Abstract
Generalized derivatives and infinitesimal spaces generalize the idea of derivatives to mappings which need not be differentiable. It is particularly powerful in the context of quasiregular mappings, where normal family arguments imply generalized derivatives always exist. The main result of this paper is to show that if f is any uniformly quasiregular mapping with \(x_0\) a topologically attracting or repelling fixed point, at which f is locally injective, then f may be conjugated to a uniformly quasiregular mapping g with fixed point 0 and so that the infinitesimal space of g at 0 contains uncountably many elements. This should be contrasted with the fact that f (and also g) is conjugate to \(x\,\mapsto\, x/2\) or \(x\,\mapsto\, 2x\) in the attracting or repelling cases respectively.
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References
Ahlfors, L. 1996. Lectures on quasiconformal mappings, 1st edn, Van Nostrand mathematical studies, vol 10. Canada: D. Van Norstrand Company.
Fletcher, A., and V. Markovic. 2007. Quasiconformal mappings and Teichmüller theory, OUP.
Gutlyanskii, V., O. Martio, V. Ryazanov, and M. Vuorinen. 2000. Infinitesimal geometry of quasiregular mappings. Ann. Acad. Sci. Fenn. 25: 101–130.
Hinkkanen, A., and G. Martin. 1996. Attractors in quasiregular semigroups. In: Proceedings of XVI Nevanlinna colloquium, eds. I. Laine and O. Martio, 135–141. Berlin: de Gruyter.
Hinkkanen, A., G. Martin, and V. Mayer. 2004. Local dynamics of uniformly quasiregular mappings. Math. Scand. 95(1): 80–100.
Iwaniec, T., and G. Martin. 2001. Geometric function theory and non-linear analysis. Oxford: Oxford University Press.
Rickman, S. 1993. Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26. Berlin: Springer.
Tukia, P., and J. Väisäla. 1982. Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(2): 303–342.
Acknowledgements
This work was supported by a Grant from the Simons Foundation (#352034, Alastair Fletcher). This work arose from a project given by the first author at a graduate level course at Northern Illinois University.
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Fletcher, A., Macclure, D., Waterman, J. et al. On the infinitesimal space of UQR mappings. J Anal 24, 67–81 (2016). https://doi.org/10.1007/s41478-016-0003-8
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DOI: https://doi.org/10.1007/s41478-016-0003-8