Uniformly quasiregular mappings on elliptic riemannian manifolds

Loading...
Thumbnail Image
Journal Title
Journal ISSN
Volume Title
Doctoral thesis (monograph)
Checking the digitized thesis and permission for publishing
Instructions for the author
Date
2008
Major/Subject
Mcode
Degree programme
Language
en
Pages
Verkkokirja (948 KB, 69 s.)
Series
Abstract
In this thesis we study uniformly quasiregular (abbreviated uqr) mappings on compact riemannian manifolds. We prove that the Julia set Jf of a uqr-mapping f : Mn → Mn on a compact riemannian manifold Mn is non-empty. We extend the rescaling principle from euclidean spaces to families of quasiregular mappings between a euclidean space and a riemannian manifold. Thus we can use the rescaling principle to obtain from the family (f j) of the iterates of the uqr-mapping f on the manifold M a quasiregular mapping g : ℝn → Mn defined in the whole space ℝn. Combining these results, we notice that if there exists a uqr-mapping on a compact riemannian manifold Mn, there exists a quasiregular mapping g : ℝn → Mn. In other words, the manifold Mn is quasiregularly elliptic. The converse result is proved in three dimensions: we construct a uqr-mapping on each oriented quasiregularly elliptic 3-dimensional compact riemannian manifold.
Description
Thesis advisor
Mathematics
Keywords
uniformly quasiregular mappings, riemannian manifolds, elliptic manifolds, Zalcman's lemma, Julia set, Lattès mappings
Other note
Citation