Generic behaviour of a measure preserving transformation

Measure preserving transformation;Measurable functions

Etedadialiabadi, Mahmood

关键词: Measure preserving transformation;    Measurable functions;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/99247/ETEDADIALIABADI-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 Junco--Lema\'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 non-zero integers. The orthogonality conditions introduced by Del Junco--Lema\'nczyk motivates a condition, which we denote by the DL-condition, on continuous unitary representations of $L^0(X,\mathbb{T})$. We show that the probabilistic (in terms of category) statement of the DL-condition 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 Junco--Lema\'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 $X$.