Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 485-495

THE ORBITS OF GENERALIZED DERIVATIVES

Alastair Fletcher and Ben Wallis

Northern Illinois University, Department of Mathematical Sciences
DeKalb, IL 60115-2888, U.S.A.; fletcher 'at' math.niu.edu

Northern Illinois University, Department of Mathematical Sciences
DeKalb, IL 60115-2888, U.S.A.; benwallis 'at' live.com

Abstract. The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al [6]. 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 Rⁿ \ {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.

2010 Mathematics Subject Classification: Primary 30C65; Secondary 30C62.

Key words: Quasiregular mapping, generalized derivative, infinitesimal space.

Reference to this article: A. Fletcher and B. Wallis: The orbits of generalized derivatives. Ann. Acad. Sci. Fenn. Math. 44 (2019), 485-495.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4429