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\title{\huge A dichotomy theorem for turbulence}
\author{Greg Hjorth} % Enter your name between curly braces
\date{\today} % Enter your date or \today between curly braces
\maketitle
%\tableofcontents
\section{Introduction}
\label{introduction}
In this note we show:
\begin{theorem}
\label{main}
\label{maintheorem}
Let $G$ be a Polish group and $X$ a Polish $G$-space with the
induced orbit equivalence relation $E_G$ Borel as a subset of $X\times X$. Then
exactly one of the following:
\leftskip 0.4in
\no (I) There is a countable language $\l$ and a Borel function
\[\theta: X\rightarrow {\rm Mod}(\l)\]
such that for all $x_1, x_2\in X$
\[x_1 E_G x_2 \Leftrightarrow \theta (x_1) \cong \theta(x_2);\]
or
\no (II) there is a turbulent Polish $G$-space $Y$ and a continuous
$G$-embedding
\[\tau: Y\hookrightarrow X.\]
\end{theorem}
There are various bows and ribbons which can be woven into these
statements. We can strengthen (I) by asking that $\theta$ also admit a
Borel orbit inverse, that is to say some Borel function
\[\rho: B \rightarrow X,\]
for some Borel set $B\subset {\rm Mod}(\l)$,
such that for all $x\in X$
\[xE_G \rho(\theta(x));\]
and then after having passed to this strengthened version of (I) we still
obtain the exact same dichotomy theorem, and hence the conclusion that
the two competing versions of (I) are equivalent.
Similarly (II) can be relaxed to just asking that $\tau$ be a Borel
$G$-embedding, or even simply a Borel reduction of the relevant orbit
equivalence relations.
It is in fact a consequence of \ref{maintheorem} that all the plausible
weakenings and strengthenings of (I) and (II) are respectively equivalent
to one another.
I will not closely examine these possible variations here.
The equivalences alluded to above follow from our main theorem
and the results of \cite{hjorth}. That monograph had previously
shown that (I) and (II) are incompatible, and proved a barbaric forerunner
of \ref{maintheorem}, and gone on to conjecture the dichotomy result
above.
\cite{hjorth} is the proper place to find further discussion of
the notation used in the proofs below. Mod$(\l)$ is the
space of $\l$-structure on $\N$ equipped with the topology generated by
quantifier
free formulas. $E_G$ refers to the orbit equivalence relation arising
from the indicated action of $G$ on the indicated space.
\section{The Gandy-Harrington topology and a derived Polish $G$-space}
\label{topology}
We recall the notions of {\it recursive} Polish space and
{\it recursive} continuous function from \cite{moschovakis}.
We assume that all our spaces are recursive
Polish spaces, our group is a recursive Polish space, and
multiplication viewed as a
function
\[G\times G\rightarrow G\]
is a recursive continuous function, as is the evaluation of the
action of $G$ on $X$ viewed as a function
\[G\times X\rightarrow X.\]
Indeed in the case of $X$ we want to do more than simply assume
that $X$ has a recursive basis; we fix from now on a sequence
$(U_\ell)_{\ell\in\omega}$ of open subsets of $X$ which forms a basis
and such that the assignment
\[\ell\mapsto U_\ell\]
is suitably effective; for instance, and this is only one method of
many to make this precise, we could assume that
\[\bigcup_{\ell\in\omega}\{\ell\}\times X\]
is a recursive open subset of $\omega\times X$.
We also fix a complete compatible metric $d_G(\cdot, \cdot)$ on
$G$ and assume it provides a recursive continuous function from
$G\times G$ to $\R$.
We assume that
there is a countable dense subgroup $G_0=\{g_n: n\in\N\}$
of $G$ such that the enumeration $n\mapsto g_n$ is recursive
continuous as a function from $\N$ to $G$ and such that
$g\cdot U_\ell\in\{U_k: k\in \N\}$ all $g\in G_0$ and $\ell\in\N$.
We then fix a basis for $G$ closed under left and
right translation by $G_0$ consisting purely of recursive
open sets. We assume it includes a neighborhood basis at the
identity of symmetric sets.
We further assume that $E_G$ is $\Delta^1_1$ as a subset of
$X\times X$.
Of course in general these assumptions might not be true,
and in the arguments below we would need
to laboriously replace each instance of ``recursive" with
``recursive in $z$" for some parameter $z$.
\def\Xl{X_{\rm low}}
\def\X*{X_{*{\rm low}}}
\begin{notation} Let $\Xl =\{x\in X: \forall A\in \Sigma^1_1
(x\in A\vee\exists B\in \Sigma^1_1(x\in B\wedge B\cap A =0))\}$.
Let $\tau_\infty$ be the topology generated by taking as a basis all
$\Sigma^1_1$ sets.
\end{notation}
In the case that $X=2^\omega$ one actually has
$\Xl=\{x\in X: \omega_1^{\rm ck}=\omega_1^{{\rm ck}(x)}\}.$
Note that $\Xl $ is a Borel subset of $X$; on the other hand, while it
is lightface $\Sigma^1_1$ it is {\it not} lightface $\Delta^1_1$.
Recall that Gandy's low basis theorem states that
any non-empty $\Sigma^1_1$ contains an element of $\Xl$.
The formulation of the next lemma was influenced by some
parallel remarks made by Ramez Sami during the course of a seminar.
It has been previously proved in \cite{kechrislouveau} using the
notion of strong Choquet space; the proof below is longer but more
direct.
\begin{lemma}
\label{polish}
$\tau_\infty$ is a Polish topology on $\Xl $.
\end{lemma}
\begin{proof}
We show $(\Xl , \tau_\infty)$ to be homeomorphic to a
$G_\delta$ subset of some well known Polish space.
For ease of exposition we assume $X=2^\omega$; it should
be clear how to extend the argument to general
recursive Polish spaces.
Let $S$ be the set of $\Sigma^1_1$ subsets of X.
To each $x\in \Xl $ associate
\[f_x: S\rightarrow 2\]
by the formulation
\[f_x(A)=1\]
if and only if
\[x\in A.\]
This map $x\mapsto f_x$ is clearly one-to-one. The preimage of a subbasic open set
in $2^S$
of the form
\[\{x: f_x(A)=1\}\]
is obviously open in $(\Xl , \tau_\infty)$; the preimage of a subbasic open
set of the form
\[\{x: f_x(A)=0\}\]
equals
\[\bigcup \{D: D\in \Delta^1_1, D\cap A=0\}\cap \Xl ,\]
and is thus again open with respect to $\tau_\infty$. Therefore
the map
\[x\mapsto f_x\]
is a homeomorphism onto its image, and we need only show that
$\{f_x: x\in \Xl \}$ is a $G_\delta$ subset of $2^S$
in the product topology to complete the proof.
For this purpose let us define at each $n\in\o$
the set $A_n^*$ to be the set of $x\in \Xl $ for which the
$n^{\rm th}$ $x$-recursive\footnote{Here for the only time
we use $X=2^\omega$; with suitable care the notion of
recursive in $x$ can be translated across to arbitrary recursive
spaces.}
linear order of $\o$ is not a wellorder of
rank less than $\omega_1^{\rm ck}$; and
then at each $m$ we let $C_{n, m}^*$ be the set of $x\in \Xl $ for which
$m$ is in the illfounded part of the $n^{\rm th}$ $x$-recursive
linear order of $\o$. These sets are of course all $\Sigma^1_1$.
