# Red workbook, p3

### Transcript

• Def.
\begin{split} x,y \text{ proximal} & \Leftrightarrow \forall V \in \mathfrak{U}(\overbrace{\Delta}^{\text{Diag. in $X\times X$}}) \exists s \in S: (sx, sy) \in V \\ & \underbrace{\Leftrightarrow}_ {\text{topologie: diese Ueberd. bilden Umg.basis von $\Delta$ in $X^2$}} \forall (U_ i)_ {i=1}^m, \bigcup U_ i = X \text{ offen} \exists s \in S, i \leq m: sx, sy \in U_ i \end{split}
• Satz
• $x,y \text{ prox.} \Leftrightarrow \exists p \in \beta S: px = py$.
• Bsp 2 (fortg)
• Sei $S\leq Q, 1_ Q \in Q$ mit $1_ q \cdot s = s \cdot 1_ Q = s \forall s \in S$
• $U: = { z \in X: z(1_ Q) = 1}$. Also $U\subseteq X$ clopen [klar?]
• Sei $C\subseteq S: x:= \chi_ C$ (char. Fkt.)
• Dann $R(x,U) = C$
• [proof] $s\in R(x,U) \Leftrightarrow s \cdot x \in U \Leftrightarrow s \cdot x (1_ Q) = 1 \Leftrightarrow x(1_ Q s) = 1 \Leftrightarrow x(s) = 1 \Leftrightarrow X \in C$
• ! Also: jede Teilmanege als Rueckkehrmenge darstellbar.
• Def. $x$ unif. rekurrent $\Leftrightarrow R(x,U) \text{ synd. } \forall U \in \mathfrak{U}$
• Satz. $X$ DS ueber $S$, $y\in X$, $L\subseteq \beta S$ min. LID
a. $y$ unif. rekurrent;
b. $\exists p \in L: p\cdot y = y$
c. $\exists \epsilon \in L\cap E(\beta S): \epsilon \cdot y = y$
d. $\exists \epsilon \in L \cap E(\beta S), x\in X: \epsilon \cdot x = y$
e. $y\in \bigcup_ {M \text{min US}} M$
f. $y \in L \cdot y$.
• Beweis
• c=> a
• $U \in \mathfrak{U}(y), V\subseteq \bar{V} \subseteq U \text{ offen}, A=R(y,V) \in \epsilon$ [$\epsilon y = y$]
• $B = {s: s\cdot \epsilon \in \hat{A}} \subseteq S \text{ syndetisch [HS 4.39]} \Rightarrow B \subseteq R(y,U)$ [$s \epsilon \in \hat{A} \Rightarrow s\epsilon y = s y \in \bar{V} \subseteq U$]

### partial translation

• Definition.\begin{split} x,y \text{ proximal} & \Leftrightarrow \forall V \in \mathfrak{U}(\Delta) \exists s \in S: (sx, sy) \in V \\ & \Leftrightarrow \forall (U_ i)_ {i=1}^m, \bigcup U_ i = X \text{ open} \exists s \in S, i \leq m: sx, sy \in U_ i \end{split}
where $\Delta$ is the diagonal in $X\times X$; note that these coverings form a neighborhood basis of $\Delta$ in $X^2$
• Theorem
• $x,y \text{ proximal} \Leftrightarrow \exists p \in \beta S: px = py$.
• Example 2 (continued)
• Let $S\leq Q, 1_ Q \in Q$ with $1_ q \cdot s = s \cdot 1_ Q = s \forall s \in S$
• $U: = { z \in X: z(1_ Q) = 1}$. Then $U\subseteq X$ is clopen
• Let $C\subseteq S: x:= \chi_ C$ (characteristic function)
• Then $R(x,U) = C$
• [proof] $s\in R(x,U) \Leftrightarrow s \cdot x \in U \Leftrightarrow s \cdot x (1_ Q) = 1 \Leftrightarrow x(1_ Q s) = 1 \Leftrightarrow x(s) = 1 \Leftrightarrow X \in C$
• ! Therefore: every subset can be a return set
• Definition. $x$ unif. recurrent $\Leftrightarrow R(x,U) \text{ syndetic } \forall U \in \mathfrak{U}$
• Theorem. $X$ dynamical system on $S$, $y\in X$, $L\subseteq \beta S$ minimal left ideal. TFAE:
a. $y$ unif. recurrent;
b. $\exists p \in L: p\cdot y = y$
c. $\exists \epsilon \in L\cap E(\beta S): \epsilon \cdot y = y$
d. $\exists \epsilon \in L \cap E(\beta S), x\in X: \epsilon \cdot x = y$
e. $y\in \bigcup_ {M \text{ minimal subsystem}} M$
f. $y \in L \cdot y$.
• Proof
• c=> a
• $U \in \mathfrak{U}(y), V\subseteq \bar{V} \subseteq U \text{ offen}, A=R(y,V) \in \epsilon$ [$\epsilon y = y$]
• $B = {s: s\cdot \epsilon \in \hat{A}} \subseteq S \text{ syndetic [HS 4.39]} \Rightarrow B \subseteq R(y,U)$ [$s \epsilon \in \hat{A} \Rightarrow s\epsilon y = s y \in \bar{V} \subseteq U$]

### Notes

The talk/lecture from the previous page continues, tackling proximality with its basic characterization in terms of $\beta S$ and starting the proof of the characterization of uniform recurrence. That’s fairly basic stuff (in the sense of necessary knowledge, not “trivial” or “easy”). The notes are a bit incomplete overall – not sure if I was too lazy (likely) or if Sabine Koppelberg jumped a bit to get to the interesting bits.

The proof that begins at the bottom of the page is, for me, a typical cases of a proof that prevents one from learning; a picture perfect proof that throws elegant arguments around but keeps from its reader the beautiful messiness of coming up with it in the first place.

The reference [HS 4.39] is alomst certainly whatever is numbered 4.39 in Hindman & Strauss, “Algebra in the Stone–Čech compactification”. (I can’t check the actual detail since my copy of H&S is still on route from LA.)

I forgot to mention in the first post that I substituted \mathfrak for Sutterlin in the transcription – Sutterlin is too hard to come by (Sutterlin U is used to indicate the neighborhood filter).