! Also: jede Teilmanege als Rueckkehrmenge darstellbar.

Def. unif. rekurrent

Satz. DS ueber , , min. LID
a. unif. rekurrent;
b.
c.
d.
e.
f. .

Beweis

c=> a

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partial translation

Definition.
where is the diagonal in ; note that these coverings form a neighborhood basis of in

Theorem

.

Example 2 (continued)

Let with

. Then is clopen

Let (characteristic function)

Then

[proof]

! Therefore: every subset can be a return set

Definition. unif. recurrent

Theorem. dynamical system on , , minimal left ideal. TFAE:
a. unif. recurrent;
b.
c.
d.
e.
f. .

Proof

c=> a

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Notes

The talk/lecture from the previous page continues, tackling proximality with its basic characterization in terms of 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).