Peter Krautzberger on the web

Red workbook, p6

Source

red workbook, p6-1
Red Workbook, p.6, part 1
red workbook, p6-2
Red Workbook, p.6, part 2

Transcript

  • Folgerung: Ann. $1_S$ ex., $X = 2^S$.
    • Prop. $C\subseteq x = \chi_C \in X$:
      • $C$ zentral <=> ex $y \in X: y$ prox. $x$, unif. rek, $y(1_S) = q$
        • [”=>” wie Satz/Beweis; “<=” Setze $U = { z: z(1_Q) = 1} \in y$
        • $\underset{Bsp.2}{\Rightarrow}$ $R(x,U) = C \stackrel{\text{Def}}{\underset{\text{Satz}}{\Rightarrow}} C$ zentral]
  • $S = (\omega, +)$
  • Wir wissen
    • (a) $x,y \in X$ prox. <=> $x,y$ stimmen auf bel. langen Int. von $\omega$ ueberein
    • (b) $y \in X$ unif. rekurrent $\Longleftrightarrow \forall U \in \mathfrak{U}(y): R(y,U)$ synd.
      • $\Longleftrightarrow \forall U_N \text{ (Basis offen) } R(y,U_n)$ syndetisch
      • $\Longleftrightarrow \forall n \exists b \forall l \exists k (l \leq k \leq l+b \text{ und } y \upharpoonright [k,k+n] = y \upharpoonright n$
  • Fuer $C \subseteq \omega$:
    • $C$ zentral $\Leftrightarrow \exists y \in 2^\omega: y$ unif. rek. und $y$ prox. $\chi_C $
    • und $y(0)=1$
  • Also zentrale Mengen analytisch!
  • Frage: echt analytisch? SK: wahrscheinlich.

  • Notiz: Rand Oben: ?: Dieses DS in irgendeinem Sinn universell?

  • Notiz: gegenueberliegende Seite:
    • anders: )offenbar ist) die eigetnlich Def. dadurch verbessert, dass klar wird, dass Obermengen wieder zentral sind
    • Frage: kann ich im selben DS bleiben, um die Def. fuer eine Obermenge zu verifizieren?
      • orig/anders: Laesst sich die Konstruktion in $2^Q$ hinueberretten durch eine geeignete Abbildung?

partial Translation

  • Conclusion: Assume an identity $1_S$ exists, $X = 2^S$.
    • (Side note: is this dynamical system universal in some sense?)
    • Proposition. $C\subseteq x = \chi_C \in X$:
      • $C$ central <=> ex $y \in X: y$ proximal $x$, uniformly recurrent, $y(1_S) = q$
        • [”=>” as in the proof of the theorem; “<=” Let $U = { z: z(1_Q) = 1} \in y$
        • $\underset{Example 2}{\Rightarrow}$ $R(x,U) = C \stackrel{\text{Definition}}{\underset{\text{Theorem}}{\Rightarrow}} C$ central]
  • $S = (\omega, +)$
  • We know
    • (a) $x,y \in X$ proximal <=> $x,y$ agree on arbitrarily long Intervals of $\omega$
    • (b) $y \in X$ unif. recurrent $\Longleftrightarrow \forall U \in \mathfrak{U}(y): R(y,U)$ syndetic
      • $\Longleftrightarrow \forall U_N \text{ (basic open) } R(y,U_n)$ syndetic
      • $\Longleftrightarrow \forall n \exists b \forall l \exists k (l \leq k \leq l+b \text{ und } y \upharpoonright [k,k+n] = y \upharpoonright n$
  • For $C \subseteq \omega$:
    • $C$ central $\Leftrightarrow \exists y \in 2^\omega: y$ unif. rec. and $y$ prox. $\chi_C $
    • and $y(0)=1$
  • So central Sets are analytic!
  • Question: properly analytic? Sabine Koppelberg: probably.

  • Note (across the page)
    • to put it differenlty: this clearly improved the initial Definition as it is now clear that supersets of central sets are central.
    • Question: can we stay in the same dynamical system to verify that supersets of a central set are central?
      • to put it differently: can we find a map to transfer the constructions in $2^Q$ over to an arbitrary DS?

Notes

This is a curious (double) page. I don’t know why Sabine Koppelberg thought it was important to add the observation that the set of central sets of $\omega$ is analytic (and I’m wondering if the question about “properly analytic” came from my PhD-sibling Gido Scharfenberger-Fabian). I don’t think I’ve ever seen this fact used in the wild. It is a nice observation though and lets me add the first entry in this transcription project with the following subsection:

Open problems

  • Is the set of central subsets of $\omega$ properly analytic? (Sabine Koppelberg was apparently leaning towards “yes!”.)