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\begin{document}


%\setbeamercovered{dynamic}

\title{Forcing, Equivalence Relations and Marker Structures}



\author{S.\ Jackson}



\institute{

(joint with {\color{red}S.\ Gao}, {\color{red}E.\ Krohne}, 
and {\color{red}B.\ Seward}) \\[10pt]
Department of Mathematics\\
University of North Texas\\
%jackson@unt.edu
}






\date{
August, 2013\\
Chapman University}


\begin{frame}
\titlepage
\end{frame}





\section{Introduction}



\begin{frame}
Basic objects of study are Borel equivalence relations $E$ on Polish spaces $X$.
We frequently regard $X$ as a standard Borel space.
\mys

The notion of complexity is provided be the concept of {\color{blue}reduction}.
\mys


\begin{defn}
\begin{itemize}
\item
We say $E$ is {\color{blue}reducible} to $F$, $E \leq F$,  if there is a Borel function $f \colon X\to Y$
such that $x E\ y \Leftrightarrow f(x) F\ f(y)$.


\item
We say $E$ is bi-reducible with $F$, $E \sim F$,  if $E \leq F$ and $F \leq E$. 

\item
We say $E$ is emdeddable into $F$, $E \sqsubseteq F$, if in addition $f$ is one-to-one.
\end{itemize}

\end{defn}


Note that a reduction gives a definable injection from $X/E$ to $Y/F$ so reduction can be viewed 
as a notion of definable cardinality for these quotient spaces. 
\end{frame}




\begin{frame}

We say $E$ is a countable (Borel) equivalence relation if all classes of $E$ are countable.
\mys


If $G$ is a Polish group and $G$ acts on $X$, then the {\color{blue}orbit equivalence relation}
$E_G$ is defined by
\[
x E_G\, y \Leftrightarrow \exists g \in G\ (g \cdot x =y).
\]


The {\color{red}Feldman-Moore} theorem says that every countable Borel equivalence 
relation is given by the Borel action of a countable group $G$. The case $G=\Z$ 
is the classical case of discrete-time dynamics. 
\mys

So, we can study the equivalence relations $E_G$ group by group.
\end{frame}


\begin{frame}
The simplest equivalence relations are the {\color{blue}smooth} or {\color{blue}tame} ones.

\begin{defn}
$E$ is smooth if there is a Borel reduction of $E$ to equality relation on a Polish space.
\end{defn}
\mys

So, for a smooth $E$, $X/E$ can be regarded as  a subset of a standard Borel space.
\mys

For countable Borel $E$, smooth is the same as saying there is a Borel selector for $E$.
\end{frame}




\begin{frame}

\begin{defn}
$E_0$ is the equivalence relation on $2^\omega$ given by 
\[
x E_0\, y \Leftrightarrow \exists n\ \forall m \geq n\ (x(m)=y(m)).
\]
\end{defn}


The {\color{red}Harrington-Kechris-Louveau} theorem says that if $E$ is a Borel equivalence
relation then either $E$ is smooth or $E_0 \sqsubseteq E$.
\mys


So, there is no complexity class of equivalence relation strictly between 
the smooth relation $E_{=}$ and $E_0$. 


\end{frame}


\begin{frame}
If $G$ is a Polish group, $G$ acts of $F(G)$ by the shift action
\[
g\cdot F=\{gf \colon f \in F\}
\]


We can view this action as being on $2^G$ by

\[
g \cdot x (h)= x(g^{-1}h)
\]
\mys


We call this the {\color{blue}Bernoulli} (left) shift action of $G$ on $2^G$. When $G$
is countable, $2^G$ is a compact Polish space in the natural product topology.
\end{frame}





\begin{frame}
\frametitle{Countable Equivalence Relations}

We let $E(2^G)$ denote the shift action of $G$ on $2^G$, 
and $F(2^G)$ denote the free part of $2^G$ with the shift action.
\mys


\begin{thm}[Dougherty-J-Kechris]
The shift action of $F_2$ on $2^{F_2}$ is a universal countable Borel equivalence relation,
that is, $E\leq E(2^{F_2})$ for any countable Borel $E$.
\end{thm}
\mys




