Fletcher, Alastair N.
Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
This dissertation investigates the role that a new tool called the Zorich transform plays in quasiregular dynamics as a generalization of the logarithmic transform in complex dynamics. In particular we use the Zorich transform to construct analogues of the logarithmic spiral maps and interpolation between radial stretch maps. These constructions are then used to completely classify the orbit space of a quasiregular map. Also, conditions are given in which a quasiregular map $f:D\to\R^n$, where $D\subset\R^n$ is a domain, that is quasiconformal in a neighborhood of a geometrically attracting fixed point can be conjugated by a quasiconformal map to the asymptotic representation of $f$ in a neighborhood of the fixed point. To find such a quasiconformal map, we construct a sequence of quasiconformal maps that converge to the desired map in a neighborhood of the fixed point.
Pratscher, Jacob A., "The Zorich Transform and Generalizing Koenigs Linearization Theorem to Quasiregular Maps" (2021). Graduate Research Theses & Dissertations. 7566.
Northern Illinois University
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