Peter Krautzberger on the web

Red workbook, p3

Source

red workbook, p3
Red Workbook, p.3

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).