitesimal space contains one generalized derivative, or contains uncountably many. This was acheived by looking at the orbit of a point under all elements in the infinitesimal space and showing that if it contains more than one point, then it contains a continuum that must necessarily be contained in R n \ {0}.…”

“…The normal family machinery mentioned above is then used to conclude that generalized derivatives always exist. Generalized derivatives have been studied in [3,5,6,13].…”

“…Conversely, it was also proved in [6] that in dimension two, any compact connected set in R 2 \ {0} arises as an orbit space of some quasiconformal map. The methods used here strongly relied on computations involving the complex dilatation of a quasiconformal map, a tool only available in the plane.…”