In general, the shift action is more or less universal for actions of $G$:

\begin{fact}
Let $E$ the the orbit equivalence relation for a Borel action of the countable group $G$
on a Polish space $X$. Then 
\[
E \leq E((2^\omega)^G) \leq E(2^{G\times \Z}).\]
\end{fact}
\end{frame}



\begin{frame}


\begin{defn}
A countable Borel equivalence relation $E$ is {\color{blue}hyperfinite}
if $E$ is the increasing union of relations $E_n$ with finite classes. 
\end{defn}


\begin{thm}[Slaman-Steel]
The following are equivalent:

\begin{itemize}
\item
$E$ is hyperfinite.
\item
$E=E_G$ where $G=\Z$.
\item
The classes of $E$ can be uniformly Borel ordered in type $\Z$ (or are finite).
\end{itemize}
\end{thm}
\end{frame}


\begin{frame}
\frametitle{Markers}

\begin{defn}
Let $E$ be a Borel equivalence relation. A marker set $M$ is a Borel set 
$M \subseteq X$ such that $M\cap [x] \neq \emptyset$, $M^c \cap [x] \neq \emptyset$
for evry $x \in X$. 
\end{defn}
\mys


Usually we require some additional properties on $M$, related to the structure of $G$. 
\mys


Many argument in dynamics/ergodic theory and descriptive dynamics use markers sets with 
certain properties (e.g., Rochlin's lemma, Ornstein's theorem, Slaman-Steel theorem).
\mys

Hyperfiniteness proofs also typically use marker arguments.
\end{frame}



\begin{frame}

\begin{thm}[Weiss] Every Borel action by $\Z^n$ is hyperfinite.
\end{thm}



\begin{thm}[Gao-J] Every Borel action by a countable abelian group is hyperfinite.
\end{thm}


Weiss' proof (and several other proofs of this result) use a basic marker lemma:
\mys



\begin{lem}
For each $n$, there is a relatively clopen $M_n\subseteq F(2^{\Z^n})$ such that 
\begin{enumerate}
\item
$\forall x\neq y \in M_n\ [\rho(x,y) >n]$
\item
$\forall x \in F(2^{\Z^n})\ \exists y \in M_nm\ [\rho(x,y) \leq n]$
\end{enumerate}
\end{lem}
\mys


For the abelian result, we need markers with more regularity. 
\end{frame}










\begin{frame}
By a set of {\color{blue}marker regions} we mean a Borel equivalence relation $\sR \subseteq E$
with $\dom(\sR)$ a complete section and all classes of $\sR$ finite. 
\mys

We say $\sR$ is clopen if for each $g \in G$ the set $\{ x \in X \colon x \sR \, g\cdot x \}$
is relatively clopen in $\dom(E)$. 
\mys

We say the marker regions from a {\color{blue}tiling} if $\dom(\sR) = \dom(E)$. 
\mys


\begin{lem}
For each $n$, there is a clopen set of markers $\sR_n$ for $F(2^{\Z^m})$ 
which form a tiling and such that each $\sR$ class is a rectangle with each 
side length in $\{ n, n+1\}$. 
\end{lem}
\mys


We call this a clopen, almost square tiling.
\end{frame}


\begin{frame}
The following question arises in several problems.



\begin{ques}
Can we get a (Borel or clopen) rectangular tiling of $F(2^{\Z^m})$ which is ``almost lined-up''?
\end{ques}


\begin{figure}[h]
\centering
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\re{0}{0}{20}{20}
\re{20}{0}{20}{22}
\re{0}{20}{20}{22}
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\re{-20}{2}{20}{20}
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\re{22}{-20}{22}{20}
\re{-24}{-40}{22}{20}
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\re{42}{-40}{22}{20}
\re{44}{-20}{22}{20}
\re{40}{0}{20}{20}
\re{40}{20}{22}{22}
\re{38}{42}{22}{22}


\end{picture}
\end{figure}
\end{frame}


\begin{frame}
Note that a (Borel or clopen) almost lined-up tiling would have the following consequences:


\begin{itemize}
\item
There would be a (Borel or clopen) ``lining'' of $F(2^{\Z \times \Z})$.
\item
There would be a (Borel or continuous) proper action of $\Z \times \Z$ on each class
of  $F(2^{\Z \times \Z})$.
\end{itemize}



