text:
\section{The recurrence spectrum} \label{recurrence}
Suppose that $G \curvearrowright X$ is a group action. We say that a set
$Y \subseteq X$ is \definedterm{complete} if $X = GY$, and
\definedterm{$\sigma$-complete} if there is a countable set $H
\subseteq G$ for which $X = HY$. The following observation
ensures that, under mild hypotheses, these notions coincide on
open sets.

\begin{proposition} \label{recurrence:stronglycomplete}
  Suppose that $G$ is a topological group, $H \subseteq G$ is
  dense, $X$ is a topological space, $G \curvearrowright X$ is
  continuous-in-$G$, and $U \subseteq X$ is open. Then $GU =
  HU$.
\end{proposition}

\begin{propositionproof}
  Suppose that $g \in G$ and $x \in U$. As $G \curvearrowright X$ is
  continuous-in-$G$, there is an open neighborhood $V \subseteq
  G$ of $\identity{G}$ for which $Vx \subseteq U$. Fix an open
  neighborhood $W \subseteq G$ of $g$ for which $\inverse{W} g
  \subseteq V$. As $H$ is dense, there exists $h \in H \intersection
  W$, and it only remains to observe that $\inverse{h} g \in V$, so
  $\inverse{h} g \cdot x \in U$, thus $g \cdot x \in hU$.
\end{propositionproof}

Let $\Ezero$ denote the equivalence relation on $\Cantorspace$
given by $c \mathrel{\Ezero} d \iff \exists n \in \amsmathbb{N} \forall m \ge n
\ c(m) = d(m)$. We will abuse language by saying that a subset
of $X$ is \definedterm{$\aleph_0$-universally Baire\xspace} if its
pre-image under every Bor\-el\xspace function from a Po\-lish\xspace space to
$X$ has the Baire\xspace property, and an
$\aleph_0$-universally-Baire\xspace equivalence relation $E$ on $X$
is \definedterm{smooth} if there is no Bor\-el\xspace function $\pi \colon
\Cantorspace \to X$ such that $c \mathrel{\Ezero} d \iff \pi(c)
\mathrel{E} \pi(d)$ for all $c, d \in \Cantorspace$. The
Har\-ring\-ton\xspace-Kech\-ris\xspace-Lou\-veau\xspace generalization of the
Glimm\xspace-Eff\-ros\xspace dichotomy (see \cite[Theorem 1.1]
{HKL90}) ensures that this is compatible
with the usual notion of smoothness for Bor\-el\xspace equivalence
relations on standard Bor\-el\xspace spaces.

When $X$ is a topological space, a continuous-in-$X$ action $G
\curvearrowright X$ is \definedterm{topologically transitive} if for all non-empty
open sets $U, V \subseteq X$ there exists $g \in G$ such that $gU
\intersection V \neq \emptyset$, and \definedterm{minimal} if every
orbit is dense.

\begin{proposition} \label{recurrence:universallyBaire}
  Suppose that $G$ is a group, $X$ is a Po\-lish\xspace space, $G \curvearrowright
  X$ is continuous-in-$X$, $B \subseteq X$ is
  $\orbitequivalencerelation{G}{X}$-invariant, $C \subseteq X$ is an
  $\orbitequivalencerelation{G}{X}$-invariant $G_\delta$\xspace set for which $G
  \curvearrowright C$ is topologically transitive and in which $B$ is comeager,
  and $F$ is a smooth $\aleph_0$-universally-Baire\xspace
  superequivalence relation of $\orbitequivalencerelation{G}{B}$.
  Then there is an $F$-class that is comeager in $C$.
\end{proposition}

\begin{propositionproof}
  Fix a dense $G_\delta$\xspace set $C' \subseteq C$ contained in $B$, and
  note that $F$ has the Baire\xspace property in $C' \times C'$, thus
  in $C \times C$. The straightforward generalization of the
  Beck\-er\xspace-Kech\-ris\xspace criterion for continuously embedding $\Ezero$
  from orbit equivalence relations induced by groups of
  homeomorphisms (see \cite[Theorem 3.4.5]{BK96}) to
  superequivalence relations of such orbit equivalence relations (see,
  for example, \cite[Theorem 2.1]{KMS14}) ensures that
  the union of $F$ and $(C \setminus B) \times (C \setminus B)$ is
  non-meager in $C \times C$, so the Kur\-at\-ow\-ski\xspace-U\-lam\xspace theorem
  (see, for example, \cite[Theorem 8.41]{Kec95}) yields an
  $F$-class that is non-meager and has the Baire\xspace property in $C$,
  thus comeager in $C$ by topological transitivity.
\end{propositionproof}

For each set $Y \subseteq X$, define $\returntimes{Y} = \set{g \in
G}[Y \intersection gY \neq \emptyset]$. The following fact is the
obvious generalization of Pet\-tis\xspace's Lemma (see, for example, \cite
[Theorem 9.9]{Kec95}) to group actions.

\begin{proposition} \label{recurrence:Pettis:global}
  Suppose that $G$ is a group, $X$ is a Baire\xspace space, $G \curvearrowright
  X$ is continuous-in-$X$, $U \subseteq X$ is non-empty and
  open, and $B \subseteq U$ is comeager. Then $\returntimes
  {U} \subseteq \returntimes{B}$.
\end{proposition}

\begin{propositionproof}
  Note that for all $g \in G$, the fact that $G \curvearrowright X$ is
  continuous-in-$X$ ensures that $gU$ is open and $gB$ is
  comeager in $gU$. In particular, it follows that if $U \intersection
  gU$ is non-empty, then $B \intersection gB$ is comeager in $U
  \intersection gU$, in which case the fact that $X$ is a Baire\xspace space
  ensures that $B \intersection gB$ is also non-empty, thus
  $\returntimes{U} \subseteq \returntimes{B}$.
\end{propositionproof}

Given an upward-closed family $\mathcal{F} \subseteq \powerset{G}$ and
any family $\Gamma \subseteq \powerset{X}$, we say that an
action $G \curvearrowright X$ is \definedterm{$\mathcal{F}$-recurrent on
$\Gamma$} if $\returntimes{B} \in \mathcal{F}$ for all $B \in \Gamma$.
This generalizes the usual notion of recurrence in topological
dynamics, where one says that a continuous-in-$X$ action of a
group on a topological space $X$ is \definedterm{$\mathcal{F}$-recurrent} if
it is $\mathcal{F}$-recurrent on non-empty open sets. We next note that,
under mild hypotheses, this latter notion propagates to recurrence
on non-meager sets with the Baire\xspace property.

