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Josia Pietsch 2023-12-14 00:50:44 +01:00
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inputs/tutorial_09.tex
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@ -14,9 +14,10 @@
\subsection{Sheet 8} \subsection{Sheet 8}
Material on topological dynamic Material on topological dynamics:
\begin{itemize} \begin{itemize}
\item Terence Tao's notes on ergodic theory 254 A. \item Terence Tao's notes on ergodic theory 254A:
\url{https://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/}
\item Furstenberg (uses very different notation!). \item Furstenberg (uses very different notation!).
\end{itemize} \end{itemize}
@ -148,7 +149,7 @@ amounts to a finite number of conditions on the preimage.
\iff&\exists x \in X.~\forall U \overset{\text{open}}{\subseteq} X.~ \iff&\exists x \in X.~\forall U \overset{\text{open}}{\subseteq} X.~
\exists z \in \Z.~f^z(x) \in U. \exists z \in \Z.~f^z(x) \in U.
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
\item Not solved. \item \todo{TODO}
\item It suffices to check the condition from part (b) \item It suffices to check the condition from part (b)
for open sets $U$ of a countable basis for open sets $U$ of a countable basis
and points $x \in X$ belonging to a countable dense subset. and points $x \in X$ belonging to a countable dense subset.
@ -156,4 +157,3 @@ amounts to a finite number of conditions on the preimage.
gives that $D$ is Borel. gives that $D$ is Borel.
\end{itemize} \end{itemize}