Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item Immersions of surfaces(Montana State University - Bozeman, College of Letters & Science, 2022) Howard, Adam Jacob; Co-chairs, Graduate Committee: David Ayala and Ryan GradyTo determine the existence of a regular homotopy between two immersions, f, g : M --> N, is equivalent to showing that they lie in the same path component of the space Imm(M, N). We identify the connected components, pi 0 Imm(W g, M), of the space of immersions from a closed, orientable, genus-g surface W g into a parallelizable manifold M. We also identify the higher homotopy groups of Imm(W g, M) in terms of the homotopy groups of M and the Stiefel space V 2 (n). We then use this work to characterize immersions from tori into hyperbolic manifolds as self covers of a tubular neighborhood of a closed geodesic up to regular homotopy. Finally, we identify the homotopy-type of the space of framed immersions from the torus to itself.Item Markov partitions and sofic codings for Anosov diffeomorphisms of nilmanifolds(Montana State University - Bozeman, College of Letters & Science, 2020) Fink, Eric Raymond; Chairperson, Graduate Committee: Jaroslaw KwapiszGiven an Anosov diffeomorphism of a compact manifold, the existence of a Markov partition and the associated conjugate symbolic dynamical system has been known for over fifty years by a celebrated result of Sinai, subsequently extended by Bowen. Building upon the work done by many authors in the context of hyperbolic toral automorphisms, we give an explicit arithmetic construction of sofic codings and Markov partitions for Anosov diffeomorphisms of nilmanifolds. Arising as quotients of nilpotent Lie groups by discrete and co-compact subgroups (lattices), nilmanifolds are conjecturally the only manifolds admitting Anosov diffeomorphisms, up to a finite covering.