Josia Pietsch bc8b5a8b6c
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\begin{theorem}[Boundedness Theorem]
\yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness}
Let $X$ be Polish, $C \subseteq X$ coanalytic,
$\phi\colon C \to \omega_1$ a coanalytic rank on $C$,
$A \subseteq C$ analytic, i.e.~$A \in \Sigma^1_1(X)$.
Let $X$ Polish, $C \in \Pi^1_1(X)$, $A \in \Sigma^1_1(X)$,
$A \subseteq C$, $\phi\colon C \to \omega_1$ a coanalytic rank.
Then $\sup \{\phi(x) : x \in A\} < \omega_1$.
Moreover for all $\xi < \omega_1$,
D_\xi \coloneqq \{x \in C : \phi(x) < \xi\}
E_\xi \coloneqq \{x \in C : \phi(x) \le \xi\}
are Borel subsets of $X$.
x \prec y&:\iff& x,y \in A \land \phi(x) < \phi(y)\\
&\iff& x,y \in A \land y \not\le_\phi^\ast x.
Since $A$ is analytic,
this relation is analytic and wellfounded on $X$.
By \yaref{thm:kunenmartin}
we get $\rho(\prec) < \omega_1$.
Thus $\sup \{\phi(x) : x \in A\} < \omega_1$.
Since $D_\xi = \bigcup_{\eta < \xi} E_\xi$,
it suffices to check $E_\xi \in \Sigma_1^1(X)$.
Let $\alpha \coloneqq \sup \{\phi(x) : x \in C\}$.
Then $E_\xi = E_\alpha$ for all $\alpha \le \xi < \omega_1$.
Consider $\xi \le \alpha$.
\item If there exists $x_0 \in C$ with $\phi(x_0) \ge \xi$,
pick such $x_0$ of minimal rank.
Then for all $y \in X$ we have
y \in E_\xi &\iff& y \in C \land \phi(y) \le \xi\\
&\iff& y \le^\ast_\phi x_0 & ~ \text{ coanalytic}\\
&\iff& x_0 \not<^\ast_\phi y & ~ \text{ analytic}\\
So $E_\xi$ is Borel.
% TODO If $\alpha < \omega_1$, this also shows that $E_\alpha$ is Borel?
\item If there exists no such $x_0$
then $\xi = \alpha$
E_\xi = E_\alpha = \bigcup_{\eta < \alpha} E_\eta
is a countable union of Borel sets by the previous case.
\section{Abstract Topological Dynamics}
% \subsection*{Basic Definitions}
% TODO: move to appendix?
Let $X$ be a set.
A \vocab{group action} of a group $G$ on $X$
is a function
$\alpha\colon G \times X \to X$
such that
\item $\forall x \in X.~\alpha(1_G,x) = x$,
\item $\forall g,h \in G, x \in X.~\alpha(gh,x) = \alpha(g,\alpha(h,x))$.
Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$.
For $x \in X$,
the \vocab{orbit} of $x$ is defined as
G\cdot x \coloneqq \{g\cdot x : g \in G\}.
A group action is called \vocab{transitive}
iff $g \mapsto g \cdot x$ is surjective for all $x \in X$,
i.e.~iff the action has exactly one orbit.
For $x \in X$,
the \vocab{stabilizer subgroup}
of $G$ with respect to $x$ is
G_x \coloneqq \{g \in G : g\cdot x = x\}.
Group actions of a group $G$ on a set $X$
correspond to group homomorphisms
$G \to \Sym(X)$.
Indeed for a group action $\alpha\colon G \times X \to X$
G&\longrightarrow & \Sym(X) \\
g&\longmapsto & (x \mapsto g \cdot x).
A group $G$ with a topology
is a \vocab{topological group}
G \times G&\longrightarrow & G \\
(x,y) &\longmapsto & x \cdot y
G&\longrightarrow & G \\
x&\longmapsto & x^{-1}
are continuous.
Let $T$ be a topological group\footnote{usually $T = \Z$ with the discrete topology}
and let $X$ be a compact metrizable space.
A \vocab{flow} $(X, T)$, sometimes denoted $T \acts X$
is a continuous action
T \times X&\longrightarrow & X \\
(t,x) &\longmapsto & tx.
A flow is \vocab{minimal} iff every orbit is dense.
$(Y,T)$ is a \vocab{subflow} of $(X,T)$ if $Y \subseteq X$
and $Y$ is invariant under $T$,
i.e.~$\forall t \in T,y \in Y.~ty \in Y$.
A flow $(X,T)$ is \vocab{isometric}
iff there is a metric $d$ on $X$ such that
for all $t \in T$ the map
a_t\colon X &\longrightarrow & X \\
x &\longmapsto & tx
is an \vocab{isometry},
i.e.~$\forall t \in T.~\forall x,y \in X.~d(a_t(x),a_t(y)) = d(x,y)$.