\begin{proposition} \label{recurrence:openBaire:global}
  Suppose that $G$ is a group, $X$ is a Baire\xspace space, $\mathcal{F}
  \subseteq \powerset{G}$ is upward closed, and $G \curvearrowright X$ is
  continuous-in-$X$ and $\mathcal{F}$-recurrent. Then $G \curvearrowright X$ is
  $\mathcal{F}$-recurrent on non-meager sets with the Baire\xspace property.
\end{proposition}

\begin{propositionproof}
  If $B \subseteq X$ is a non-meager set with the Baire\xspace property,
  then there is a non-empty open set $U \subseteq X$ in which $B$
  is comeager, and Proposition \ref{recurrence:Pettis:global}
  ensures that $\returntimes{U} \subseteq \returntimes{B}$, thus
  $\returntimes{B} \in \mathcal{F}$.
\end{propositionproof}

When $X$ is a Po\-lish\xspace space, the \definedterm{decomposition
into minimal components} of a continuous action $G \curvearrowright X$
is the equivalence relation on $X$ given by $x \mathrel
{\minimaldecomposition{G}{X}} y \iff \closure{\orbit{x}{G}} = \closure
{\orbit{y}{G}}$. It is easy to see that $\minimaldecomposition{G}{X}$
is $G_\delta$\xspace and smooth (although the latter is also a direct consequence
of the former and \cite[Corollary 1.2]{HKL90}), and
that for each $\minimaldecomposition{G}{X}$-class $C$, the action $G
\curvearrowright C$ is minimal.

When $G$ is Bor\-el\xspace, the \definedterm{recurrence spectrum} of a
Bor\-el\xspace action $G \curvearrowright X$ is the collection of all upward-closed
families $\mathcal{F} \subseteq \powerset{G}$ such that every smooth
Bor\-el\xspace superequivalence relation $F$ of $\orbitequivalencerelation
{G}{X}$ has an equivalence class $C$ for which $G \curvearrowright C$ is
$\mathcal{F}$-recurrent on $\sigma$-complete Bor\-el\xspace sets.  As a theorem
of Beck\-er\xspace-Kech\-ris\xspace ensures that every Bor\-el\xspace action of a Po\-lish\xspace
group on a standard Bor\-el\xspace space is Bor\-el\xspace isomorphic to a
continuous action on a Po\-lish\xspace space (see \cite[Theorem 5.2.1]
{BK96}), the following observation ensures that, under
mild hypotheses, the notion of recurrence spectrum is robust, in
the sense that it does not depend on the particular underlying notion
of definability, and in the sense that it is invariant under passage to
sufficiently large $\orbitequivalencerelation{G}{X}$-invariant subsets.

\begin{proposition} \label{recurrence:characterization:global}
  Suppose that $G$ is a separable group, $X$ is a
  Po\-lish\xspace space, $G \curvearrowright X$ is continuous, $\mathcal{F} \subseteq
  \powerset{G}$ is upward closed, and $B \subseteq X$ is
  $\orbitequivalencerelation{G}{X}$-invariant and comeager in every
  $\minimaldecomposition{G}{X}$-class. Then the following are
  equivalent:
  \begin{enumerate}
    \item Every smooth $\aleph_0$-universally-Baire\xspace
    superequivalence relation $F$ of $\orbitequivalencerelation{G}
      {B}$ has a class $C$ for which $G \curvearrowright C$ is
      $\mathcal{F}$-recurrent on $\sigma$-complete
      $\aleph_0$-universally-Baire\xspace sets.
    \item There is an $\minimaldecomposition{G}{X}$-class $C$ for
      which $G \curvearrowright C$ is $\mathcal{F}$-recurrent.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  To see $(1) \implies (2)$, fix an $\minimaldecomposition{G}
  {X}$-class $C$ for which $G \curvearrowright B \intersection C$ is
  $\mathcal{F}$-recurrent on $\sigma$-complete open sets. To see that $G
  \curvearrowright C$ is $\mathcal{F}$-recurrent, suppose that $U \subseteq C$ is a
  non-empty open set, and note that the minimality of $G \curvearrowright C$
  ensures that $U$ is complete, and therefore $\sigma$-complete by
  Proposition \ref{recurrence:stronglycomplete}, thus $\returntimes{U}
  \in \mathcal{F}$.

  To see $(2) \implies (1)$, fix an $\minimaldecomposition{G}
  {X}$-class $C$ for which $G \curvearrowright C$ is $\mathcal{F}$-recurrent, and
  suppose that $F$ is a smooth $\aleph_0$-universally-Baire\xspace
  superequivalence relation of $\orbitequivalencerelation{G}{B}$.
  Proposition \ref{recurrence:universallyBaire} then yields an
  $F$-class $D$ that is comeager in $C$. To see that $G \curvearrowright D$
  is $\mathcal{F}$-recurrent on $\sigma$-complete
  $\aleph_0$-universally-Baire\xspace sets, suppose that $A \subseteq D$
  is such a set, and note that $\sigma$-completeness ensures that
  $A$ is non-meager in $C$. Fix a dense $G_\delta$\xspace set $C' \subseteq
  C$ contained in $D$, and note that $A \intersection C'$ has the
  Baire\xspace property in $C'$, so $A$ has the Baire\xspace property in $C$,
  thus Proposition \ref{recurrence:openBaire:global} ensures that
  $\returntimes{A} \in \mathcal{F}$.
\end{propositionproof}

Let $\dual{\Gamma}$ denote the family of sets whose complements
are in $\Gamma$, let $\Dtwo{\Gamma}$ denote the family of
differences of sets in $\Gamma$, and let $\unions{\Gamma}$
denote the family of countable unions of sets in $\Gamma$.

The \definedterm{horizontal sections} of a set $R \subseteq X \times
Y$ are the sets of the form $\horizontalsection{R}{y} = \set{x \in X}
[x \mathrel{R} y]$ for $y \in Y$, whereas the \definedterm{vertical
sections} of a set $R \subseteq X \times Y$ are the sets of the form
$\verticalsection{R}{x} = \set{y \in Y}[x \mathrel{R} y]$ for $x \in
X$. We say that $\mathcal{F}$ is \definedterm{$\Gamma$-on-open} if
$\set{x \in X}[\horizontalsection{U}{x} \in \mathcal{F}] \in \Gamma$ for all
open sets $U \subseteq G \times X$.

Given a superequivalence relation $E$ of $\orbitequivalencerelation
{G}{X}$, we say that an action $G \curvearrowright X$ is \definedterm
{$E$-locally $\mathcal{F}$-recurrent on $\Gamma$} if for all $B \in
\Gamma$, there is an $E$-class $C$ such that $\returntimes{B
\intersection C} \in \mathcal{F}$. We next note that, under mild hypotheses,
the recurrence spectrum can also be characterized in terms of local
recurrence of $G \curvearrowright X$ itself.

\begin{proposition} \label{recurrence:complete}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace space, $G
  \curvearrowright X$ is continuous, $\Gamma \subseteq \powerset{X}$ is a
  family of $\aleph_0$-universally-Baire\xspace sets containing the open
  sets and closed under finite intersections and finite unions, and
  $\mathcal{F} \subseteq \powerset{G}$ is upward closed and $\dual
  {\Gamma}$-on-open. Then the following are equivalent:
  \begin{enumerate}
    \item There is an $\minimaldecomposition{G}{X}$-class $C$ for
      which $G \curvearrowright C$ is $\mathcal{F}$-recurrent.
    \item The action $G \curvearrowright X$ is $\minimaldecomposition{G}
      {X}$-locally $\mathcal{F}$-recurrent on $\sigma$-com\-plete
      $\unions{(\Dtwo{\Gamma})}$ sets.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  To see $\neg (2) \implies \neg (1)$, observe that if $B \subseteq X$
  is a $\sigma$-complete $\unions{(\Dtwo{\Gamma})}$ set such
  that $\returntimes{B \intersection C} \notin \mathcal{F}$ for every
  $\minimaldecomposition{G}{X}$-class $C$, then it is non-meager
  and has the Baire\xspace property in every such class, so there is no
  $\minimaldecomposition{G}{X}$-class $C$ for which $G \curvearrowright C$
  is $\mathcal{F}$-recurrent by Proposition \ref{recurrence:openBaire:global}.

  To see $\neg (1) \implies \neg (2)$, fix a basis $\sequence{U_n}[n
  \in \amsmathbb{N}]$ for $X$. For all $n \in \amsmathbb{N}$, define $V_n = \set{\pair{g}{x} \in
  G \times X}[U_n \intersection gU_n \intersection \equivalenceclass
  {x}{\minimaldecomposition{G}{X}} \neq \emptyset]$. Observe
  that if $\pair{g}{x} \in V_n$, then the minimality of $G \curvearrowright
  \equivalenceclass{x}{\minimaldecomposition{G}{X}}$ yields $h \in
  G$ for which $h \cdot x \in U_n \intersection gU_n$, so the
  continuity of $G \curvearrowright X$ yields open neighborhoods $U_g
  \subseteq G$ of $g$ and $U_x \subseteq X$ of $x$ such that
  $hU_x \union \inverse{U_g}hU_x \subseteq U_n$, thus $U_g
  \times U_x \subseteq V_n$, hence $V_n$ is open. It follows that
  the $\minimaldecomposition{G}{X}$-invariant sets $A_n = \set{x
  \in G U_n}[\horizontalsection{V_n}{x} \notin \mathcal{F}]$ are in
  $\Gamma$, so the sets $B_n = A_n \setminus \union[m < n]
  [A_m]$ are in $\Dtwo{\Gamma}$, thus the set $B = \union[n \in
  \amsmathbb{N}][B_n \intersection U_n]$ is in $\unions{(\Dtwo{\Gamma})}$.
  But if there is no $\minimaldecomposition{G}{X}$-class $C$ for
  which $G \curvearrowright C$ is $\mathcal{F}$-recurrent, then $B$ is complete,
  and therefore $\sigma$-complete by Proposition \ref
  {recurrence:stronglycomplete}, thus $G \curvearrowright X$ is not
  $\minimaldecomposition{G}{X}$-locally $\mathcal{F}$-recurrent on
  $\sigma$-complete $\unions{(\Dtwo{\Gamma})}$ sets.
\end{propositionproof}

We next show that, under an additional mild hypothesis on $G$,
Propositions \ref{recurrence:Pettis:global}, \ref
{recurrence:openBaire:global}, and \ref
{recurrence:characterization:global} can be strengthened so as to
show that the recurrence spectrum is also robust in the sense that
it does not depend on whether the underlying notion of recurrence
is local.

The following fact is a somewhat more intricate generalization of
the special case of Pet\-tis\xspace's Lemma for second-countable groups
to continuous actions of such groups.

\begin{proposition} \label{recurrence:Pettis:local}
  Suppose that $G$ is a second-countable Baire\xspace group, $X$ is a
  second-countable Baire\xspace space, $G \curvearrowright X$ is continuous, $U
  \subseteq X$ is non-empty and open, and $B \subseteq U$ is
  comeager. Then $\returntimes{U \intersection \orbit{x}{G}}
  \subseteq \returntimes{B \intersection \orbit{x}{G}}$ for comeagerly
  many $x \in X$.
\end{proposition}

\begin{propositionproof}
  We write $\forall^* x \in X \ \phi(x)$ to indicate that
  $\set{x \in X}[\phi(x)]$ is comeager. As the fact that $G \curvearrowright X$
  is continuous-in-$X$ ensures that it is open, it follows that $\set
  {\pair{g}{x} \in G \times X}[g \cdot x \notin U \setminus B]$ is
  comeager, so the set $C = \set{x \in X}[\forall^* g \in
  G \ g \cdot x \notin U \setminus B]$ is comeager by the
  Kur\-at\-ow\-ski\xspace-U\-lam\xspace theorem. Observe that if $h \in G$ and $x \in X$,
  then $\set{g \in G}[g \cdot x \notin h (U \setminus B)] = h \set{g \in
  G}[g \cdot x \notin U \setminus B]$, so the fact that $G \curvearrowright X$ is
  continuous-in-$X$ also ensures that if $x \in C$, then
  $\forall^* g \in G \ g \cdot x \notin (U \setminus B)
  \union h(U \setminus B)$, in which case the fact that $(U
  \intersection hU) \setminus (B \intersection hB) \subseteq (U
  \setminus B) \union h(U \setminus B)$ therefore implies that
  $\forall^* g \in G \ g \cdot x \notin (U \intersection hU)
  \setminus (B \intersection hB)$. Note now that for all $h \in G$,
  the fact that $G \curvearrowright X$ is continuous-in-$X$ ensures that $U
  \intersection hU$ is open, so the fact that $G \curvearrowright X$ is
  continuous-in-$G$ implies that if $x \in X$ and $U \intersection
  hU \intersection \orbit{x}{G}$ is non-empty, then there are
  non-meagerly many $g \in G$ for which $g \cdot x \in U
  \intersection hU$. In particular, it follows that if $x \in C$ and $U
  \intersection hU \intersection \orbit{x}{G}$ is non-empty, then so
  too is $B \intersection hB \intersection \orbit{x}{G}$, hence
  $\returntimes{U \intersection \orbit{x}{G}} \subseteq \returntimes
  {B \intersection \orbit{x}{G}}$ for all $x \in C$.
\end{propositionproof}

We next note that, under mild hypotheses, $\mathcal{F}$-recurrence of
topologically transitive actions not only propagates to
$\mathcal{F}$-recurrence on non-meager sets with the Baire\xspace property,
but to its $\orbitequivalencerelation{G}{X}$-local strengthening.