The existence of a lining seems to be related to the (Borel, continuous) 
chromatic number problem for $F(2^{\Z^m})$.



\begin{thm}[Kechris-Soleci-Todorcevic] 
$3\leq \chi_b(m) \leq m+1$.
\end{thm}


\begin{thm}[Gao-J]
$3 \leq \chi_b(m) \leq \chi_c(m)\leq 4$.
\end{thm}
\end{frame}

\begin{frame}
\frametitle{2-colorings and minimality}


\begin{defn}
A {\color{blue}$2$-coloring} of a group $G$ is an $x\colon G\to \{0,1\}$ 
satisfying the following: for every $s \neq 1_G$, there is a finite $T=T(s) \subseteq G$
such that: 
\[
\forall g \in G\ \exists t \in T\ (x(gt)\neq x(gst)).
\]
\end{defn}
\mys

The notion of a 2-coloring was formulated independently by {\color{red}Pestov}, and 
{\color{red}Glassner-Uspensky} independently showed many groups admit $2$-colorings.


\begin{fact}
$x \in 2^G$ is a $2$-coloring iff $\overline{[x]}\subseteq F(2^G)$. 
\end{fact}

\end{frame}



\begin{frame}

\begin{defn}
$x \in 2^G$ is {\color{blue}minimal} if $\overline{[x]}$ is a minimal closed invariant set (subflow), 
that is, $\forall y \in \overline{[x]}\ (  \overline{[y]}=  \overline{[x]})$.
\end{defn}
\mys


Being minimal has a combinatorial reformulation.
\mys



\begin{fact}
$x \in 2^G$ is minimal iff for every $A \in G^{<\omega}$ there is a $T \in G^{<\omega}$
such that 
\[
\forall g \in G\ \exists t \in T\ \forall a \in A\ ( x(gta)=x(a)).
\]
\end{fact}
\mys


\begin{rem}
Minimal $x$ exist in any subflow of any $2^G$ (don't need $\ac$ in fact).
\end{rem}

\end{frame}


\begin{frame}
\begin{thm}[Gao-J-Seward]
Every countable group $G$ has a $2$-coloring.
\end{thm}
\mys


So, there is a compact invariant set $\overline{[x]} \subseteq F(2^G)$.
\mys


An early consequence of this was the following. Recall ({\color{red}Slaman-Steel})
that for any countable equivalence relation there are Borel complete sections
$B_n$ such that $\bigcap_n B_n =\emptyset$. 
\mys


\begin{cor}
Let $B_n \subseteq F(2^G)$ be relatively clopen complete sections. Then 
$\bigcap_n B_n \neq \emptyset$.
\end{cor}

\end{frame}






\begin{frame}
\frametitle{minimal 2-coloring forcing}


\begin{thm}[GJS; minimal $2$-coloring forcing] 
For any countable group $\Gamma$ there is separative forcing notion
$\mP_{mc}$ on which $\Gamma$ acts by automorphisms and such that 
\[
\emptyset \Vdash (x_G \text{ is a minimal  2-coloring of } \Gamma ).
\]
\end{thm}
\mys


The forcing can be described directly, or an instance of {\color{blue}orbit-forcing}.
\mys



\begin{defn}
Let $x \in F(2^\Gamma)$. $\mP_x$ is the forcing notion 
\[
\mP_x=\{ p \in 2^{< G} \colon \exists g \in \Gamma \ (p= g \cdot x \res \dom(p)) \}
\]
\end{defn}
\end{frame}


\begin{frame}

A generic $G$ for $\mP_x$ produces an $x_G \in \overline{[x]}$.
\mys


If $x$ is a minimal $2$-coloring, then $x_G$ will also be a minimal $2$-coloring.
\mys



\begin{itemize}
\item
Varying $x$ can produce different forcing effects. 
\item
The forcings can also be described 
directly by (usually) finitary $\hat{p} \in 2^{<G}$ with extra side-conditions. 
\end{itemize}
\mys

