# Red workbook, p2

25 Feb 2014### Source

### Transcript

**Koppelberg**20. Aug. 2006- Wied.

**1. Dyn. System**- Ziel: zentral = dyn. zentral
- Def.: DS mit kompatk, , Op. von S auf X (stetig in komp. ), schreibe einfach statt .
- Bsp:
- (),
- , (Prod.raumder diskreten ),
**?**stetig & assoziativ (nachrechnen / klar)- [z.B. , -> Kap. 8 Skript]

- Def.
- Untersystem, min US, US min.
- operiert auch auf (wie immer)
- aber
**nicht**DS [Stetigkeit!)

a. stetig, aber nicht !

b. gilt. (i.A.)

- Bsp. (als DS).

- [margin note, top]
- Notizen:
- mit Identitaet dazu
- [also immer Idenitaeten adjungieren] (sup)

- Hawaiian earring als DS => wie sehen zentrale aus?
- [margin note, right]
- Notiz?Prod. top = nur endlich viele
**entweder**0 oder 1- sonst Umgebung
- -> “Wie ” Stetigkeit.

### partial translation

**Koppelberg**20. Aug. 2006- Repetition.

**1. Dynamical System**(DS)- Goal: central = dynamically central
- Def.: DS with compact, , Op. from S to X (cts, in compact ), we write (short for ).
- Ex:
- (),
- , (with Prod.topology),
**?**cts & associative (obvious)- [e.g. , -> Ch. 8 lecture notes]

- Def.
- dyn. subsystem, min. subsystem, min. subsystem.
- operates on (as usual)
- but
**not**DS [continuity!)

a. cts, but not !

b. gilt. (i.A.)

- Example: (as DS).

- [margin note, top]
- Notes:
- mit Identitaet dazu
- kepe adjoining identities => (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 . 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…