\begin{proposition} \label{recurrence:openBaire:local}
  Suppose that $G$ is a second-countable Baire\xspace group, $X$ is a
  second-countable Baire\xspace space, $\mathcal{F} \subseteq \powerset{G}$ is
  upward closed, and $G \curvearrowright X$ is continuous, $\mathcal{F}$-recurrent,
  and topologically transitive. Then $G \curvearrowright X$ is
  $\orbitequivalencerelation{G}{X}$-locally $\mathcal{F}$-recurrent on
  non-meager sets with the Baire\xspace property.
\end{proposition}

\begin{propositionproof}
  Suppose that $B \subseteq X$ is a non-meager set with the Baire\xspace
  property, and fix a non-empty open set $U \subseteq X$ in
  which $B$ is comeager. The topological transitivity of $G \curvearrowright X$
  ensures that the set $C = \set{x \in X}[\orbit{x}{G} \text{ is dense}]$
  is comeager, and Proposition \ref{recurrence:Pettis:local} implies
  that the set $D = \set{x \in X}[\returntimes{U \intersection \orbit{x}{G}}
  \subseteq \returntimes{B \intersection \orbit{x}{G}}]$ is comeager.
  So it only remains to observe that if $x \in C \intersection D$, then
  $\returntimes{U} \subseteq \returntimes{U \intersection \orbit{x}{G}}
  \subseteq \returntimes{B \intersection \orbit{x}{G}}$, thus
  $\returntimes{B \intersection \orbit{x}{G}} \in \mathcal{F}$.
\end{propositionproof}

We can now establish the promised robustness result.

\begin{proposition} \label{recurrence:characterization:local}
  Suppose that $G$ is a second-countable Baire\xspace group, $X$ is a
  Po\-lish\xspace space, $G \curvearrowright X$ is continuous, $\mathcal{F} \subseteq
  \powerset{G}$ is upward closed, and $B \subseteq X$ is
  $\orbitequivalencerelation{G}{X}$-invariant and comeager in
  every $\minimaldecomposition{G}{X}$-class. Then the following
  are equivalent:
  \begin{enumerate}
    \item Every smooth $\aleph_0$-universally-Baire\xspace superequivalence
      relation $F$ of $\orbitequivalencerelation{G}{B}$ has a class $C$
      for which $G \curvearrowright C$ is $\orbitequivalencerelation{G}
      {C}$-locally $\mathcal{F}$-recurrent on $\sigma$-complete
      $\aleph_0$-universally-Baire\xspace sets.
    \item There is an $\minimaldecomposition{G}{X}$-class $C$ for
      which $G \curvearrowright C$ is $\mathcal{F}$-recurrent.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  Exactly as in the proof of Proposition \ref
  {recurrence:characterization:global}, but replacing the use of
  Proposition \ref{recurrence:openBaire:global}
  with that of Proposition \ref{recurrence:openBaire:local}.
\end{propositionproof}

\section{The strongest notion of recurrence} \label{maximal}

Given a set $S \subseteq G$, we say that a set $Y \subseteq X$ is
\definedterm{$S$-transient} if $Y \intersection SY = \emptyset$.

\begin{proposition} \label{maximal:cocompact:transient}
  Suppose that $G$ is a topological group, $X$ is a Haus\-dorff\xspace
  space, $G \curvearrowright X$ is continuous, $K \subseteq G$ is compact,
  and $x \in X$ is not fixed by any element of $K$. Then there is a
  $K$-transient open neighborhood of $x$.
\end{proposition}

\begin{propositionproof}
  For each $g \in K$, the fact that $X$ is Haus\-dorff\xspace yields disjoint
  open neighborhoods $V_g \subseteq X$ of $x$ and $W_g
  \subseteq X$ of $g \cdot x$, and the continuity of $G \curvearrowright X$
  yields open neighborhoods $U_g \subseteq G$ of $g$ and $V_g'
  \subseteq V_g$ of $x$ for which $U_g V_g' \subseteq W_g$,
  thus $U_g V_g' \intersection V_g' = \emptyset$. The compactness
  of $K$ then yields a finite set $F \subseteq K$ for which $K
  \subseteq \union[g \in F][U_g]$, in which case $\intersection[g \in
  F][V_g']$ is a $K$-transient open neighborhood of $x$.
\end{propositionproof}

It follows that upward-closed families whose corresponding notions
of recurrence are realizable by suitable free actions necessarily
contain all co-compact neighborhoods of the identity.

\begin{proposition}
  Suppose that $G$ is a topological group and $\mathcal{F} \subseteq
  \powerset{G}$ is an upward closed family for which there is an
  $\mathcal{F}$-recurrent continuous free action of $G$ on a Haus\-dorff\xspace
  space. Then $\mathcal{F}$ contains every co-compact neighborhood
  of $\identity{G}$.
\end{proposition}

\begin{propositionproof}
  This is a direct consequence of Proposition \ref
  {maximal:cocompact:transient}.
\end{propositionproof}

Given sets $Y, Z \subseteq X$, define $\hittingtimes{Y}{Z} =
\set{g \in G}[gY \intersection Z \neq \emptyset]$. When $X$ is a
topological space, we say that a continuous-in-$X$ action $G
\curvearrowright X$ is \definedterm{topologically mixing} if $\hittingtimes
{U}{V}$ is co-compact for all non-empty open sets $U, V \subseteq
X$.

\begin{proposition} \label{examples:cocompact:topological}
  Suppose that $G$ is a topological group, $X$ is a topological
  space, $\mathcal{F} \subseteq \powerset{G}$ is the family of
  co-pre-compact subsets of $G$ containing $\identity{G}$, and
  $G \curvearrowright X$ is continuous-in-$X$, $\mathcal{F}$-recurrent, and
  topologically transitive. Then $G \curvearrowright X$ is topologically
  mixing.
\end{proposition}

\begin{propositionproof}
  Given non-empty open sets $U, V \subseteq X$, the topological
  transitivity of $G \curvearrowright X$ yields $g \in G$ for which $gU
  \intersection V \neq \emptyset$, so the fact that $G
  \curvearrowright X$ is $\mathcal{F}$-recurrent ensures that $\returntimes{gU
  \intersection V}$ is co-compact. As $h \in \returntimes
  {gU \intersection V} \implies hgU \intersection V \neq \emptyset
  \implies hg \in \hittingtimes{U}{V}$, it follows that $\returntimes{gU
  \intersection V} g \subseteq \hittingtimes{U}{V}$, so $\hittingtimes
  {U}{V}$ is co-compact.
\end{propositionproof}

It follows that the existence of a suitable free Bor\-el\xspace action of $G$
whose recurrence spectrum contains the family $\mathcal{F}$ of
co-pre-compact subsets of $G$ containing $\identity{G}$ is
equivalent to the existence of a suitable continuous
topologically-mixing free action of $G$.