I claim that
\[f\in\{f_x: x\in \Xl \}\]
if and only if the following four things happen:
\begin{enumerate}
\item for all $n\in\o$ and basic open $W$ with
\[ f(W)=1,\]
there is some $U\subset X$ whose diameter is less than $\frac{1}{n}$
with
\[\overline{U}\subset W\]
and
\[f(U)=1;\]
\item for all $k\in\o$,
$A_0,A_1,...,A_k,C_0, C_1,...,C_k\in\Sigma^1_1$, with
\[\forall i\leq k(f(A_i)=1, f(C_i)=0),\]
there is some $x\in \Xl$ with
\[\forall i\leq k(x\in A_i, x\notin C_i);\]
(in particular this rules out grievously
absurd choices for
$f: S\rightarrow \{0, 1\}$, such as the two constant functions);
\item for all $n$ with
\[f(A_n^*)=1\]
there exists $m$ with
\[f(C^*_{n,m})=1;\]
\item for each $\Sigma^1_1$ set $A$ we fix a Lusin scheme
\[(C_s^A)_{s\in\omega^{<\omega}}\]
of $\Pi_0^1$ sets such that
\[A=\{x\in X| \exists h\in\omega^\omega\forall n\in \omega
(x\in C^A_{h|_n})\};\]
we then demand that
\[f(A)=1\Rightarrow f(C_0^A)=1,\]
and that for all $s\in\omega^\omega$
\[f(C_s^A)=1\Rightarrow \exists k\in\omega(f(C^A_{s^\smallfrown k})=1).\]
\end{enumerate}
The first, third, and fourth of these conditions correspond to $G_\delta$ sets, while
the second is closed.
The purpose of the first condition is to ensure that
we can indeed find a point $x_f$ in some intersection of basic open $V$ with $f(V)=1$;
and in this light the second condition ensures that this $x_f$ meets all the
open and closed sets promised by $f$, and from this we can obtain by the fourth
condition and the usual
kinds of arguments that $x_f$ actually meets every $\Sigma^1_1$ set with
$f(A)=1$. The third condition is the key one. It guarantees that no recursive in
$x_f$ wellorder has rank exactly $\omega_1^{\rm ck}$.
\end{proof}
\begin{notation}
We let $\X*$ be the set of all $x\in X$ such that there exists
some $g\in G$ with
\[g\cdot x\in \Xl.\]
We endow this with the topology $\tau^*_\infty$ generated by
basic open sets of the form
\[(A\cap \Xl)^{\Delta V}\]
where $A\subset X$ is $\Sigma^1_1$ and $V\subset G$ is open.
\end{notation}
In fact if $x\in \X*$ one does not just obtain one $g\in G$
with $g\cdot x\in \Xl$ but a comeager set of $g$ which so move
$x$ to a real with $\omega_1^{{\rm ck}(g\cdot x)}=\omega_1^{\rm ck}$:
We first use that
\[\omega_1^{{\rm ck}(x)}=\omega_1^{\rm ck}\]
to conclude that there will be a comeager collection of $g$ with
$\omega_1^{{\rm ck}(g,x)}=\omega_1^{\rm ck}$,
and any such $g$ will do.
Thus $\X*$ equals
\[(\Xl)^{*G}=(\Xl)^{\Delta G},\]
and is consequently Borel.
Note moreover that in this space, for any $A\subset \Xl$ which is
$\Sigma^1_1$ and for any $V\subset G$ open
we in fact have a sequence of $\Delta^1_1$ sets
\[(D_\alpha)_{\alpha\in \omega_1^{\rm ck}}\]
such that
\[A^{*V}=\X*\setminus
(\bigcup_{\alpha\in \omega_1^{\rm ck}} (D_\alpha)^{\Delta V}),\]
and thus $A^{*V}$ is closed.
The next proof crumbles into a sequence of calculations. The first
few of these resemble similar points made in the course of the proof
of 5.1.5 from \cite{beckerkechris}.
\begin{lemma} $(\X*, \tau^*_\infty)$ is a Polish $G$-space.
\end{lemma}
\begin{proof}
The action of $G$ on this space is obviously continuous.
\smallskip
Claim: $(\X*, \tau^*_\infty)$ is Hausdorff.
Proof of claim: $\tau^*_\infty$ includes the subspace topology
on $\X*$ induced by the original topology on $X$. Since this
original topology
is Hausdorff, so is $\tau^*_\infty$.
\hfill ($\Box$Claim)
\smallskip
Claim: $(\X*, \tau^*_\infty)$ is regular.
Proof of claim: Let $A^{\Delta V}$ be an open subset of
$(\X*, \tau^*_\infty)$
and $x\in A^{\Delta V}$. Then we may find some $W_0\subset V$ open and
non-empty with $x\in A^{*W_0}$.
Appealing to the continuity of the group operations we may find non-empty
open sets
\[W_2\subset W_1\subset W_0\]
and an open neighborhood $W_3$ of the identity with
\[W_1W_3^{-1}\subset W_0\]
and
\[W_2W_3^{-1}\subset W_1.\]
$x$ is then a member of the open set $(A^{*W_1})^{\Delta W_3}$ since for
{\it any} $g\in W_3$ we have
\[W_1g^{-1}\subset W_0\]
and
\[\forall^*h\in W_1g^{-1}(h\cdot (g\cdot x)\in A).\]
Thus the claim will be proved if we show
$(A^{*W_1})^{\Delta W_3}$ is included in the $\tau_\infty^*$-closed
set $A^{*W_2}$. But here we have that for any $h\in W_3$ and
$z$ with $h\cdot z\in A^{*W_1}$
\[\forall^*g\in W_1(gh\cdot z\in A)\]
\[\therefore \forall^*g\in W_1h(g\cdot z\in A).\]
But
\[W_2W_3^{-1}\subset W_1\]
\[\therefore W_2\subset W_1h\]
\[\therefore z\in A^{*W_2}.\]
\hfill ($\Box$Claim)
\smallskip
Since $(\X*, \tau^*_\infty)$ is clearly separable, we thus have that
it is metrizable by Urysohn.
To see that the
space is Polish we first define a continuous map
\[\pi: G\times \Xl \rightarrow \X*\]
by
\[(g, x)\mapsto g\cdot x.\]
\smallskip
Claim: $\pi$ is continuous.
Proof of claim: We have $g\cdot x\in A^{\Delta V}$ if and only
if $x\in A^{\Delta Vg}$.
\hfill ($\Box$Claim)
\smallskip
Claim: $\pi$ is an open map.
Proof of claim: It suffices to show that whenever $A\subset \Xl$ is
$\Sigma^1_1$ and $V\subset G$ is open,
\[V\cdot A=\bigcup_{W_0, W_1 {\rm \: open,\:} W_0W_1\subset V^{-1}}
(\{y: \exists h\in W_0(h\cdot y\in A)\})^{\Delta W_1}.\]
There are two parts to demonstrating this. We show these
two
inclusions separately.
\smallskip
$\subset$: Let
\[g\cdot x=z,\]
$g\in V$, $x\in A$. Then considering the continuity of the
action we may choose open $V_0$ and $V_1$ containing $g^{-1}$ and $1$ which satisfy:
\[V_0(V_1)^{-1}V_1\subset V^{-1}.\]
We can then let $W_0=V_0V_1^{-1}$ and $W_1=V_1$.
With this arranged we have that {\it for all} $\epsilon \in W_1$
\[(g^{-1}\epsilon^{-1})\epsilon\cdot z=(g^{-1}\epsilon^{-1})\epsilon g\cdot x =x\in A.\]
Noting that any such $g^{-1}\epsilon^{-1}$ will be in $W_0$ we have
in particular that
\[z\in (\{y: \exists h\in W_0(h\cdot y\in A)\})^{* W_1}
\subset (\{y: \exists h\in W_0(h\cdot y\in A)\})^{\Delta W_1}.\]
\smallskip
$\supset$: Consider some
\[z\in (\{y: \exists h\in W_0(h\cdot y\in A)\})^{\Delta W_1},\]
where $W_0W_1\subset V^{-1}$.
Then in particular, we can find $g_1\in W_1$ with
\[g_1\cdot z\in \{y: \exists h\in W_0(h\cdot y\in A)\}.\]
And then we may find some $g_2\in W_0$ with
\[g_2g_1\cdot z\in A.\]
But then
\[g_2g_1\in V^{-1}\]
by assumption on $W_0$ and $W_1$, and hence
\[x=g_2g_1\cdot z\]
and
\[g=(g_2g_1)^{-1}\]
witness that $z\in V\cdot A$.
