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Title:  Generic behaviour of a measure preserving transformation 
Author(s):  Etedadialiabadi, Mahmood 
Director of Research:  Solecki, Slawomir 
Doctoral Committee Chair(s):  van den Dries, Lou 
Doctoral Committee Member(s):  Hieronymi, Philipp; Tserunyan, Anush 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Measure preserving transformation
Measurable functions 
Abstract:  We study two different problems: generic behavior of a measure preserving transformation and extending partial isometries of a compact metric space. In Chapter $1$, we consider a result of Del JuncoLema\'nczyk [\ref{DL_B}] which states that a generic measure preserving transformation satisfies a certain orthogonality conditions, and a result of Solecki [\ref{S1_B}] which states that every continuous unitary representations of $L^0(X,\mathbb{T})$ is a direct sum of action by multiplication on measure spaces $(X^{\kappa},\lambda_\kappa)$ where $\kappa$ is an increasing finite sequence of nonzero integers. The orthogonality conditions introduced by Del JuncoLema\'nczyk motivates a condition, which we denote by the DLcondition, on continuous unitary representations of $L^0(X,\mathbb{T})$. We show that the probabilistic (in terms of category) statement of the DLcondition translates to some deterministic orthogonality conditions on the measures $\lambda_\kappa$. Also, we show a certain notion of disjointness for generic functions in $L^0(\mathbb{T})$ and a similar orthogonality conditions to the result of Del JuncoLema\'nczyk for a generic unitary operator on a Hilbert space $H$. In Chapter $2$, we show that for every $\epsilon>0$, every compact metric space $X$ can be extended to another compact metric space, $Y$, such that every partial isometry of $X$ extends to an isometry of $Y$ with $\epsilon$distortion. Furthermore, we show that the problem of extending partial isometries of a compact metric space, $X$, to isometries of another compact metric space, $X\subseteq Y$, is equivalent to extending partial isometries of $X$ to certain functions in $\operatorname{Homeo}(Y)$ that look like isometries from the point of view of $X$. 
Issue Date:  20171208 
Type:  Text 
URI:  http://hdl.handle.net/2142/99247 
Rights Information:  Copyright 2017 Mahmood Etedadialiabadi 
Date Available in IDEALS:  20180313 20200314 
Date Deposited:  201712 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois