Red workbook, p2

Source

Transcript

• Koppelberg 20. Aug. 2006
  • Wied.
  \begin{split} C \subseteq S \text{ zentral} & :\Longleftrightarrow & \exists p \in E(\beta S) \cap K(\beta S): C \in p \\ & \phantom{:}\Longleftrightarrow & C \text{ IP-Menge & PWS}\end{split}
• 1. Dyn. System
  • Ziel: zentral = dyn. zentral
  • Def.: DS (X,\varphi) mit X kompatk, T_ 2, \varphi Op. von S auf X (stetig in komp. (S,X)), schreibe einfach s\cdot x statt \varphi(s,x).
  • Bsp:
    1. W (\to t^n), S \to \beta S
    2. S\leq Q, X=2^Q (Prod.raumder diskreten {0,1}),
    • s\cdot x = x (q\cdot s) \forall s \in S, q\in Q, x\in X
    • ? stetig & assoziativ (nachrechnen / klar)
    • [z.B. S=Q=W, shift -> Kap. 8 Skript]
  • Def.
    • Untersystem, min US, \exists US min.
    • \beta S operiert auch auf X (wie immer)
    • aber nicht DS [Stetigkeit!)
      a. p\mapsto p \cdot x stetig, aber nicht x \mapsto p \cdot x!
      b. (p \cdot q) \cdot x = p \cdot (q \cdot z) gilt. (i.A.)
  • Bsp. Y\subseteq \beta S (als DS). Y \beta S\text{-invariant} \Leftrightarrow Y \text{ Linksid. von } \beta S
  • Y \text{ min. } US \Leftrightarrow Y \text{min. LID}
• [margin note, top]
  • Notizen: S_ 0={e_ 0}, e_ 0 \cdot e_ 0 = e_ 0
  • S_ {i+1} = S_ i \cup {e_ {i+1}} mit e_ {i+1} Identitaet dazu
  • [also immer Idenitaeten adjungieren] \Rightarrow \bigcup S_ i \equiv (\mathbb{N}, \vee) (sup)
• Hawaiian earring als DS => wie sehen zentrale aus?
• [margin note, right]
  • Notiz?Prod. top = nur endlich viele
  • entweder 0 oder 1
  • sonst Umgebung = {0,1}
  • -> "Wie 2^{\mathbb{N}}" Stetigkeit.

partial translation

• Koppelberg 20. Aug. 2006
  • Repetition.
  \begin{split} C \subseteq S \text{ central} & :\Longleftrightarrow & \exists p \in E(\beta S) \cap K(\beta S): C \in p \\ & \phantom{:}\Longleftrightarrow & C \text{ IP-set & piecewise syndetic}\end{split}
• 1. Dynamical System (DS)
  • Goal: central = dynamically central
  • Def.: DS (X,\varphi) with X compact, T_ 2, \varphi Op. from S to X (cts, in compact (S,X)), we write s\cdot x (short for \varphi(s,x)).
  • Ex:
    1. W ($ \to t^n$), S \to \beta S
    2. S\leq Q, X=2^Q (with Prod.topology),
    • s\cdot x = x (q\cdot s) \forall s \in S, q\in Q, x\in X
    • ? cts & associative (obvious)
    • [e.g. S=Q=W, shift -> Ch. 8 lecture notes]
  • Def.
    • dyn. subsystem, min. subsystem, \exists min. subsystem.
    • \beta S operates on X (as usual)
    • but not DS [continuity!)
      a. p\mapsto p \cdot x cts, but not x \mapsto p \cdot x!
      b. (p \cdot q) \cdot x = p \cdot (q \cdot z) gilt. (i.A.)
  • Example: Y\subseteq \beta S (as DS). Y \beta S\text{-invariant} \Leftrightarrow Y \text{ left ideal (LID) of } \beta S
  • Y \text{ min. subsystem} \Leftrightarrow Y \text{mininmal LID}
• [margin note, top]
  • Notes: S_ 0={e_ 0}, e_ 0 \cdot e_ 0 = e_ 0
  • S_ {i+1} = S_ i \cup {e_ {i+1}} mit e_ {i+1} Identitaet dazu
  • kepe adjoining identities => \bigcup S_ i \equiv (\mathbb{N}, \vee) (sup)
  • Hawaiian earring as DS => what do central sets look like?

Notes

My first workbook starts likemost would -- with lecture notes.

IIRC, these notes come from series of talks Sabine Koppelberg (my PhD advisor at FU Berlin) gave over the summer 2006 to a small audience (possibly just me? I don't remember). These talks followed her lecture notes for the course "Ultrafilter, Topologie und Kombinatorik" she gave in the previous semester on all things \beta S. The content is mainly based on Hindman, Strauss, Algebra in the Stone–Čech Compactification, greatly improved by Sabine's own style.

The next two pages will continue this talk and ~20 pages will follow on the subject (interrupted by exercises and other notes). The topic are dynamical systems and recurrence, the famous Bergelman-Hindman result (as indicated: central = dynamically central), some notes on thick, pieceswise syndetic and the combinatorial description of central as well as the Central Sets Theorem.

It's funny to see how very inexperienced I was, e.g., the note on the product topology -- I really didn't know that? Wow. Then again, I never took a topology course while getting my Diplom (I could have used a better advisory infrastructure).

It's also funny (and somewhat alarming) to see how many subjects came up this early. But we'll get to that...