=> C \cap R(x,V) \neq \emptyset, say s \in \ldots \cap \ldots
=> 0 = s \cdot x(1_ Q) \underset{\text{Def}}{=} x(1_ Q \cdot s ) = x(s) = 1 ↯ ]
Nimm U = { z \in X: z(1_ Q) = 1} clopen, \in \mathfrak{U}(y).
Nach Beispiel C = R(x,U). ☐
partial Translation
Theorem (Auslander-Ellis) X dynamical system over S, x\in X => \exists y \in X: x \text{ proximal } y & uniformly recurrent
[Take \epsilon \in E_ \min(\beta S), and let y= \epsilon \cdot x \stackrel{\text{Corollary}}{\rightarrow} confirm that this works.]
Definition. C dynanmically central :\Leftrightarrow there exists. dynamical system X over S, x,y \in X, x proximal y, y unif. recurrent, and \exists U \in \mathfrak{U}(y): C = R(x,U).
Theorem (Bergelson, Hindman etc) C \subseteq S central <==> C dyn. central.
Proof.
"<=": C = R(x,U) (U\in \mathfrak{U} etc)
\underset{\text{Cor. above}}{\Rightarrow} there exists \epsilon \in E_ \min(\beta S): y = \epsilon \cdot x => \underbrace{R(x,U)}_ {=C} \in \epsilon
"=>": Let Q := \begin{cases} S & \text{ if } 1_ S \text{ ex} \\ S \cup {1_ Q} & \text{ else } 1_ Q \text( identity ) \end{cases}
X = 2^Q (see example earlier) etc, x = \chi_ C (since C \subseteq S \subseteq Q)
By our assumptions choose \epsilon \in E_ \min(\beta S) with C \in \epsilon.
let y:= \epsilon \cdot x => x proximal y, y uniformly recurrent
Claim: y(1_ Q) = 1
[else =0, V :={ z : z(1_ Q) = 0} \in \mathfrak{U}(y)
y = \epsilon x, therefore R(x,V) \in \epsilon
=> C \cap R(x,V) \neq \emptyset, say s \in \ldots \cap \ldots
=> 0 = s \cdot x(1_ Q) \underset{\text{Def}}{=} x(1_ Q \cdot s ) = x(s) = 1 ↯ ]
Take U = { z \in X: z(1_ Q) = 1} clopen, \in \mathfrak{U}(y).
As in the example: C = R(x,U). ☐
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 Auslander-Ellis now looks almost distressingly easy, with a terribly arbitrary choice of \epsilon.
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).