If $(X,T)$ is a flow, then a pair $(x,y)$, $x \neq y$
is \vocab{proximal} iff
\exists z \in X.~\exists (t_n)_{n < } \in T^{\omega}.~t_n x \xrightarrow{n \to \infty} z \land t_n y \xrightarrow{n \to \infty} z.
A flow is \vocab{distal} iff
it has no proximal pair.
Note that a flow is minimal iff it has no proper subflows.
Let $(T,X)$ and $(T,Y)$ be flows.
A \vocab{factor map} $\pi\colon (T,X) \to (T,Y)$
is a continuous surjection $X \twoheadrightarrow Y$
that is $T$-equivariant,
i.e.~$\forall t \in T, x \in X.~\pi(t\cdot x) = t\cdot \pi(x)$.
If such a factor map exists,
we also say that $(T,Y)$ is a \vocab{factor}
of $(T,X)$.
An \vocab{isomorphism} from $(T,X)$ to $(T,Y)$ is
a homeomorphism $X \leftrightarrow Y$
commuting with the group action.
What is called ``factor'' here is called ``subflow''
by Furstenberg.
Recall that $S_1 = \{z \in \C : |z| = 1\}$.
Let $X = S_1$, $T = S_1$
$(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.%
\footnote{Note that here we consider the abelian group structure of $S^1$
and $\alpha + \beta$ denotes the addition of \emph{angles},
i.e.~$\alpha \cdot \beta$ in complex numbers.}
Let $X,Y$ be compact metric spaces
and $\pi\colon (X,T) \to (Y,T)$ a factor map.
Then $(X,T)$ is an \vocab{isometric extension}
of $(Y,T)$ if there is
$\rho\colon X\times_Y X \to \R$%
\footnote{Recall that in the category of topological spaces
the \vocab{fiber product} of
$A \xrightarrow{f} C$, $B \xrightarrow{g} C$
is $A \times_C B = \{(a,b) \in A \times B: f(a) = g(b)\}$,
i.e. $X \times_Y X = \{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$.}
such that
\item $\rho$ is continuous.
\item For each $y \in Y$, $\rho$ is a metric on the fiber
$X_y \coloneqq \{x \in X: \pi(x) = y\}$.
\item $\forall t \in T.~\rho(tx_1,tx_2) = \rho(x_1,x_2)$.
\item $\forall y,y' \in Y.~$
the metric spaces $(X_y, \rho)$ and $(X_{y'}, \rho)$
are isometric.
A flow is isometric iff it is an isometric extension
of the trivial flow,
i.e.~the flow acting on a singleton.
Indeed maps $\rho\colon X\times_\star X = X^2 \to \R$
as in \yaref{def:isometricextension}
correspond to metrics witnessing that the flow is isometric.
An isometric extension of a distal flow is distal.
Let $\pi\colon X\to Y$ be an isometric extension.
Towards a contradiction,
suppose that $x_1,x_2 \in X$ are proximal.
Take $z \in X$ and a sequence $(g_n)_{n < \omega}$ in $T$
such that $g_n x_1 \to z$ and $g_n x_2 \to z$.
Then $g_n \pi(x_1) \to \pi(z)$
and $g_n \pi(x_2) \to \pi(z)$,
so by distality of $Y$
we have $\pi(x_1) = \pi(x_{2})$.
Then $\rho(g_n x_1, g_n x_2)$
is defined and equal to $\rho(x_1,x_2)$.
By the continuity of $\rho$,
we get $\rho(g_n x_1, g_n x_2) \to \rho(z,z) = 0$.
Therefore $\rho(x_1,x_2) = 0$.
Hence $x_1 = x_2$ $\lightning$.
}{Let $\pi\colon X \to Y$ isometric extension.
Suppose $x_1,x_2 \in X$ is proximal.
Then $\pi(x_1) = \pi(x_2)$.
But there exists a $T$-equivariant metric on the fibers.
Let $\Sigma = \{(X_i, T) : i \in I\} $
be a collection of factors of $(X,T)$.
Let $\pi_i\colon (X,T) \to (X_i, T)$ denote the factor map.
Then $(X, T)$ is a \vocab{limit}%
\footnote{This is not a limit in the category theory sense and not uniquely determined.}
% TODO THE inverse limit is A limit
of $\Sigma$ iff
\forall x_1 \neq x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2).
A limit of distal flows is distal.
Let $(X,T)$ be a limit of $\Sigma = \{(X_i, T) : i \in I\}$.
Suppose that each $(X_i, T)$ is distal.
If $(X,T)$ was not distal,
then there were $x_1, x_2, z \in X$
and a sequence $(g_n)$ in $T$
with $g_n x_1 \to z$ and $g_n x_2 \to z$.
Take $i \in I$ such that $\pi_i(x_1) \neq \pi_i(x_2)$.
But then $g_n \pi_i(x_1) \to \pi_i(z)$
and $g_n \pi_i(x_2) \to \pi_i(z)$,
which is a contradiction since $(X_i, T)$ is distal.
}{Suppose there is a proximal pair $x_1,x_2$.
Take $i$ such that $\pi_i(x_1) \neq \pi_i(x_2) \lightning$.