\hfill (Claim$\Box$)
\smallskip
In summary, the above claims have shown
\[\pi: G\times \Xl\rightarrow \X*\]
to be an open continuous map from a Polish space onto a
metric space. Thus by Hausdorff's theorem,
$(\X*, \tau^*_\infty)$ is Polish.
\end{proof}
\section{The split in cases}
\label{setup}
As above, assume that $G$ is a recursive Polish group and
$X$ is a recursive Polish $G$-space.
\begin{definition} A sequence
\[\vec\b =(\b_\alpha)_{\alpha\in \delta}\]
is {\it suitable} if
\begin{enumerate}
\item each $\b_\alpha$ is a subset of the Borel
subsets of $X$;
\item $\delta<\omega_1^{\rm ck}$;
\item the function
\[\alpha\mapsto \b_\alpha\]
is $\Delta^1_1$ in the codes for Borel sets;
\item \label{a} in particular each $\b_\alpha$ is countable;
%\item \label{b} each
%\[\tau_{\b_\alpha}=_{\rm df} \{\bigcup C| C\subset \b_\alpha\}\]
%generates a finer Polish topology on $X$;
\item \label{c} each $\b_\alpha$ includes the basic open sets in the original
topology on $X$;
%\item each \label{d} $\b_\alpha$ is closed under finite Boolean combinations;
\item each \label{e} $\b_\alpha$ is closed under $G_0$-translation.
%\item \label{f} if $B\in \b_\alpha$ and $V\subset G$ is basic open, then
%\[B^{\Delta V}, B^{* V}\in \b_\alpha.\]
\end{enumerate}
\end{definition}
This definition has glossed over the details of being {\it $\Delta^1_1$ in the
codes for Borel sets}; a protracted discussion is unlikely to
be helpful. The reader will be able to provide their own coding for
Borel sets, and any reasonable understanding of ``in the codes"
should suffice; for instance we can ask that there be some recursive wellorder
$\prec$ of length $\delta$ and $\Delta^1_1$ function
\[F: \omega\times \omega \rightarrow \oo\]
such that for any $n\in\omega$ and
$\alpha=|\!| n|\!|_\prec$ we have that
\[\{F(n, m): m\in\omega\}\]
gives a set of codes for the Borel sets appearing in the
$\b_\alpha$.
%Various closure conditions are placed on the sets $\b_\alpha$. They are
%motivated by \cite{beckerkechris}. Among other results
%it had been there shown that given a collection of Borel sets $\b$ satisfying
%\ref{a}-\ref{f} above,
%\[\{B^{\Delta V}: B\in\b, V\subset G {\rm \: basic \: open}\}\]
%generates a Polish topology on $X$ under which the $G$-action is still continuous.
Given such an admissible sequence, we define a sequence of objects
\[\hat{\varphi}_\alpha^{\vec \b}(x, B^{\Delta V_0}, V_1)\]
around each point $x\in X$ and at each $\alpha<\delta$.
\begin{definition} Given an admissible sequence
$\vec\b =(\b_\beta)_{\beta\in \delta}$ as above, $x\in X$, $B\in \b_\alpha$,
$V_0, V_1\subset G$ basic open, $V_1$ containing $1_G$, $V_1^{-1}=V_1$, we
define $\hat{\varphi}_\beta^{\vec \b}(x, B^{\Delta V_0}, V_1)$ by induction on
$\beta$:-
\smallskip
\no{BASE CASE:} for $x\in B^{\Delta V_0}$
\[\hat{\varphi}_0^{\vec \b}(x, B^{\Delta V_0}, V_1)\]
equals the set of $\ell\in \omega$ such that
there exists $k\in\o$, $x_0, x_1,..., x_k\in X$ such that
\leftskip 0.4in
\no $x_0=x$;
\no $x'=x_k\in U_\ell$;
\no each $x_{i+1}\in V_1\cdot x_i\cap B^{\Delta V_0}$.
\leftskip 0in
\smallskip
\no{INDUCTIVE STEP:} If $\beta$ is an ordinal lying strictly between 0 and
$\delta$, then
\[\hat{\varphi}_\beta^{\vec \b}(x, B^{\Delta V_0}, V_1)\]
equals the set of all
\[\langle \hat{\varphi}_\alpha^{\vec \b}(x', C^{\Delta W_0}, W_1),
C, W_0, W_1\rangle ,\]
where
\leftskip 0.4in
\no $\alpha< \beta$;
\no $C\in \b_\alpha$;
\no $W_0, W_1\subset G$ basic open;
\no $W_1$ contains $1_G$, $W_1^{-1}=W_1$,
\no $x_0=x$;
\no $x'=x_k\in C^{\Delta W_0}$;
\no each $x_{i+1}\in V_1\cdot x_i\cap B^{\Delta V_0}$.
\leftskip 0in
\end{definition}
\begin{notation} For $\epsilon >0$ we let $W_\epsilon$ be the
set
\[\{g\in G| d_G(g, 1_G)<\epsilon\}.\]
\end{notation}
\begin{definition} For $A\subset X$, $A\in \Sigma^1_1$, $V_0, V_1\subset G$
basic open, $V_1^{-1}=V_1, 1_G\in V_1$, and $x\in A^{\Delta V_0}$,
we let
\[{\cal O}(x, A^{\Delta V_0}, V_1)\]
be the set of $x'$ for which there exists $k, x_0, x_1, ..., x_k$ such that
\leftskip 0.4in
\no $x_0=x$;
\no $x'=x_k$;
\no each $x_{i+1}\in V_1\cdot x_i\cap A^{\Delta V_0}$.
\leftskip 0in
\end{definition}
\begin{definition} For $\vec\b=(\b_\alpha)_{\alpha<\delta}$
suitable, $\epsilon>0$, $A\in \Sigma^1_1$, $V_0, V_1\subset G$
basic open, $V_1^{-1}=V_1, 1_G\in V_1$, we say that
\smallskip
\centerline{\it $(A, V_0, V_1)$ is $\epsilon$-stable for $\vec \b$}
\smallskip
\no if for all $x, x', x'', \alpha<\delta, V_2,V_3, B$ with
\leftskip 0.4in
\begin{enumerate}
\item $x\in A^{\Delta V_0}$,
\item $x', x''\in {\cal O}(x, A^{\Delta V_0}, V_1)$,
\item $\alpha<\delta$,
\item $B\in\b_\alpha$,
\item $V_2, V_3\subset G$ basic open,
\item $V_3^{-1}=V_3$,
\item $1_G\in V_3$,
\end{enumerate}
\leftskip 0in
\no $x'\in B^{\Delta V_2}$ implies
\[\exists^* g\in W_\epsilon (g\cdot x''\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec \b}_\alpha(x', B^{\Delta V_2}, V_3)=
\hat{\varphi}^{\vec \b}_\alpha(g\cdot x'', B^{\Delta V_2}, V_3)).\]
\end{definition}
There are a few remarks about this definition. Firstly that if
$(A, V_0, V_1)$ is $\frac{1}{n}$-stable then it is $\epsilon$-stable for any
larger $\epsilon$.
%; and instead of
%considering arbitrary $x', x''\in {\cal O}(x, A^{\Delta V_0}, V_1)$ we can restrict
%our attention only to those $x', x''\in G_0\cdot x$ and obtain an equivalent
%definition; these points are important only in
%that they will later reduce quantifier which seems to range over the reals or some other %large
%Polish space by ones ranging over a countable set.
The second observation is that we may replace
\[\exists^* g\in W_\epsilon (g\cdot x''\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec \b}_\alpha(x', B^{\Delta V_2}, V_3)=
\hat{\varphi}^{\vec \b}_\alpha(g\cdot x'', B^{\Delta V_2}, V_3))\]
by
\[\exists g\in W_\epsilon (g\cdot x''\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec \b}_\alpha(x', B^{\Delta V_2}, V_3)=
\hat{\varphi}^{\vec \b}_\alpha(g\cdot x'', B^{\Delta V_2}, V_3));\]
if one $g\in W_\epsilon$ brings about the desired result then it follows
from the structure of the definitions that an open set of them will, and hence
certainly a non-meager set will as well.
\begin{lemma}
\label{reflect}
If $\vec\b$ is a suitable sequence, $n\in\o$, $V_0, V_1\subset G$
basic open, $V_1^{-1}=V_1$, $1_G\in V_1$, $A\in\Sigma^1_1$, and if
\[(A, V_0, V_1)\]
is $\frac{1}{n}$-stable for $\vec \b$, then there is a $\Delta^1_1$ set
$D\supset A$ with
\[(D, V_0, V_1)\]
$\frac{1}{n}$-stable for $\vec \b$.
\end{lemma}
\begin{proof}
For any $\Delta^1_1(x)$ set $C$ we have that the set of $x''$ with
\[\exists g^*\in W_{\frac{1}{n}}(g\cdot x'' \in C)\] is uniformly
$\Delta^1_1(x)$; in particular,
the set of $x''$ with
\[\exists^* g\in W_\epsilon (g\cdot x''\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec \b}_\alpha(x', B^{\Delta V_2}, V_3)=
\hat{\varphi}^{\vec \b}_\alpha(g\cdot x'', B^{\Delta V_2}, V_3))\]
is uniformly, in the indices for $V_0$, $V_1$, and $B$,
$\Delta^1_1(x)$.
On the other hand, the set
\[{\cal O}(x, A^{\Delta V_0}, V_1)\]
is uniformly
$\Sigma^1_1(x)$.
Thus the collection of $\Sigma^1_1$ sets
$A$ with $(A, V_0, V_1)$ $\frac{1}{n}$-stable is uniformly
$\Pi^1_1$ in the indices.
And thus the lemma follows from $\Pi^1_1$ reflection.\footnote{See
\cite{hamash} for the statement and proof of this theorem.}
\end{proof}
\begin{definition} For $\vec \b=(\b_\alpha)_{\alpha\in\delta}$ suitable
and $n\in\o$, let $A_{\vec \b, n}$ be the set
of all $x\in X$ for which there is a non-meager collection of $g\in G$ with
\[\forall D\in \Delta^1_1, V_0, V_1\subset G{\rm \: basic\: open}\]
\[(g\cdot x\in D^{\Delta V_0}\Rightarrow ((D, V_0, V_1) {\rm \: not\:}
\frac{1}{n}-{\rm stable\: for \:}\vec \b))).\]
\end{definition}
For a given $n$ and $\vec \b$,
the set $A_{\vec \b, n}$ is $\Sigma^1_1$.
\begin{notation}
From now on let us endow $\X* \cap A_{\vec \b, n}$ with the topology
$\tau^*_{n, \infty}$
generated by all sets of the form
\[A^{\Delta V}\]
where $A\in\Sigma^1_1$, $A\subset \X* \cap A_{\vec \b, n}$, and
$V\subset G$ basic open. Let
\[(\hat{U}_\ell)_{\ell\in\o}\]
enumerate a basis for $\tau^*_{n, \infty}$.
\end{notation}
\begin{lemma} $(\X* \cap A_{\vec b, n},\tau^*_{n, \infty})$ is a
Polish $G$-space.
\end{lemma}
\begin{proof}
Since it is an open subset of $(\X*, \tau^*_\infty)$.
\end{proof}
At this point the proof splits into two cases. Case(1) below
leads to a turbulent Polish $G$-space $Y$ as in (II) above.
Case(2), on the other hand, will finish with a classification of the
$G$-orbits by countable models considered up to isomorphism.
\section{Case(1)}
For some suitable $\vec \b=(\b_\alpha)_{\alpha\in\delta}$ and $n\in\o$,
$A_{\vec \b, n}$ is non-empty.
\bigskip
Let us fix such a $\vec \b$ and $n$. We will press forward, and eventually show
that option (II) in \ref{main} holds.
Following $\S$6.2 of \cite{hjorth} we may define ``local invariants" for
the Polish $G$-space $(\X* \cap A_{\vec\b, n}, \tau^*_{n, \infty})$.
\begin{notation} For $x\in \X* \cap A_{\vec\b, n}$ and $U$ $\tau^*_{n,\infty}$-open,
$V\subset G$ a symmetric open neighborhood of the identity, we let
\[{\cal O}(x, U, V)\]
be the set of all $x'$ such that there exists $k, x_0, x_1,..., x_k$
with
\[x_0=x,\]
\[x_k=x',\]
and at each $i0$ the local invariant
$\varphi_\alpha(x, U, V)$ encodes the set of all possible
$\varphi_\beta(x', U', V')$ we can obtain by reducing $\alpha$
to $\beta$, $U$ to $U'$. $V$ to $V'$, and passing to some other
$x'\in {\cal O}(x, U, V)$.
Let us fix some particular $x\in \X* \cap A_{\vec \b, n}$;
it in fact does not matter which.
Still following the relevant $\S$6.2 of \cite{hjorth} we come, for any
given $x$ in the space, to
some $\gamma(x)<\omega_1$ such that for all $U$ that is $\tau^*_{n,\infty}$-open,
for all $V\subset G$ that is a
symmetric basic open neighborhood of the identity,
and for all $x'. x''\in [x]_G$
\[\varphi_{\omega_1}(x', U, V)=\varphi_{\omega_1}(x'', U, V)\]
\[{\rm iff}\]
\[\varphi_{\gamma(x)}(x', U, V)=\varphi_{\gamma(x)}(x'', U, V).\]
We then obtain a Polish topology on the set
\[X^x=_{\rm df}\{y\in \X* \cap A_{\vec \b, n}|
\varphi_{\gamma(x)+2}(x, \X* \cap A_{\vec\b, n}, G)=
\varphi_{\gamma(x)+2}(y, \X* \cap A_{\vec\b, n}, G)\}\]
by taking as a basis all sets of the form
\[\{y|\varphi_{\gamma(x)}(y, U, V)=\varphi_{\gamma(x)}(x', U, V)\},\]
as $U$ ranges over $\tau^*_{n,\infty}$-open sets,
$V$ ranges over symmetric basic open neighborhoods of the identity,
and $x'$ ranges over $[x]_G$.
From this point on let us understand that $X^x$ always refers to
$X^x$ in the indicated Polish topology.
\begin{lemma}
\label{density}
For any $y\in X^x$,
\[\overline{[x]_G}=\overline{[y]_G}.\]
\end{lemma}
Here we understand that the closure operation refers to the indicated
topology on the space $X^x$.
\begin{proof} An immediate consequence of the definitions.
\end{proof}
\begin{corollary}
\label{meet}
For any $y\in X^x$, $A\subset X$ a $\Sigma^1_1$ set,
\[[x]_G\cap A\neq 0 {\rm \: iff \:} [y]_G\cap A\neq 0.\]
\end{corollary}
Of course in this corollary it is important that we restrict ourselves
to sets $A$ which are lightface $\Sigma^1_1$ with respect to the
original space $X$.
\begin{lemma}
\label{localdensity}
For any $y\in X^x$,
\[\gamma(x)=\gamma(y).\]
Moreover for any $U$ that is $\tau^*_{n,\infty}$ basic open and $V\subset G$ a
symmetric basic open neighborhood of the identity,
\[{\cal O}(y, U, V) \]
is dense (with respect to the topology on $X^x$) in some basic open
set of the form
\[\{y|\varphi_{\gamma(x)}(y, U, V)=\varphi_{\gamma(x)}(x', U, V)\}.\]
\end{lemma}
\begin{proof} These facts can be verified from the definitions in a reasonably
routine manner; or see $\S$6.2 \cite{hjorth} for the details.
\end{proof}
From now until the end of the section, let us just fix some
$y\in X^x$. We will try to show that $[y]_G$ is meager in the
space $X^x$.
