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Red workbook, p8

Source

red workbook, p8
Red Workbook, p.8

Transcript

  • Lemma A \subseteq S (a) A \in p \in K(\beta S) (also pws) => \exists g \subseteq_ e S: \stackrel{\stackrel{[unreadable]}{\downarrow}}{\beta S \cdot p} \subseteq \widehat{ \bigcup_ {x \in g} x^{-1} A }
    • (b) T := \bigcup_ {x \in g} x^{-1} A dick, g \subseteq_ e S (also A pws)
      • => \beta S \cdot q \subseteq \widehat{T} fuer ein q \in \beta S
      • => A \in x \cdot q fuer ein q\in \beta S , ein x\in g
  • Beweis
    • (b): q \in K(\beta S) mit \beta S \cdot q \subseteq \widehat{T} (Prop)
      • q \in \beta S \cdot q \subseteq \widehat{T} \Rightarrow \exists x \in g: x^{-1}A \in q \Rightarrow x \cdot q \in \widehat{A}
    • (a) L := \beta S \cdot p min. LID , p \in L .
      • Beh: L \subseteq \bigcup_ {t \in S} \widehat{t^{-1}A } [ x \in L, p \in \beta S \cdot x = cl_ {\beta S} (s \cdot x)
        • da p\in \widehat{A} offen => \exists t \in S: t\cdot x \in \widehat{A} \Rightarrow x \in \widehat{t^{-1}A} ]
      • L kompakt, also \underset{\stackrel{=}{\beta S \cdot p} }{L} \subseteq \bigcup_ {t \in g} \widehat{ t^{-1}A} fuer ein g \subseteq_ e S . □
  • Prop A\subseteq S ; AeQ: (a) A pws (b) \exists y \in S: y^{-1}A zentral
    • (c) \exists D \subseteq S \text{ synd} \forall d \in D: d^{-1}A zentral
  • Beweis
    • (b) => (a): nimm \epsilon \in E_ \min(\beta S): y^{-1}A \in \epsilon \Rightarrow A \in y \cdot \epsilon \in K(\beta S)
    • (a) => (c): Sei e \subseteq_ e S mit \bigcup_ {t \in e} t^{-1}A = T dick;
      • sei L\subseteq \widehat{T} min LID, sei \epsilon \in L \cap E(\beta S)
      • Sei y\in \epsilon mit \epsilon \in \widehat{ y^{-1} A } . Im DS \beta S :
      • \epsilon uniform rekurrent, B = R(\epsilon, \widehat{ y^{-1}}A) syndetisch
      • D: y \cdot B (syndetisch! (Bem.)) und
      • \forall d\in D: d^{-1}A \in \epsilon (also zentral) [ d = y\cdot b fuer ein b \in B , b\epsilon \in y^{-1}A
        • y b \epsilon \in \widehat{A}, d\epsilon \in \widehat{A}, d^{-1}A \in \epsilon ]

partial Translation

  • Lemma A \subseteq S
    • (a) A \in p \in K(\beta S) (hence piecewise syndetic (pws)) => \exists g \subseteq_ e S: \beta S \cdot p \subseteq \widehat{ \bigcup_ {x \in g} x^{-1} A } [[so this union is thick!]]
    • (b) T := \bigcup_ {x \in g} x^{-1} A thick, g \subseteq S finite (so A is pws)
      • => \beta S \cdot q \subseteq \widehat{T} for some q \in \beta S
      • => A \in x \cdot q for some q\in \beta S and some x\in g
  • Proof
    • (b): q \in K(\beta S) with \beta S \cdot q \subseteq \widehat{T} (by the previous proposition) [[workbook p 7]]
      • q \in \beta S \cdot q \subseteq \widehat{T} \Rightarrow \exists x \in g: x^{-1}A \in q \Rightarrow x \cdot q \in \widehat{A}
    • (a) L := \beta S \cdot p minimal left ideal (LID) , p \in L .
      • Claim: L \subseteq \bigcup_ {t \in S} \widehat{t^{-1}A }
        • x \in L, p \in \beta S \cdot x = cl_ {\beta S} (s \cdot x) since p\in \widehat{A} open => \exists t \in S: t\cdot x \in \widehat{A} \Rightarrow x \in \widehat{t^{-1}A} ]
      • L compact, hence \underset{\stackrel{=}{\beta S \cdot p} }{L} \subseteq \bigcup_ {t \in g} \widehat{ t^{-1}A} for some finite g \subseteq S . □
  • Proposition A\subseteq S ; TFAE
    • (a) A pws
    • (b) \exists y \in S: y^{-1}A central
    • (c) \exists D \subseteq S \text{ syndetic} \forall d \in D: d^{-1}A central
  • Proof
    • (b) => (a): take \epsilon \in E_ \min(\beta S): y^{-1}A \in \epsilon \Rightarrow A \in y \cdot \epsilon \in K(\beta S)
    • (a) => (c): Let e \subseteq S finite with \bigcup_ {t \in e} t^{-1}A = T thick;
      • Let L\subseteq \widehat{T} min. LID, let \epsilon \in L \cap E(\beta S)
      • Let y\in \epsilon with \epsilon \in \widehat{ y^{-1} A } . In the dynamical system \beta S :
      • \epsilon is uniformly recurrent, B = R(\epsilon, \widehat{ y^{-1}}A) syndetic
      • D: y \cdot B (syndetic! (by the above remark.)) undand
      • \forall d\in D: d^{-1}A \in \epsilon (hence central)
        • [proof]: d = y\cdot b for some b \in B , b\epsilon \in y^{-1}A , y b \epsilon \in \widehat{A}, d\epsilon \in \widehat{A}, d^{-1}A \in \epsilon

Notes

More wonderful stuff about thick, piecewise syndetics, and central sets.

The lemma tells us that pws could be called “almost thick” – a finite set of translations is enough to make a pws set thick. The proposition on the other hand tells us that pws is surprisingly close to being central – just one translation! (just keep in mind they are very much not the same notion). In addition, such a translation happens very, very frequently (a syndetic set!).

Somehow, I find this to be a lot of fun even if it’s not particularly surprising – minimal idempotent ultrafilters are just so incredibly rich.