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2 changed files with 22 additions and 14 deletions
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@ -24,7 +24,7 @@ $X^{X}$ is a compact Hausdorff space.
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is continuous:
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is continuous:
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Consider $\{f : f \circ f_0 \in U_{\epsilon}(x,y)\}$.
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Consider $\{f : f \circ f_0 \in U_{\epsilon}(x,y)\}$.
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We have $ff_0 \in U_{\epsilon}(x,y)$
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We have $f \circ f_0 \in U_{\epsilon}(x,y)$
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iff $f \in U_\epsilon(x,f_0(y))$.
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iff $f \in U_\epsilon(x,f_0(y))$.
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\item Fix $x_0 \in X$.
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\item Fix $x_0 \in X$.
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Then $f \mapsto f(x_0)$ is continuous.
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Then $f \mapsto f(x_0)$ is continuous.
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@ -43,12 +43,15 @@ and take the closure in $X^X$.
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\end{definition}
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\end{definition}
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$E(X,T)$ is compact and Hausdorff,
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$E(X,T)$ is compact and Hausdorff,
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since $X^X$ has these properties.
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since $X^X$ has these properties.
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% TODO THINK ABOUT THIS
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Properties of $(X,T)$ translate to properties of $E(X,T)$:
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\gist{
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\begin{goal}
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Properties of $(X,T)$ translate to properties of $E(X,T)$:
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We want to show that if $(X,T)$ is distal,
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\begin{goal}
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then $E(X,T)$ is a group.
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We want to show that if $(X,T)$ is distal,
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\end{goal}
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then $E(X,T)$ is a group.
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\end{goal}
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}{}
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\begin{proposition}
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\begin{proposition}
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$E(X,T)$ is a semigroup,
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$E(X,T)$ is a semigroup,
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@ -59,11 +62,14 @@ Properties of $(X,T)$ translate to properties of $E(X,T)$:
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Take $t \in T$. We want to show that $tG \subseteq G$,
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Take $t \in T$. We want to show that $tG \subseteq G$,
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i.e.~for all $h \in G$ we have $th \in G$.
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i.e.~for all $h \in G$ we have $th \in G$.
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We have that $t^{-1}G$ is compact,
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\gist{
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since $t^{-1}$ is continuous
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We have that $t^{-1}G$ is compact,
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and $G$ is compact.
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since $t^{-1}$ is continuous
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and $G$ is compact.
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}{$t^{-1}G$ is compact.}
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It is $T \subseteq t^{-1}G$ since $T \ni s = t^{-1}\underbrace{(ts)}_{\in G}$.
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Then $T \subseteq t^{-1}G$ since $T \ni s = t^{-1}\underbrace{(ts)}_{\in G}$.
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So $G = \overline{T} \subseteq t^{-1}G$.
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So $G = \overline{T} \subseteq t^{-1}G$.
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Hence $tG \subseteq G$.
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Hence $tG \subseteq G$.
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@ -91,8 +97,8 @@ Properties of $(X,T)$ translate to properties of $E(X,T)$:
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\begin{definition}
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\begin{definition}
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A \vocab{compact semigroup} $S$
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A \vocab{compact semigroup} $S$
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is a nonempty semigroup with a compact
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is a nonempty semigroup\footnote{may not contain inverses or the identity}
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Hausdorff topology,
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with a compact Hausdorff topology,
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such that $S \ni x \mapsto xs$ is continuous for all $s$.
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such that $S \ni x \mapsto xs$ is continuous for all $s$.
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\end{definition}
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\end{definition}
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\begin{example}
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\begin{example}
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@ -125,8 +131,8 @@ Properties of $(X,T)$ translate to properties of $E(X,T)$:
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\end{proof}
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\end{proof}
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The \yaref{lem:ellisnumakura} is not very interesting for $E(X,T)$,
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The \yaref{lem:ellisnumakura} is not very interesting for $E(X,T)$,
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since we already know that it has an identity,
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since we already know that it has an identity.
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in fact we have chosen $R = \{1\}$ in the proof.
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%in fact we might have chosen $R = \{1\}$ in the proof.
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But it is interesting for other semigroups.
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But it is interesting for other semigroups.
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@ -59,6 +59,8 @@
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and since $B$ is Hausdorff, compact subsets of $B$ are closed.
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and since $B$ is Hausdorff, compact subsets of $B$ are closed.
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\end{subproof}
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\end{subproof}
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\nr 1
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Let $(X,T)$ be a flow and $G = E(X,T)$ its Ellis semigroup.
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Let $(X,T)$ be a flow and $G = E(X,T)$ its Ellis semigroup.
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Let $d$ be a compatible metric on $X$.
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Let $d$ be a compatible metric on $X$.
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\begin{enumerate}[(a)]
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\begin{enumerate}[(a)]
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