If we succeed in this task, for some arbitrarily chosen $y\in X^x$, then
we will have obtained that every orbit in the space must be meager. And
then by \ref{density} and \ref{localdensity} we have that the definition of
turbulence is satisfied.
Now turning our attention back towards the fixed, but arbitrary, $y\in X^x$
we have from \ref{meet} that there will be some
\[\hat{y}\in [y]_G\]
such that for all\footnote{Here,
and everywhere else the issue might emerge, $\Sigma^1_1$,
$\Delta^1_1$, and $\Pi^1_1$ refer to $\Sigma^1_1(X)$,
$\Delta^1_1(X)$, and $\Pi^1_1(X)$ respectively.}
$D\in \Delta^1_1$, $V_0, V_1\subset G$ basic open, if
\[\hat{y}\in D^{\Delta V_0},\]
then $(D, V_0, V_1)$ is not $\frac{1}{n}$-stable for $\vec \b$.
Hence by \ref{reflect}, for all $A\in \Sigma^1_1$,
$V_0, V_1\subset G$ basic open, if
\[\hat{y}\in A^{\Delta V_0},\]
then $(A, V_0, V_1)$ is not $\frac{1}{n}$-stable for $\vec \b$.
Fix this $\hat{y}$. We need to ``blur" this element of $[y]_G$
making it slightly more generic, without altering its
important properties; for this purpose let us fix some $m>n$
such that
\[(W_{\frac{1}{m}})^\pm (W_{\frac{1}{m}})^\pm (W_{\frac{1}{m}})^\pm
\subset W_{\frac{1}{n}}.\]
\begin{claim} There exists $y'\in[y]_G$ such that:
\leftskip 0.4in
\no (i) for all $A\in \Sigma^1_1$, if $y'\in A$, then
\[y'\in A^{*W}\]
for some basic open neighborhood $W$ of $1_G$;
\no (ii) $\omega_1^{\rm ck}=\omega_1^{{\rm ck}(y')}$; and hence, for
all $A\in \Sigma^1_1$, either
\[y'\in A\] or there is a $\Delta^1_1$ set $D$ with
\[y'\in D\]
and
\[D\cap A=0;\]
\no (iii) for all $A\in \Sigma^1_1$, $V_0, V_1\subset G$ basic
open,
if
\[{y'}\in A^{\Delta V_0},\]
then $(A, V_0, V_1)$ is not $\frac{1}{m}$-stable for $\vec \b$.
\end{claim}
\begin{proof} Note first of all that since $y\in X^x$ we
may certainly find $g\in W_{\frac{1}{m}}$ such that
\[y'=g\cdot \hat{y}\]
satisfies (i) and (ii); any sufficiently generic
$g\in W_{\frac{1}{m}}$ will do. For (iii) we need a further
subclaim.
\smallskip
Subclaim: If $(A, V_0, V_1)$ is $\frac{1}{m}$-stable and $g\in
W_{\frac{1}{m}}$ with
\[y'=g\cdot \hat{y}\in A^{\Delta V_0},\]
then there is some $\frac{1}{n}$-stable $(C, V_0', V_1')$
with
\[\hat{y}\in C^{\Delta V_0'}.\]
Proof of subclaim: Assume not, and then we may assume further
that $g\in G_0$, since slightly nudging $g$ into $G_0$ should
not affect ${\cal O}(y', A^{\Delta V_0}, V_1)$.
Now consider some arbitrary $\hat{z}\in X$ with
\[z'=g\cdot \hat{z}\in A^{\Delta V_0},\]
and
\[\hat{z}^*\in {\cal O}(\hat{z}, A^{\Delta V_0g}, g^{-1}V_1g).\]
Let
\[z^*=g\cdot \hat{z}^*\in {\cal O}(z', A^{\Delta V_0}, V_1).\]
By assumption on $(A, V_0, V_1)$, for any $\alpha<\delta$, $B\in \Sigma^1_1$,
and $V_2, V_3$ basic open in $G$, with $B^{\Delta V_2}$ containing $z$,
(and hence $B^{\Delta V_2 g^{-1}}$ containing $z^*=g\cdot \hat{z}^*$)
we may find
\[h\in W_\frac{1}{m}\]
with
\[\hat{\varphi}^{\vec {\cal B}}_\alpha(hg\cdot \hat{z}, B^{\Delta V_2 g^{-1}},
gV_3 g^{-1})\]
(which by definition of $z'$ equals
\[\hat{\varphi}^{\vec {\cal B}}_\alpha(h\cdot z', B^{\Delta V_2 g^{-1}},
gV_3 g^{-1}))\]
equal to
\[\hat{\varphi}^{\vec {\cal B}}_\alpha(z^*, B^{\Delta V_2 g^{-1}},
gV_3 g^{-1})\]
\[\therefore
\hat{\varphi}^{\vec {\cal B}}_\alpha(g^{-1}hg\cdot \hat{z},
B^{\Delta V_2},
gV_3 g^{-1})
=\hat{\varphi}^{\vec {\cal B}}_\alpha(g^{-1}z^*,
B^{\Delta V_2},
gV_3 g^{-1}),\]
which in turn by the definition of $z^*$ equals
\[\hat{\varphi}^{\vec {\cal B}}_\alpha( \hat{z}^*,
B^{\Delta V_2},
gV_3 g^{-1}).\]
Thus we have shown $(A, V_0g, g^{-1}V_1g)$ to be
$\frac{1}{n}$-stable.
\hfill (Claim$\Box$)
But this subclaim proves the claim, since $\hat{y}$ was assumed not
to be so captured by any $\frac{1}{n}$-stable set.
\end{proof}
Let us then fix $y'$ as above.
\begin{claim} For all $A\in \Sigma^1_1$, $V_0, V_1\subset G$ basic open
with $1_G\in V_1$, $V_1^{-1}=V_1$, if
\[y'\in A^{\Delta V_0}\]
then
\[\exists y_1, y_2\in {\cal O}(y', A^{\Delta V_0}, V_1)\]
\[\exists \alpha<\delta \exists B\in \b_\alpha\]
\[\exists V_2, V_3\subset G {\rm \: basic \:
open, } 1_G\in V_3, V_3^{-1}=V_3\]
with
\[y_1\in B^{\Delta V_2}\]
but
\[\forall g\in W_{\frac{1}{m}}
(g\cdot y_2\in B^{\Delta V_2}\Rightarrow
\hat{\varphi}^{\vec {\cal B}}_\alpha(y_1, B^{\Delta V_2}, V_3)
\neq
\hat{\varphi}^{\vec {\cal B}}_\alpha(g\cdot y_2, B^{\Delta V_2}, V_3)).\]
\end{claim}
\begin{proof} Otherwise applying (ii) from our assumptions on $y'$ we
find $D\in \Delta^1_1$ containing $y'$ such that
\[\forall z'\in A^{\Delta V_0}\cap D\]
\[\forall z_1, z_2\in {\cal O}(z', A^{\Delta V_0}, V_1)\]
\[\forall \alpha<\delta \forall B\in \b_\alpha\]
\[\forall V_2, V_3\subset G {\rm \: basic \:
open,\: }1_G\in V_3, V_3^{-1}=V_3\]
if
\[z_1\in B^{\Delta V_2}\]
then
\[\exists g\in W_{\frac{1}{n}}
(g\cdot y_1\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec {\cal B}}_\alpha(z_1, B^{\Delta V_2}, V_3)
=
\hat{\varphi}^{\vec {\cal B}}_\alpha(g\cdot z_2, B^{\Delta V_2}, V_3)),\]
and hence as prior to \ref{reflect}
\[\exists^* g\in W_{\frac{1}{n}}
(g\cdot y_1\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec {\cal B}}_\alpha(z_1, B^{\Delta V_2}, V_3)
=\hat{\varphi}^{\vec {\cal B}}_\alpha(g\cdot z_2, B^{\Delta V_2}, V_3)).\]
By (i) we obtain
\[y'\in D^{*W}\]
some basic open neighborhood $W$ of $1_G$. After possibly shrinking $W$ and
$V_0$ we can assume
\[W\subset V_1\]
and that there is some $g_0\in G_0$ with
\[g_0W=V_0.\]
Here it will be easier if we make a harmless simplifying assumption. I claim
that we may assume $V_0=W$ and that $g_0=1_G$. The point is that we can always
replace $A$ by the set
\[g_0^{-1}A,\]
and observe that
\[A^{\Delta g_0 W}=(g_0^{-1}A)^{\Delta W}.\]
So after making this minor notational reorganization let us assume that
\[1_G=g_0,\]
\[W=V_0,\]
\[1_G\in V_0,\]
\[V_0\subset V_1,\]
\[y'\in A^{\Delta V_0}\cap D^{*V_0}\subset (A\cap D)^{\Delta V_0}.\]
Then for any
\[z\in (A\cap D)^{\Delta V_0},\]
\[z_1, z_2\in {\cal O}(z, (A\cap D)^{\Delta V_0}, V_1),\]
and for any
\[\alpha<\delta\]
\[B\in \b_\alpha\]
\[V_2, V_3\subset G {\rm \: basic \:
open}\]
with
\[1_G\in V_3, V_3^{-1}=V_3\]
\[z_1\in B^{\Delta V_2}\]
we may find $h_0\in V_0\subset V_1$ with
\[z'=_{\rm df}h_0\cdot z\in D\cap (A\cap D)^{\Delta V_0},\]
\[\therefore {\cal O}(z', (A\cap D)^{\Delta V_0}, V_1)
={\cal O}(z, (A\cap D)^{\Delta V_0}, V_1),\]
\[\therefore z_1, z_2\in {\cal O}(z', (A\cap D)^{\Delta V_0}, V_1)
\subset {\cal O}(z', A^{\Delta V_0}, V_1),\]
and hence by the assumptions on $D$ and the fact that $z'\in D$ we have
\[\exists^*g\in W_{\frac{1}{m}}
(g\cdot y_1\in B^{\Delta V_2}\wedge
\hat{\varphi}^{\vec {\cal B}}_\alpha(z_1, B^{\Delta V_2}, V_3)
=
\hat{\varphi}^{\vec {\cal B}}_\alpha(g\cdot z_2, B^{\Delta V_2}, V_3)).\]
And thus we have seen that $(A\cap D, V_0, V_1)$ is $\frac{1}{m}$-stable. Which
contradicts assumption (iii) on $y'$.
\end{proof}
But the statement of this lemma is catastrophic with respect to $y$'s hope of having
a $G_\delta$ orbit. It implies in particular
that for all $A\in \Sigma^1_1$, $V_0, V_1\subset G$ basic open
with $1_G\in V_1$, $V_1^{-1}=V_1$, if
\[y'\in A^{\Delta V_0}\]
then
\[\exists y_1, y_2\in A^{\Delta V_0}\cap[y]_G\]
with
\[{\cal O}(y_1, A^{\Delta V_0}, V_1)
={\cal O}(y_2, A^{\Delta V_0}, V_1),\]
\[\therefore \varphi_{\gamma(x)}(y_1, A^{\Delta V_0}, V_1)=
\varphi_{\gamma(x)}(y_2, A^{\Delta V_0}, V_1)
=\varphi_{\gamma(x)}(y', A^{\Delta V_0}, V_1)\]
{\it but}
\[\forall g\in G(d_G(g, 1)<\frac{1}{m}\Rightarrow g\cdot y_1\neq y_2).\]
But now since the sets of the form
\[\{z: \varphi_{\gamma(x)}(z, A^{\Delta V_0}, V_1)
=\varphi_{\gamma(x)}(x', A^{\Delta V_0}, V_1)\}\]
(as $x'$ ranges over $[x]_G$, $A$ ranges over $\Sigma^1_1$, $V_0, V_1$ range
over basic open sets with $V_1$ a symmetric neighborhood of the identity)
form a basis of the topology on $X_x$, we have that the induced map
\[G\rightarrow [y]_G\]
\[g\mapsto g\cdot y\]
is not open. Thus by the Effros lemma, $[y]_G$ is not $G_\delta$.
Thus we have shown that no orbit in $X_x$ is $G_\delta$. And hence all the orbits
are meager.
And hence as in $\S$6.2 of \cite{hjorth}, all the orbits in $X_x$ are dense, the local
orbits of the form
\[{\cal O}(y, A^{\Delta V_0}, V_1)\] are all dense inside the corresponding
open set
\[\{z: \varphi_{\gamma(x)}(z, A^{\Delta V_0}, V_1)
=\varphi_{\gamma(x)}(x', A^{\Delta V_0}, V_1)\}\]
for some $x'\in [x]_G$ with $\varphi_{\gamma(x)}(y, A^{\Delta V_0}, V_1)
=\varphi_{\gamma(x)}(x', A^{\Delta V_0}, V_1)$, and by the calculation above every orbit
is meager. Thus the action is turbulent, and we are done with this case.
\section{Case (2)}
Not case (1).
%And thus in particular we may use transfinite induction and
%the number uniformization
%theorem\footnote{I.e. that if $A\subset \omega\times\omega$ is $\Pi^1_1$ with
%all vertical sections non-empty, then there is $\pi: \omega\rightarrow \omega$
%such that for all $n$ we have $(n, \pi(n))\in A$.}
%for $\Pi^1_1$ to find some sequence
%\[\vec \b=(\b_\alpha)_{\alpha\in\omega_1^{\rm ck}}\]
%such that
\begin{claim} There is a sequence
\[\vec \b=(\b_\alpha)_{\alpha\in\omega_1^{\rm ck}}\]
such that
\leftskip 0.4in
\noindent (a) $\alpha\mapsto \b_\alpha$ is $\Delta^1_1$ in the codes;
\noindent (b) for each $\delta<\omega_1^{\rm ck}$ the sequence
$\vec \b|_\delta = (\b_\alpha)_{\alpha\in\delta}$ is suitable;
\noindent (c) for each $\delta<\omega_1^{\rm ck}$, $\epsilon>0$,
$x\in X$ there is some $D\in \b_\delta$ along with basic open
\[V_0, V_1\subset G,\]
$V_1$ a symmetric open neighborhood of the identity, $(D, V_0, V_1)$ $\epsilon$-stable
for $\vec \b|_\delta$
and
\[x\in D^{\Delta V_0}.\]
\leftskip 0in
\end{claim}
\begin{proof} Recalling from Spector-Gandy that a subset of
$\Lck$ is $\Pi^1_1$ in the codes if and only if it is
$\Sigma_1^{\Lck}$, it suffices to show that there is a
total $\Sigma_1^{\Lck}$ function
\[\alpha \mapsto B_\alpha,\]
with each $B_\alpha$ included in the indices for
$\Delta^1_1$ sets, such that if we let $\b_\alpha$ be the
corresponding collection of actual $\Delta^1_1$ sets encoded
by the $B_\alpha$, then $(\b_\alpha)_{\alpha\in\delta}$
satisfies (c) above and 1, 2, 4, 5, 6 from the definition of suitability
all $\delta<\omega_1^{\rm ck}$; note that these
requirements are all $\Pi^1_1$
and hence $\Sigma_1^\Lck$. Since the
total
$\Sigma_1^\Lck$ functions are closed under transfinite
$\Sigma_1^\Lck$ recursion, it suffices to show that given
any $(B_\alpha)_{\alpha\in \delta}\in \Lck$ which gives
rise to $(\b_\alpha)_{\alpha\in \delta}$ with the desired
properties we can
find some $B_\delta\in \Lck$ such that the slightly extended
corresponding $(\b_\alpha)_{\alpha\in \delta+1}$ still has
(c) above and 1, 2, 4, 5, 6.
We know from the case assumption that given any $x\in X, \epsilon >0$
there will be some $D\in \Delta^1_1$ and basic open $V_0, V_1$
with $x\in D^{\Delta V_0}$ and $(D, V_0, V_1)$ having the
$\Pi^1_1(\epsilon)$ property of $\epsilon$-stability. The problem
is to find a single set of such indices in $\Lck$ which works
for all such $x$ and $\epsilon$.