\begin{proposition}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace
  space, $G \curvearrowright X$ is continuous, and $\mathcal{F} \subseteq
  \powerset{G}$ is the family of co-pre-compact subsets of $G$
  containing $\identity{G}$. Then $\mathcal{F}$ is in the recurrence
  spectrum of $G \curvearrowright X$ if and only if there is an equivalence
  class $C$ of $\minimaldecomposition{G}{X}$ for which $G
  \curvearrowright C$ is topologically mixing.
\end{proposition}

\begin{propositionproof}
  This is a direct consequence of Propositions \ref
  {recurrence:characterization:global} and \ref
  {examples:cocompact:topological}.
\end{propositionproof}

To our suprise, we were unable to find a proof in the literature of
the fact that locally-compact non-compact Po\-lish\xspace groups have
free topologically-mixing continuous actions on Po\-lish\xspace spaces. In a
pair of private emails, Glas\-ner\xspace-Weiss\xspace suggested that the
strengthening in which the underlying space is compact should be a
consequence of generalizations of the results of \cite{Wei12} to
locally compact groups, and that a substantially simpler construction
should yield the aforementioned result. However, we give an
elementary proof by checking that the action of $G$ by left
multiplication on the space $\closedsets{G}$ of closed subsets of
$G$ is topologically mixing, where $\closedsets{G}$ is equipped
with the \definedterm{Fell\xspace topology} generated by the sets $V_K =
\set{F \in \closedsets{G}}[F \intersection K = \emptyset]$ and
$W_U = \set{F \in \closedsets{G}}[F \intersection U \ne \emptyset]$,
where $K \subseteq G$ compact and $U \subseteq G$ open. It is
well-known that $\closedsets{G}$ is a compact Po\-lish\xspace space (see,
for example, \cite[Exercise 12.7]{Kec95}).

\begin{proposition}
  Suppose that $G$ is a locally-compact non-compact Po\-lish\xspace group.
  Then there is a Po\-lish\xspace space $X$ for which there is a free
  topologically-mixing continuous action $G \curvearrowright X$.
\end{proposition}

\begin{propositionproof}
  While it is well-known that $G \curvearrowright \closedsets{G}$ is
  continuous, we will provide a proof for the reader's convenience.
  Towards this end, it is sufficient to show that if $g \in G$, $F \in
  \closedsets{G}$, and $U_{gF} \subseteq \closedsets{G}$ is an
  open neighborhood of $gF$, then there are open neighborhoods
  $U_g \subseteq G$ of $g$ and $U_F \subseteq \closedsets{G}$ of
  $F$ for which $U_g U_F \subseteq U_{gF}$. Clearly we can
  assume that $U_{gF} = V_K$ for some compact $K \subseteq G$,
  or $U_{gF} = W_U$ for some open set $U \subseteq G$. In the
  former case, it follows that $F \intersection \inverse{g}K =
  \emptyset$, so the local compactness of $G$ ensures that for all
  $h \in K$ there are a pre-compact open neighborhood $U_{g,h}
  \subseteq G$ of $g$ and an open neighborhood $V_{g,h} \subseteq
  G$ of $h$ such that $F \intersection \inverse{\closure{U_{g,h}}}
  V_{g,h} = \emptyset$, and the compactness of $K$ yields a finite
  set $L \subseteq K$ such that $K \subseteq \union[h \in L][V_{g,h}]$,
  in which case the sets $U_g = \intersection[h \in L][U_{g,h}]$ and
  $U_F = V_{\inverse{\closure{U_g}} K}$ are as desired. In the latter
  case, there exists $h \in F$ for which $gh \in U$, so there are open
  neighborhoods $U_g \subseteq G$ of $g$ and $U_h \subseteq G$
  of $h$ such that $U_g U_h \subseteq U$, thus the sets $U_g$ and
  $U_F = W_{U_h}$ are as desired.

  To see that $G \curvearrowright \closedsets{G}$ is topologically mixing, it is
  sufficient to show that if $U = V_K \intersection \intersection[i < m]
  [W_{U_i}]$ and $U' = V_{K'} \intersection \intersection[j < n]
  [W_{U_j'}]$ are non-empty, where $K, K' \subseteq G$ are
  compact and $U_i \subseteq \setcomplement{K}$ and $U_j'
  \subseteq \setcomplement{K'}$ are open for all $i < m$ and $j <
  n$, then $\set{g \in G}[g U \intersection U' = \emptyset]$ is
  compact. Towards this end, note that for all $i < m$ and $j < n$,
  the sets
  \begin{equation*}
    \textstyle
    L_i
      = \set{g \in G}[g U_i \subseteq K']
      = \intersection[h \in U_i][{\set{g \in G}
        [g h \in K']}]
      = \intersection[h \in U_i][K' \inverse{h}]
  \end{equation*}
  \centerline{and}
  \begin{equation*}
    \textstyle
    L_j'
      = \set{g \in G}[U_j' \subseteq g K]
      = \intersection[h \in U_j'][{\set{g \in G}
        [h \in g K]}]
      = \intersection[h \in U_j'][h \inverse{K}]
  \end{equation*}
  are compact, set $L = \union[i < m][L_i] \union \union[j < n][L_j']$,
  and observe that if $g \in \setcomplement{L}$, then there exist
  $g_i \in gU_i \setminus K'$ for all $i < m$ and $g_j' \in U_j'
  \setminus gK$ for all $j < n$. But then the set $F = \set{g_i}[i
  < n] \union \set{g_j'}[j < n]$ is in $gU \intersection U'$.

  The \definedterm{free part} of the action $G \curvearrowright \closedsets{G}$
  is the set $X$ of $F \in \closedsets{G}$ that are not fixed by any
  non-identity element of $G$. The local-compactness and
  separability of $G$ ensure that $X$ is the intersection of
  countably-many sets of the form $X_K = \set{F \in \closedsets{G}}
  [\forall g \in K \ gF \neq F]$, where $K \subseteq G \setminus
  \set{\identity{G}}$ is compact. As Proposition \ref
  {maximal:cocompact:transient} ensures that each $X_K$ is open,
  it follows that $X$ is $G_\delta$\xspace, and therefore Po\-lish\xspace. To see that $G
  \curvearrowright X$ is the desired action, it only remains to establish that
  $X$ is comeager. And for this, it is sufficient to show that if $K
  \subseteq G \setminus \set{\identity{G}}$ is compact, then $X_K$
  is dense. Towards this end, suppose that $U = V_L \intersection
  \intersection[i < n][W_{U_i}]$ is non-empty, where $L \subseteq G$
  is compact and $U_i \subseteq \setcomplement{L}$ is open for all
  $i < n$, and fix $g_i \in U_i$ for all $i < n$. As $G$ is locally compact,
  by passing to open neighborhoods of $g_i$ contained in $U_i$, we
  can assume that each of the sets $U_i$ is pre-compact. As $G$ is
  not compact, there exists $g \in \setcomplement{(L \union \union[i
  < n][\inverse{K} U_i])}$. Then the set $F = \set{g} \union \set{g_i}[i
  < n]$ is in $U$, and the fact that $F \intersection Kg = \emptyset$
  ensures that $F \in X_K$.
\end{propositionproof}

