Uniformly quasiregular mappings on elliptic riemannian manifolds
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Doctoral thesis (monograph)
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2008
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en
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Verkkokirja (948 KB, 69 s.)
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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
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MathematicsKeywords
uniformly quasiregular mappings, riemannian manifolds, elliptic manifolds, Zalcman's lemma, Julia set, Lattès mappings