Towards this purpose let $P\subset \omega\times X$ be $\Pi^1_1$ and
universal for $\Pi^1_1$ subsets of $X$. We let
\[f:\omega\times X\rightarrow {\rm LO}\]
be recursive continuous such that $(n, x)\in P$ if and only if
$f(n, x)\in $ WO. At each $\beta <\omega_1^{\rm ck} $ and $n\in \omega$
we let $P_{\beta, n}=
\{x\in X: f(x) \in {\rm WO}, |\!|f(x)|\!|<\beta\}$\footnote{Here
$|\!|w|\!|$ denotes the ordinal isomorphic to $w$ for $w\in$ WO.}.
Note that the $\Delta^1_1$ subsets of $X$ are exactly the
$\{P_{\alpha, n}: \alpha<\omega_1^{\rm ck}, n\in\N\}$. By the case assumption
and the number uniformization theorem\footnote{I.e. that if $A\subset
\omega^\omega\times\omega$ is $\Pi^1_1$ with
all vertical sections non-empty, then there is $\Delta^1_1$
$\pi: \omega^\omega\rightarrow \omega$
such that for all $z$ we have $(z, \pi(n))\in A$.}, we may find a
$\Delta^1_1$ function
\[(\epsilon, x)\mapsto (\alpha_{\epsilon, x}, n_{\epsilon, x})\]
such that each $(P_{\alpha_{\epsilon, x}, n_{\epsilon, x}}, V_0, V_1)$
is $\epsilon$-stable (some choice of
basic open $V_0, V_1$)
for $(\b_\alpha)_{\alpha\in\delta}$, and
\[x\in P_{\alpha_{\epsilon, x}, n_{\epsilon, x}}^{\Delta V_0}.\]
Boundedness gives
\[{\rm sup}\{\alpha_{\epsilon, x}: \epsilon > 0, x\in X\}<\omega_1^{\rm ck},\]
and so some such recursive ordinal
$\gamma> {\rm sup}\{\alpha_{\epsilon, x}: \epsilon > 0, x\in X\}$ will
give $\{P_{\beta, n}: n\in \N, \beta\leq \gamma\}$ as required.
Under any reasonable system of indexing $\Delta^1_1$ sets we can
find $B_\delta\in \Lck$ such that $B_\delta$ exactly gives the
indices of the $\Delta^1_1$ sets appearing in
$\{P_{\beta, n}: n\in \N, \beta\leq \gamma\}$.
\end{proof}
\begin{claim}
\label{equivalence}
For all $x, y\in X$
\[\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(x, X, G)=
\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(y, X, G)\Rightarrow x E_G y.\]
\end{claim}
\begin{proof} Suppose not. Then we may find $x, y\in X$ with
\[\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(x, X, G)=
\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(y, X, G)\]
but
\[x\neg E_G y.\]
But then using for the first time
that $E_G$ is $\Delta^1_1$ we may apply the low basis theorem\footnote{I.e. any
non-empty $\Sigma^1_1$ contains an $x$ with $\omega_1^{{\rm ck}(x)}=\omega_1^{\rm ck}.$}
for
$\Sigma^1_1$ and find $x, y\in X$ with
\[\omega_1^{{\rm ck}(x, y)}=\omega_1^{\rm ck},\]
\[\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(x, X, G)=
\hat{\varphi}_{\omega_1^{\rm ck}}^{\vec\b}(y, X, G),\]
\[x\neg E_G y.\]
Now choose a recursive $\epsilon_0>0$ such that
\[\epsilon_0<\frac{1}{2}\]
and
\[\forall g\in W_{\epsilon_0}(d_X(y, g\cdot y)<\frac{1}{2}).\]
\smallskip
\noindent{\bf Subclaim:} For all $\beta<\omega_1^{\rm ck}$ there exists
\[x_0\in [x]_G\]
such that for all $\alpha<\beta$, $B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\epsilon_0}(\hat{\varphi}_{\alpha}^{\vec\b}
(x_0, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot y, B^{\Delta V_2}, V_3)).\]
\smallskip
\no {\bf Proof of subclaim:} Choose $A\in\b_{\beta}$, $V_0, V_1\subset G$
basic open sets with $V_1$ a symmetric neighborhood of the identity, $(A, V_0, V_1)$
$\epsilon_0$-stable for $\vec\b|_{\beta}$, and
\[y\in A^{\Delta V_0}.\]
Then choose $x_0E_G x$ with
\[\hat{\varphi}_{\beta}^{\vec\b}
(x_0, A^{\Delta V_0}, V_1)=
\hat{\varphi}_{\beta}^{\vec\b}(y, A^{\Delta V_0}, V_1).\]
It follows from the iterative
structure of the definition of the $\hat{\varphi}_{\beta}^{\vec\b}(\cdot, \cdot, \cdot)$
invariants that for each $B, V_2, V_3,\alpha$ as in statement of subclaim that
\[\exists\hat{y}\in{\cal O}(y, A^{\Delta V_0}, V_1)\]
with
\[\hat{\varphi}_{\alpha}^{\vec\b}
(x_0, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(\hat{y}, B^{\Delta V_2}, V_3).\]
Then by definition of stability we have that
\[\exists^*h\in W_{\epsilon_0}
(\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot y, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(\hat{y}, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha^{\rm ck}}^{\vec\b}
(x_0, B^{\Delta V_2}, V_3)).\]
\hfill (Subclaim$\Box$)
Now we may apply {\it overspill}\footnote{I.e. the converse of the usual
boundedness theorem for $\Sigma^1_1$ subset of the recursive ordinals.} to find some
\[x_0\in [x]_G\]
such that for all
$\alpha<\omega_1^{\rm ck}$, $B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\epsilon_0}(\hat{\varphi}_{\alpha}^{\vec\b}
(x_0, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot y, B^{\Delta V_2}, V_3)).\]
By the low basis theorem we may assume
\[\omega_1^{{\rm ck}(x_0, x, y)}=\omega_1^{{\rm ck}(x, y)}
=\omega_1^{\rm ck}.\]
At this stage we can choose a recursive real
$\delta_0$ such that
\[\delta_0<\frac{1}{2}\]
\[\forall g\in W_{\delta_0}(d_X(x_0, g\cdot x_0)<\frac{1}{2}).\]
\smallskip
\noindent{\bf Subclaim:} For all $\beta<\omega_1^{\rm ck}$ there exists
\[y_0\in W_{\epsilon_0}\cdot y\]
such that for all $\alpha<\beta$,
$B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\delta_0}(\hat{\varphi}_{\alpha}^{\vec\b}
(y_0, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot x_0, B^{\Delta V_2}, V_3)).\]
\smallskip
\noindent{\bf Proof of subclaim:} Again we may choose some $(A, V_0, V_1)$
which is $\delta_0$-stable for $\vec\b|_{\beta}$ with $x_0\in A^{\Delta V_0}$.
And then we may choose
\[y_0\in W_{\epsilon_0}\cdot y\]
with
\[\hat{\varphi}_{\beta}^{\vec\b}
(x_0, A^{\Delta V_0}, V_1)=
\hat{\varphi}_{\beta}^{\vec\b}(y, A^{\Delta V_0}, V_1),\]
and we may complete the argument as in the proof of the
last subclaim.