\section{Transience} \label{transience}

Following the usual convention, let $\Sigmaclass[0][1]$ denote
the pointclass of open sets, and recursively define $\Piclass[0][n]$
to be the pointclass of complements of $\Sigmaclass[0][n]$ sets,
and $\Sigmaclass[0][n+1]$ to be the pointclass of countable unions
of $\Piclass[0][n]$ sets. It is well known that $\unions{(\Dtwo
{\Sigmaclass[0][n]})}$, $\unions{(\Dtwo{\Piclass[0][n]})}$, and
$\Sigmaclass[0][n+1]$ coincide on metric spaces (see,
for example, \cite[\S11.B]{Kec95}).

Given a set $S \subseteq G$, we use $\principleF{S}$ to denote
the family of sets $T \subseteq G$ for which $S \intersection T
\neq \emptyset$.

\begin{proposition} \label{transience:Stransient:global}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace
  space, $G \curvearrowright X$ is continuous, and $S \subseteq G$. Then
  the following are equivalent:
  \begin{enumerate}
    \item The family $\principleF{S}$ is not in the recurrence spectrum
      of $G \curvearrowright X$.
    \item There is a smooth $\aleph_0$-universally Baire\xspace
      superequivalence $F$ of $\orbitequivalencerelation{G}{X}$
      for which each action $G \curvearrowright \equivalenceclass{x}{F}$
      has an $S$-transient $\sigma$-complete
      $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has an $S$-transient
      $\sigma$-complete $\Sigmaclass[0][2]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  As a set $Y \subseteq X$ is $S$-transient if and only if
  $\returntimes{Y} \notin \principleF{S}$, Proposition \ref
  {recurrence:characterization:global} yields $(1) \iff (2)$. Note that if
  $U \subseteq G \times X$ and $x \in X$, then $\horizontalsection
  {U}{x} \in \principleF{S} \iff x \in \union[g \in S][\verticalsection{U}
  {g}]$, so $\principleF{S}$ is $\Sigmaclass[0][1]$-on-open. As a
  set $Y \subseteq X$ is $S$-transient if and only if $\returntimes{C
  \intersection Y} \notin \principleF{S}$ for all equivalence classes $C$
  of $\minimaldecomposition{G}{X}$, Proposition \ref
  {recurrence:complete} yields $(1) \iff (3)$.
\end{propositionproof}

Given a set $\mathcal{S} \subseteq \powerset{G}$, we say that a set $Y
\subseteq X$ is \definedterm{$\mathcal{S}$-transient} if there is a set $S
\in \mathcal{S}$ for which $Y$ is $S$-transient.

\begin{proposition} \label{transience:calStransient:global}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace
  space, $G \curvearrowright X$ is continuous, and $\mathcal{S} \subseteq
  \powerset{G}$. Then the following are equivalent:
  \begin{enumerate}
    \item There exists $S \in \mathcal{S}$ for which the family $\principleF
      {S}$ is not in the recurrence spectrum of $G \curvearrowright X$.
    \item There exist $S \in \mathcal{S}$ and a smooth
      $\aleph_0$-universally Baire\xspace superequivalence $F$ of
      $\orbitequivalencerelation{G}{X}$ for which each action $G
      \curvearrowright \equivalenceclass{x}{F}$ has an $S$-transient $\sigma$-complete
      $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has an $\mathcal{S}$-transient $\sigma$-complete
      $\Sigmaclass[0][2]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  This is a direct consequence of Proposition \ref
  {transience:Stransient:global}.
\end{propositionproof}

Given a set $S \subseteq G$, we say that a set $Y \subseteq X$ is
\definedterm{$S$-wandering} if $gY \intersection hY = \emptyset$
for all distinct $g, h \in S$, and \definedterm{weakly wandering} if
there exists an infinite set $S \subseteq G$ for which $Y$ is
$S$-wandering.

\begin{proposition} \label{transience:weakwandering:global}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace space,
  $G \curvearrowright X$ is continuous, and $\mathcal{S}$ is the family of sets of
  the form $\inverse{S} S \setminus \set{\identity{G}}$, where $S
  \subseteq G$ is infinite. Then the following are equivalent:
  \begin{enumerate}
    \item There exists $S \in \mathcal{S}$ for which the family $\principleF
      {S}$ is not in the recurrence spectrum of $G \curvearrowright X$.
    \item There exist an infinite set $S \subseteq G$ and a smooth
      $\aleph_0$-universally Baire\xspace superequivalence $F$ of
      $\orbitequivalencerelation{G}{X}$ for which each action $G
      \curvearrowright \equivalenceclass{x}{F}$ has an $S$-wandering
      $\sigma$-complete $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has a weakly-wandering
      $\sigma$-complete $\Sigmaclass[0][2]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  Observe that if $S \subseteq G$, then a set $Y \subseteq X$ is
  $S$-wandering if and only if it is $(\inverse{S} S \setminus
  \set{\identity{G}})$-transient, and appeal to Proposition \ref
  {transience:calStransient:global}.
\end{propositionproof}

A subset of a standard Bor\-el\xspace space is \definedterm{analytic} if it is
the image of a standard Bor\-el\xspace space under a Bor\-el\xspace function, and
\definedterm{co-analytic} if its complement is analytic. A result of
Lu\-sin\xspace-Sier\-pi\'{n}\-ski\xspace (see, for example, \cite[Theorem 21.6]{Kec95})
ensures that analytic subsets of standard Bor\-el\xspace spaces are
$\aleph_0$-universally Baire\xspace. Let $\Sigmaclass[1][1]$ denote the
pointclass of analytic sets, and let $\Piclass[1][1]$ denote the
pointclass of co-analytic sets.

