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}
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
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).
