Generalized Ulam-von Neumann transformations.
Item
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Title
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Generalized Ulam-von Neumann transformations.
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Identifier
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AAI9029947
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identifier
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9029947
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Creator
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Jiang, Yunping.
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Contributor
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Adviser: Dennis P. Sullivan
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Date
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1990
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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The first part of this thesis uses a singular change of metric on an interval to study mappings on the boundary of hyperbolicity. The change of metric makes it possible to apply the theory of expanding mappings. We classify these mappings up to smooth equivalence, by showing that all the eigenvalues at the periodic points, the type of power law at the critical point and a quantity which we call the asymmetry at the critical point form a complete set of invariants.;In the second part of this thesis, we study hyperbolic mappings depending on a parameter {dollar}\varepsilon{dollar}. Each of them has an invariant Cantor set. As {dollar}\varepsilon{dollar} tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function of the invariant Cantor set as {dollar}\varepsilon{dollar} goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed {dollar}\sqrt{lcub}\varepsilon{rcub}{dollar}. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting mapping with respect to the Ulam-von Neumann type metric.;A key technical result of this thesis is the uniform ({dollar}\alpha{dollar} + {dollar}\gamma{dollar})-Koebe distortion lemma (Lemma 2.7). Its proof combines the ideas of the distortion lemmas of Denjoy and Koebe.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