Given a superequivalence relation $E$ of $\orbitequivalencerelation
{G}{X}$, we say that a set $Y \subseteq X$ is \definedterm
{$E$-locally weakly-wandering} if its intersection with each
$E$-class is weakly wandering. Given a set $\mathcal{S} \subseteq
\powerset{G}$, define $\principleF{\mathcal{S}} = \intersection[S \in \mathcal{S}]
[\principleF{S}]$.

\begin{proposition} \label{transience:weakwandering:local}
  Suppose that $G$ is a Po\-lish\xspace group, $X$ is a Po\-lish\xspace space, $G
  \curvearrowright X$ is continuous, and $\mathcal{S}$ is the family of sets of the
  form $\inverse{S} S \setminus \set{\identity{G}}$, where $S
  \subseteq G$ is infinite. Then the following are equivalent:
  \begin{enumerate}
    \item The family $\principleF{\mathcal{S}}$ is not in the
      recurrence spectrum of $G \curvearrowright X$.
    \item There is a smooth $\aleph_0$-universally Baire\xspace
      superequivalence $F$ of $\orbitequivalencerelation{G}{X}$ for
      which each action $G \curvearrowright \equivalenceclass{x}{F}$ has an
      $\orbitequivalencerelation{G}{X}$-locally-weakly-wandering
      $\sigma$-complete $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has an $\minimaldecomposition
      {G}{X}$-locally-weakly-wandering $\sigma$-complete $\unions
      {(\Dtwo{\Sigmaclass[1][1]})}$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  As a set $Y \subseteq X$ is $\orbitequivalencerelation{G}
  {X}$-locally weakly-wandering if and only if $\returntimes{C
  \intersection Y} \notin \principleF{\mathcal{S}}$ for all equivalence classes
  $C$ of $\orbitequivalencerelation{G}{X}$, Proposition \ref
  {recurrence:characterization:local} yields $(1) \iff (2)$.
  Note that if $U \subseteq G \times X$ and $x \in X$,
  then $\horizontalsection{U}{x} \in \mathcal{F}_{\mathcal{S}} \iff \forall
  \sequence{g_i}[i \in \amsmathbb{N}] \in \functions{\amsmathbb{N}}{G} \exists i \neq j
  \ ( g_i = g_j \text{ or } \inverse{g_i} g_j \in \horizontalsection
  {U}{x})$, so $\principleF{\mathcal{S}}$ is $\Piclass[1][1]$-on-open. As a
  set $Y \subseteq X$ is $\minimaldecomposition{G}{X}$-locally
  weakly-wandering if and only if $\returntimes{C \intersection Y}
  \notin \principleF{\mathcal{S}}$ for all equivalence classes $C$ of
  $\minimaldecomposition{G}{X}$, Proposition \ref
  {recurrence:complete} yields $(1) \iff (3)$.
\end{propositionproof}

We say that a set $Y \subseteq X$ is \definedterm{very weakly
wandering} if there are arbitrarily large finite sets $S \subseteq G$
for which $Y$ is $S$-wandering.

\begin{proposition} \label{transience:veryweakwandering:global}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace space,
  $G \curvearrowright X$ is continuous, and $\mathcal{S}$ is the family of sets of
  the form $\union[n \in \amsmathbb{N}][\inverse{S_n} S_n \setminus \set{\identity
  {G}}]$, where $S_n \subseteq G$ has cardinality $n$ for all $n
  \in \amsmathbb{N}$. Then the following are equivalent:
  \begin{enumerate}
    \item There exists $S \in \mathcal{S}$ for which the family $\principleF
      {S}$ is not in the recurrence spectrum of $G \curvearrowright X$.
    \item There exist sets $S_n \subseteq G$ of cardinality $n$ and a
      smooth $\aleph_0$-universally Baire\xspace superequivalence relation
      $F$ of $\orbitequivalencerelation{G}{X}$ for which each action
      $G \curvearrowright \equivalenceclass{x}{F}$ has a $\sigma$-complete
      $\aleph_0$-universally-Baire\xspace set that is $S_n$-wandering for all
      $n \in \amsmathbb{N}$.
    \item The action $G \curvearrowright X$ has a very-weakly-wandering
      $\sigma$-complete $\Sigmaclass[0][2]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  Observe that if $S_n \subseteq G$ for all $n \in \amsmathbb{N}$, then a set
  $Y \subseteq X$ is $S_n$-wandering for all $n \in \amsmathbb{N}$ if and
  only if it is $(\union[n \in \amsmathbb{N}][\inverse{S_n} S_n \setminus
  \set{\identity{G}}])$-transient, and appeal to Proposition \ref
  {transience:calStransient:global}.
\end{propositionproof}

Although we are already in position to establish the analog of
Proposition \ref{transience:weakwandering:local} for very weak
wandering, the following observations will allow us to obtain a
substantially stronger complexity bound.

\begin{proposition} \label{transience:openneighborhoods}
  Suppose that $G$ is a topological group, $X$ is a topological
  space, $G \curvearrowright X$ is continuous, and $U \subseteq X$ is a
  non-empty open set. Then there exist a non-empty open set $V
  \subseteq U$ and an open neighborhood $W \subseteq G$ of
  $\identity{G}$ for which $W \returntimes{V} \inverse{W} \subseteq
  \returntimes{U}$.
\end{proposition}

\begin{propositionproof}
  The continuity of $G \curvearrowright X$ yields a non-empty open set $V
  \subseteq U$ and an open neighborhood $W \subseteq G$ of
  $\identity{G}$ for which $WV \subseteq U$. To see that $W
  \returntimes{V} \inverse{W} \subseteq U$, note that if $g \in
  \returntimes{V}$ and $g_x, g_y \in W$, then there exist $x, y \in V$
  for which $g \cdot x = y$, in which case the points $x' = g_x \cdot
  x$ and $y' = g_y \cdot y$ are in $U$, so the fact that $g_y g
  \inverse{g_x} \cdot x' = y'$ ensures that $g_y g \inverse{g_x} \in
  \returntimes{U}$.
\end{propositionproof}

\begin{proposition} \label{transience:dense}
  Suppose that $G$ is a topological group, $X$ is a topological
  space, $G \curvearrowright X$ is continuous, and $S \subseteq G$. Then
  every $S$-wandering non-empty open set $U \subseteq X$ has a
  non-empty open subset $V \subseteq U$ such that for all dense sets
  $H \subseteq G$, there is an injection $\phi \colon S \to H$ with the
  property that $V$ is $\image{\phi}{S}$-wandering.
\end{proposition}

\begin{propositionproof}
  By Proposition \ref{transience:openneighborhoods}, there exist a
  non-empty open set $V \subseteq U$ and an open neighborhood
  $W \subseteq G$ of $\identity{G}$ for which $W \returntimes{V} \inverse{W}
  \subseteq \returntimes{U}$. Note that if $g,h \in S$ and $\inverse
  {(gW)}(hW) \intersection \returntimes{V} \neq \emptyset$, then
  the fact that $\inverse{(gW)}(hW) = \inverse{W} \inverse{g} h W$
  yields that $\inverse{g} h \in W \returntimes{V} \inverse{W}
  \subseteq \returntimes{U}$, thus $g = h$. But if $H \subseteq G$
  is dense, then there is a function $\phi \colon S \to H$ with
  the property that $\phi(g) \in gW$ for all $g \in S$, and it follows
  that $\phi$ is injective and $V$ is $\image{\phi}{S}$-wandering.
\end{propositionproof}