To illustrate the give the direct definition of $\mP_{mc}$ for the case $\Gamma=\Z\times \Z$.
\end{frame}



\begin{frame}
$\mP_{mc}$ consists of conditions 
\[p=(\hat{p}; s_0,\dots, s_n; T_0 ,\dots,T_n; 
A_0,\dots,A_m; U_0,\dots,U_n)
\]


\noindent
satisfying the following:

\begin{enumerate}
\item
$\hat{p} \in 2^R$ where $R=[a,b]\times [c,d] \subseteq \Z \times Z$. 
\item
$T_0,\dots,T_n$, $U_0,\dots,U_m \in 2^{<(\Z\times \Z)}$.
\item
$A_i \in 2^{<(\Z\times \Z)}$ and $\exists h\ [\hat{p}\res (h \cdot (\dom(A_i))) =A_i$.
\item
$\forall g \in \dom(\hat{p})\ \forall i \leq n\ \exists t \in T_i\ 
[ gt, gst \in \dom(\hat{p}) \wedge \hat{p}(gt)\neq \hat{p}(gst)]$
\item
$\forall g \in \dom(\hat{p})\ \forall i \leq m\  \exists t \in U_i\ 
[ \hat{p}\res (gt \cdot (\dom(A_i))) =A_i]$

and

$\forall g \in \dom(\hat{p})\ \forall i \leq m\  \exists t \in U_i\ 
[ \hat{p}\res (gt \cdot (\dom(A_i))) =1-A_i]$
\end{enumerate}
\end{frame}


\begin{frame}
We have the following facts about $\mP_{mc}$.
\mys


\begin{lem}
For any $g \in \Z\times \Z$, $D_g=\{ p \colon g \in \dom(\hat{p})$ is dense.
\end{lem}


\begin{lem}
For each $s\neq (0,0)$ in $\Z \times \Z$, $D_s=\{ p \colon \exists i\ (s=s_i)\}$ is dense.
\end{lem}


\begin{lem}
$\forall p \in \mP_{mc}\ \forall A \subseteq \hat{p}$
\[  D_{p,A}= \{ q \colon \exists i \leq m_q\ A \subseteq A_i(q) \} 
 \text{is dense below } p].
\]
\end{lem}
\end{frame}


\begin{frame}
Let $G$ be a generic for $\mP_{mc}$, and let $x_g=\cup \{ \hat{p} \colon p \in G\}$.
So, $x_G \in 2^{\leq (\Z \times \Z)}$. 
\mys



The first lemma shows that $x_G=2^{\Z \times \Z}$,
the second lemma shows that $x_G$ is a $2$-coloring, and
the third lemma shows that $x_G$ is minimal.
\mys


For example, to show second lemma, copy the domain $R$ of $\hat{p}$ to a larger rectangular domain 
using copies of $\hat{p}$ and $1-\hat{p}$ in such a way that we block the shift $s$.
\mys


\begin{figure}[h]
\centering
%\setlength{\unitlength}{0.4mm}
\begin{tikzpicture}[scale=0.04]

\draw [fill=gray] (0,0) rectangle (20,20);
\te{20}{0}{40}{20}
\te{0}{20}{20}{40}
\draw [fill=gray] (20,20) rectangle (40,40);
\draw [fill=gray] (20,40) rectangle (40,60);

\te{40}{20}{60}{40}
\te{40}{40}{60}{60}
\te{40}{0}{60}{20}
\te{0}{20}{20}{40}
\te{0}{40}{20}{60}
\node at (30,30){$\hat{p}$};
\node at (50,10){\small{$1-\hat{p}$}};

\draw[fill] (40,40) circle (10mm);
\draw[fill] (40,40) circle (10mm);
\draw[fill] (50,50) circle (10mm);
\draw[->] (40,40) to [out=60,in=180] (50,50);
\node[below right] at (40,40) {\small{$g$}};
\node[right] at (50,50) {\hspace{-5pt} \small{$gs$}};

