Red workbook, p5
04 Mar 2014Source
Transcript
 Satz (AuslanderEllis) DS ueber , => & uniform rekurrent
 [Nimm , setze dies tut’s.]

Def. dyn. zentral es ex. DS ueber S, , prox , unif. rek., und .
 Satz (Bergelson, Hindman etc) zentral <==> dyn. zentral.
 Beweis
 ”<=”: ( etc)
 ex. =>
 ”=>”: Setze
 (siehe Beispiel) etc, (da )
 Nach Vors. waehle mit .
 setze => prox. , unif. rekurrent
 Beh.:
 [sonst ,
 , also
 => , say
 => ↯ ]
 [sonst ,
 Nimm clopen, .
 Nach Beispiel . ☐
 Beweis
partial Translation
 Theorem (AuslanderEllis) dynamical system over , => & uniformly recurrent
 [Take , and let confirm that this works.]

Definition. dynanmically central there exists. dynamical system over S, , proximal , unif. recurrent, and .
 Theorem (Bergelson, Hindman etc) central <==> dyn. central.
 Proof.
 ”<=”: ( etc)
 there exists =>
 ”=>”: Let
 (see example earlier) etc, (since )
 By our assumptions choose with .
 let => proximal , uniformly recurrent
 Claim:
 [else ,
 , therefore
 => , say
 => ↯ ]
 [else ,
 Take clopen, .
 As in the example: . ☐
 Proof.
Notes
This is the continuation of lecture with notes on the previous pages and contains the next notes from the next lecture.
This lecture starts where the last one left off, reaping the rewards – the famous theorem by AuslanderEllis now looks almost distressingly easy, with a terribly arbitrary choice of .
The theorem by Hindman and Bergelson (citation needed) is less known perhaps. It greatly simplifies thinking about central sets and is really quite central (pardon the pun).