\hfill (Subclaim$\Box$)
\smallskip
Once more overspill allows us to find $y_0\in W_{\epsilon_0}$ such that
for all
$\alpha<\omega_1^{\rm ck}$, $B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\delta_0}(\hat{\varphi}_{\alpha}^{\vec\b}
(y_0, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot x_0, B^{\Delta V_2}, V_3)).\]
Continuing in a like minded fashion we may produce sequences
\[g_0, g_1, \cdots,\]
\[x_0, x_1, \cdots,\]
\[g'_0, g'_1,\cdots,\]
\[y_0, y_1,\cdots,\]
\[\delta_0, \delta_1,\cdots,\]
\[\epsilon_0, \epsilon_1,\cdots,\]
such that:
\leftskip 0.4in
\noindent (0) for all $j\in\omega$,
\[\omega_1^{{\rm ck}(x_j, y_j)}=\omega_1^{{\rm ck}(x_{j+1}, y_j)}
=\omega_1^{\rm ck};\]
\noindent (i) for all $j\in\omega$,
$\alpha<\omega_1^{\rm ck}$, $B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\delta_j}(\hat{\varphi}_{\alpha}^{\vec\b}
(y_j, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot x_j, B^{\Delta V_2}, V_3));\]
\noindent (ii) for all $j\in\omega$,
$\alpha<\omega_1^{\rm ck}$, $B\in \b_\alpha$, $V_2, V_3\subset G$ basic open
with $V_3$ a symmetric neighborhood of $1_G$,
\[\exists^*h\in W_{\delta_{j+1}}(\hat{\varphi}_{\alpha}^{\vec\b}
(x_{j+1}, B^{\Delta V_2}, V_3)=
\hat{\varphi}_{\alpha}^{\vec\b}(h\cdot y_j, B^{\Delta V_2}, V_3));\]
\noindent (iii) each $x_j$ equals $g_{j-1}g_{j-2}\cdots g_0\cdot x_0$;
\noindent (iv) each $y_j$ equals $g_{j-1}'g_{j-2}'\cdots g_0'\cdot y_0$;
\noindent (v) each $g_j\in W_{\delta_j}$;
\noindent (vi) each $g_j'\in W_{\epsilon_{j+1}}$;
\noindent (vii) at each $j$ we have $\forall g\in W_{\delta_j}$
\[d_G(gg_{j-1}g_{j-2}\cdots g_0, g_{j-1}\cdots g_0)<2^{-j-1};\]
\noindent (viii) at each $j$ we have $\forall g\in W_{\epsilon_j}$
\[d_G(gg_{j-1}'g_{j-2}'\cdots g_0', g_{j-1}'\cdots g_0')<2^{-j-1};\]
\noindent (ix) at each $j$
\[\forall g\in W_{\delta_j}(d_X(x_j, g\cdot x_j)<2^{-j-1});\]
\noindent (x) at each $j$
\[\forall g\in W_{\epsilon_{j+1}}(d_X(y_j, g\cdot y_j)<2^{-j-1}).\]
All the relevant sequences are Cauchy, so we can find $g_\infty$,
$g_\infty '$ with
\[g_j\rightarrow g_\infty,\]
\[g_j'\rightarrow g_\infty',\]
and hence by the continuity of the action
\[x_j\rightarrow g_\infty x_0,\]
\[y_j\rightarrow g_\infty ' y_0.\]
We will be finished with the proof of the claim if we show that
\[g_\infty x_0 = g_\infty ' y_0.\]
But note that at any $j$ we can apply (i) to find $h\in W_{\delta_j}$ with
\[d_X(h\cdot x_j, y_j)< 2^{-j-1};\]
which by (v) yields
\[d_X(x_j, y_j)\leq d_X(x_i, h\cdot x_i) +
d_X(h\cdot x_i, y_i)< 2^{-j-1}+ 2^{-j-1}=2^{-j}.\]
And so
\[x_0 E_G y_0\]
\[\therefore x E_G y,\]
after all.
\end{proof}
\begin{notation} Let HC denote the collection of hereditarily countable sets.
TC$(a)$ denotes the transitive closure of a set $a$.
\end{notation}
\begin{definition} Let us say that
\[\theta: X\rightarrow {\rm HC}\]
is $\Delta^1_1$ {\it in the codes} if there is an actual $\Delta^1_1$
function
\[\Theta:X\rightarrow 2^{\omega\times\omega}\]
such that for all $x\in X$ we have that
$\Theta(x)$ is a well founded and extensional relation
on $\omega$ whose transitive collapse is isomorphic to
\[{\rm TC}(\theta(x))\cup\{\theta(x)\};\]
that is to say, $\Theta(x)$ encodes the smallest transitive set containing
$\theta(x)$ as an element.
\end{definition}
\begin{claim}
For all $\alpha<\omega_1^{\rm ck}$, $B\in\b_\alpha$, $V_0, V_1\subset G$ basic open with
$V_1$ a symmetric neighborhood of the identity, the function
\[x\mapsto \hat{\varphi}_\alpha^{\vec\b}(x, B^{\Delta V_0}, V_1)\]
is $\Delta^1_1$
in the codes.
\end{claim}
\begin{proof} This follows routinely by induction on $\alpha$.
\end{proof} Thus we may obtain a Borel function
\[\theta: X\rightarrow 2^{\omega\times\omega}\]
such that for all $x\in X$ the relation $\theta(x)$ on $\omega$ is wellfounded,
extensional, and has transitive collapse equal to
\[\{\hat{\varphi}_\alpha^{\vec\b}(x, X, G): \alpha<\omega_1^{\rm ck}\}
\cup\{{\rm TC}(\hat{\varphi}_\alpha^{\vec\b}(x, X, G)):
\alpha<\omega_1^{\rm ck}\}.\]
Hence
\[xE_G y\]
if and only if $\theta(x)$ and $\theta(y)$ are isomorphic as relations on
$\omega$. Thus $E_G$ is reducible to
\[\cong\!|_{{\rm Mod}(\l)},\]
for $\l$ a language generated by a single binary relation.
\bigskip
Finally some words on how we can extend the conclusion of this case
to obtain a reduction
that allows an orbit inverse.
To start we may apply theorem 5.1.5 of \cite{beckerkechris} to produce a Polish topology
$\hat{\tau}$ on $X$ which is compatible with the original Borel structure,
includes all sets of the form
\[B^{\Delta V}\]
where $B\in\b_\alpha$ some $\alpha<\omega_1^{\rm ck}$, and is such that the
action of $G$ is still continuous with respect to this space.
From here we can consider the Polish $G$-space $(X, \hat{\tau})$. For this space we
may directly evoke the generalized Scott invariants
\[x\mapsto \varphi_x\]
from $\S$6.2 of \cite{hjorth}.
Traipsing through the definitions there one sees that
\[\varphi_x=\varphi_y\Rightarrow \forall \alpha <\omega_1
(\hat{\varphi}_\alpha^{\vec \b}(x, X, G)=
\hat{\varphi}_\alpha^{\vec \b}(y, X, G)),\]
and so by \ref{equivalence}
\[x\mapsto \varphi_x\]
gives a complete invariant.
It was shown in $\S$6.2\cite{hjorth} that whenever we have these $\varphi_x$'s serving
as complete invariants then we can indeed obtain a Borel reduction to isomorphism
on a class of countable structures which admits an orbit inverse.
And thus case(2) leads to a strengthened version of (I) as its outcome.
\newpage
\begin{thebibliography}{99}
\bibitem{beckerkechris}
H. Becker, A.S. Kechris,
{\bf The descriptive set theory of Polish group actions,}
London Mathematical Society.
\bibitem{hamash} L.A. Harrington, D. Marker, S. Shelah,
{\it Borel orders,} {\bf Transactions of the American
Mathematical Society,}
vol. 310(1988), pp. 293-302.
\bibitem{hjorth} G. Hjorth,
{\bf Classification and orbit equivalence relations,}
American Mathematical Society, Rhode Island, 2000.
\bibitem{kechrislouveau}
A.S. Kechris, A. Louveau, {\it The classification of hypersmooth Borel equivalence
relations,}
{\bf Journal of the American Mathematical Society,}
vol. 10(1997), pp. 215--242.
\bibitem{moschovakis} Y.N. Moschovakis,
{\bf Descriptive set theory,} North-Holland, Amsterdam,
1980.
\end{thebibliography}
\leftskip 0.5in
greg@math.ucla.edu
\bigskip
MSB 6363
UCLA
CA 90095-1555
\leftskip 0in
\end{document}
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