\end{tikzpicture}
\end{figure}

\end{frame}


\begin{frame}
\frametitle{Two theorems for general groups}

The following two theorems are proved using $\mP_{mc}$.
\mys

\begin{thm}[GJS]
Let $G$ be a countable group and $E_G$ the equivalence relation generated by the 
shift action of $G$ on $F(2^G)$.
Let $B_n \subseteq X$ be Borel complete sections, and let $f \colon \omega \to \omega$
with $\limsup f=\infty$. There there an $x \in F(2^G)$ such that 
$\exists^\infty n\ \rho(x,B_n)<f(n)$. 
\end{thm}


\begin{rem}
The Slaman-Steel markers are Borel complete sections $B_n \subseteq F(2^\Z)$
with $\bigcap_n B_n=\emptyset$. 
\end{rem}


\begin{rem}
There does exists a sequence $B_n \subseteq F(2^{\Z^n})$ of relatively clopen 
complete sections such that for all $x \in  F(2^{\Z^n})$ we have 
$\rho(x,B_n)\to \infty$. 
\end{rem}


%The theorem shows we cannot control the rate of increase of the distances from 
%$x$ to the $B_n$.
\end{frame}


\begin{frame}
\begin{thm}[GJS]
Let $G$ be a countable group and $E_G$ the equivalence relation generated by the 
shift action of $G$ on $F(2^G)$. Let $f \colon (F(2^G),E_G)\to (Y,F)$ be a Borel 
invariant map (i.e., $F$ is a factor of $E_G$). Then $F$ has a recurrent 
point.
\end{thm}
\mys



By a {\color{blue}recurrent} point $y \in Y$ we mean that for every non-empty open set $U \subseteq Y$
there is a $A \in G^{<\omega}$ such that $\forall z \in [y]\ \exists g \in A\ 
g \cdot y \in U$.
\mys

In fact, for any non-empty Borel set $B \subseteq Y$, there is a $y \in Y$
which is recurrent for $B$. 
\end{frame}



\begin{frame}
\frametitle{Special Groups}
We specialize to the groups $G=\Z^n$.
\mys

Some of these results are related to the {\color{blue}coloring problem}
for $\Z^n$.
\mys


\begin{ques}[Kechris-Solecki-Todorcevic]
What the Borel/clopen chromatic number of $F(2^{\Z^n})$?
\end{ques}
\mys

It is known (Gao-Jackson) that 
\[
3 \leq \chi_b(F(2^{\Z^n})) \leq \chi_c (F(2^{\Z^n})) \leq 4
\]
\end{frame}



\begin{frame}
\begin{thm}
There does not exists a Borel coloring $c \colon F(2^{\Z^n}) \to k$
such that for every $x \in  F(2^{\Z^n})$ there are arbitrarily large regions
in $[x]$ which are $2$-colored by $c$.
\end{thm}
\mys


To prove this we need a variation of the minimal $2$-coloring forcing
which we call the {\color{blue}odd minimal $2$-coloring forcing}.
\mys


Conditions in this forcing $\mP_o$ are just like those of $\mP$ (the minimal
$2$-coloring forcing) except we require that the domain of $\hat{p}$ 
have odd side lengths. 
\mys



Previous density lemmas go through just as before.
\end{frame}



\begin{frame}
Suppose $c \colon F(2^{\Z^n})\to k$ is Borel. Let $x=x_G$ where 
$G$ is generic for $\mP_o$.
\mys


Suppose $p=(\hat{p}; \cdots)\in G$ and $p \Vdash c(x_G)=0$, say. 
\mys


Let $q \leq p$, $q \in G$,  be such that $\hat{p} \subseteq A_i$ for some 
$A_i \in \vec{A}(q)$.
\mys


Let $r \leq q$, $r \in G$ be such that there are copies of $\hat{q}$ an odd distance 
apart in $\hat{r}$ (such sets are dense).
\mys


Let $g \in \Z^n$ be such that $g \cdot x\res R$ is $2$-colored by $c$, where 
$R$ is sufficiently large (say twice the size of $R$).
\mys



For some $h \in \Z^n$ we have $hg \cdot x \res \dom(r)=\hat{r}$ and $hg (\dom(r))\subseteq R$.
This is a contradiction as $gh \cdot x$ is still generic.\end{frame}



\begin{frame}
\frametitle{A Ramsey-type result}

\begin{thm}
Let $B \subseteq F(2^{\Z^n})$ be Borel. Then there is an $x \in  F(2^{\Z^n})$
and a rectangular lattice $L\subseteq [x]$ such that either $L \subseteq B$ or 
$L\subseteq B^c$.

If $B$ is a complete section, then we have $L \subseteq B$.
\end{thm}
\mys


We use another variation of the minimal $2$-coloring forcing. We use a forcing which builds a 
minimal $2$-coloring but all conditions have a periodicity requirement.
\mys


Conditions of the form

\[
p=(R,\Delta, \{ a, b\}, c, \Lambda)
\]
\end{frame}



\begin{frame}
\begin{itemize}
\item
$R \subseteq \Z \times \Z$ is a rectangle.
\item
$\Delta$ is a translate of a rectangular lattice $L$ and $\Z^2$ is the disjoint union of $\delta R$ 
for $\delta \in \Delta$.
\item
$\{ a, b\} \subseteq R$
\item
$c \colon (\cup_{\delta \in \Delta} \delta (R-\{ a,b\}) \to \{ 0,1\}$.
\item
$\Lambda \subseteq L$ is a rectangular lattice and $c$ has period $\Lambda$. 
\item
(local recognizability)
If $x \in \Delta$, $y \notin \Delta$, then there is a $g \in R$ such that 
$c(gx)\neq c(gy)$ and both are defined. 
\end{itemize}


\begin{rem}
The local recognizability condition is not necessary as it will hold generically.
\end{rem}
\end{frame}



\begin{frame}
\begin{figure}
\centering

\begin{tikzpicture}[scale=0.03]

\draw[step=20]  (-100,-100) grid (100,100);

\draw[fill, lightgray] (0,0) rectangle (20,20);


\foreach \i in {-5,...,4}
{
\foreach \j in {-5,...,4}
{
\draw[fill] (\i*20+15,\j*20+15)  circle (5mm);
\draw[fill] (\i*20+18,\j*20+13)  circle (5mm);
}
}

\end{tikzpicture}
\caption{a condition in the forcing}
\end{figure}
\end{frame}







\begin{frame}

\begin{figure}
\centering

\begin{tikzpicture}[scale=0.03]

\draw[step=20]  (-100,-100) grid (100,100);

\draw[fill, lightgray] (0,0) rectangle (20,20);


\foreach \i in {-2,...,1}
{
\foreach \j in {-2,...,1}
{
\draw[fill] (\i*60-20+40+15,\j*60-20+40+15)  circle (5mm);
\draw[fill] (\i*60-20+40+18,\j*60-20+40+13)  circle (5mm);
}
}

\draw[fill, color=orange,opacity=0.5] (-20,-20) rectangle (40,40);

\foreach \i in {-5,...,4}
{
\foreach \j in {-5,...,4}
{
\draw[fill,color=orange,opacity=0.2] (\i*20,\j*20)  rectangle (\i*20+20,\j*20+20);

}
}
\end{tikzpicture}
\caption{the extension relation}
\end{figure}


\end{frame}

\begin{frame}
\frametitle{Recent Results}

Using variations of minimal $2$-colorings we have the following.
\mys


\begin{thm}
There is no continuous ``lining'' of $F(2^{\Z\times \Z})$.
\end{thm}


\begin{cor}
This is no clopen, almost lined up rectangular marker regions for $F(2^{\Z\times \Z})$.
\end{cor}

Extending (and simplifying) these arguments {\color{red}Ed Krohne} has shown:
\mys

\begin{thm}
There is no continuous $3$-coloring of $F(2^{\Z\times \Z})$.
\end{thm}
\end{frame}


\begin{frame}
So we have:
\mys


\[
\chi_c(F(2^{\Z^n}))= \begin{cases} 3 & \text{ if } n=1\\
4 & \text{ if } n \geq 2
\end{cases}
\]
\mys

For $n \geq 2$ we still don't know $\chi_b(F(2^{\Z^n}))$. 
\mys














\end{frame}




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