Given a superequivalence relation $E$ of $\orbitequivalencerelation
{G}{X}$, we say that a set $Y \subseteq X$ is \definedterm{$E$-locally
very-weakly-wandering} if its intersection with each $E$-class is
very weakly wandering.

\begin{proposition} \label{transience:veryweakwandering:local}
  Suppose that $G$ is a Po\-lish\xspace group, $X$ is a Po\-lish\xspace space, $G
  \curvearrowright X$ is continuous, and $\mathcal{S}$ is the family of sets of the
  form $\union[n \in \amsmathbb{N}][\inverse{S_n} S_n \setminus \set{\identity
  {G}}]$, where $S_n \subseteq G$ has cardinality $n$ for all $n
  \in \amsmathbb{N}$. Then the following are equivalent:
  \begin{enumerate}
    \item The family $\principleF{\mathcal{S}}$ is not in the recurrence
      spectrum of $G \curvearrowright X$.
    \item There is a smooth $\aleph_0$-universally Baire\xspace
      superequivalence $F$ of $\orbitequivalencerelation{G}{X}$ for
      which each action $G \curvearrowright \equivalenceclass{x}{F}$ has an
      $\orbitequivalencerelation{G}{X}$-locally very-weakly-wandering
      $\sigma$-complete $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has an $\minimaldecomposition{G}
      {X}$-locally very-weakly-wandering $\sigma$-complete
      $\Sigmaclass[0][4]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  As a set $Y \subseteq X$ is $\orbitequivalencerelation{G}
  {X}$-locally very-weakly-wandering if and only if $\returntimes{C
  \intersection Y} \notin \principleF{\mathcal{S}}$ for all equivalence classes
  $C$ of $\orbitequivalencerelation{G}{X}$, Proposition \ref
  {recurrence:characterization:local} yields $(1) \iff (2)$. The fact that
  every $\minimaldecomposition{G}{X}$-locally
  very-weakly-wandering set $Y \subseteq X$ is
  $\orbitequivalencerelation{G}{X}$-locally very-weakly-wandering
  yields $(3) \implies (2)$. To see $(1) \implies (3)$, fix a countable
  dense set $H \subseteq G$, and let $\mathcal{T}$ denote the family of
  sets of the form $\union[n \in \amsmathbb{N}][\inverse{T_n} T_n \setminus \set
  {\identity{H}}]$, where $T_n \subseteq H$ has cardinality $n$ for
  all $n \in \amsmathbb{N}$. Now observe that if condition (1) holds, then
  Proposition \ref{recurrence:characterization:global} ensures that
  there is no equivalence class $C$ of $\minimaldecomposition{G}
  {X}$ for which $G \curvearrowright C$ is $\principleF{\mathcal{S}}$-recurrent, so
  Proposition \ref{transience:dense} implies that there is no
  equivalence class $C$ of $\minimaldecomposition{G}{X}$ for which
  $G \curvearrowright C$ is $\principleF{\mathcal{T}}$-recurrent, thus $\principleF
  {\mathcal{T}}$ is not in the recurrence spectrum of $G \curvearrowright X$. Note
  that if $U \subseteq G \times X$ and $x \in X$, then
  $\horizontalsection{U}{x} \in \principleF{\mathcal{T}} \iff \exists n \in \amsmathbb{N}
  \forall \sequence{h_i}[i < n] \in \functions{n}{H} \exists i \neq j \ (h_i
  = h_j \text{ or } \inverse{h_i} h_j \in \horizontalsection{U}{x})$, so
  $\principleF{\mathcal{T}}$ is $\Sigmaclass[0][3]$-on-open. As every set
  $Y \subseteq X$ with the property that $\returntimes{C \intersection
  Y} \notin \principleF{\mathcal{T}}$ for all equivalence classes $C$ of
  $\minimaldecomposition{G}{X}$ is $\minimaldecomposition{G}
  {X}$-locally very-weakly-wandering, Proposition \ref
  {recurrence:complete} yields condition (3).
\end{propositionproof}

We say that a set $Y \subseteq X$ is \definedterm{non-trivially
transient} if there is a non-pre-compact set $S \subseteq G$
for which $Y$ is $S$-transient.

\begin{proposition} \label{transience:minimallytransient:global}
  Suppose that $G$ is a separable group, $X$ is a Po\-lish\xspace
  space, $G \curvearrowright X$ is continuous, and $\mathcal{S}$ is the family
  of non-pre-compact sets $S \subseteq G$. Then the following
  are equivalent:
  \begin{enumerate}
    \item There exists $S \in \mathcal{S}$ for which the family $\principleF
      {S}$ is not in the recurrence spectrum of $G \curvearrowright X$.
    \item There exist a non-pre-compact set $S \subseteq G$ and
      a smooth $\aleph_0$-universally Baire\xspace superequivalence $F$
      of $\orbitequivalencerelation{G}{X}$ for which each action $G
      \curvearrowright \equivalenceclass{x}{F}$ has an $S$-transient
      $\sigma$-complete $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has a non-trivially-transient
      $\sigma$-complete $\Sigmaclass[0][2]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  Observe that a set $Y \subseteq X$ is non-trivially transient if and
  only if it is $\mathcal{S}$-transient, and appeal to Proposition \ref
  {transience:calStransient:global}.
\end{propositionproof}

Although we are already in position to establish the analog of
Proposition \ref{transience:weakwandering:local} for non-trivial
transience, we will again first establish an observation yielding
a substantially stronger complexity bound.

\begin{proposition} \label{transience:nonprecompact}
  Suppose that $G$ is a locally compact group, $X$ is a
  topological space, $G \curvearrowright X$ is continuous, and $S \subseteq
  G$ is not pre-compact. Then every $S$-transient non-empty open
  set $U \subseteq X$ has a non-empty open subset $V \subseteq
  U$ such that for all dense sets $H \subseteq G$, there is a function
  $\phi \colon S \to H$ with the property that $\image{\phi}{S}$ is
  not pre-compact and $V$ is $\image{\phi}{S}$-transient.
\end{proposition}

\begin{propositionproof}
  By Proposition \ref{transience:openneighborhoods}, there exist a
  non-empty open set $V \subseteq U$ and an open neighborhood
  $W \subseteq G$ of $\identity{G}$ for which $\returntimes{V}
  \inverse{W} \subseteq \returntimes{U}$. As $G$ is locally compact,
  we can assume that $W$ is pre-compact. As $H \subseteq G$ is
  dense, there is a function $\phi \colon S \to H$ with the property
  that $\phi(g) \in gW$ for all $g \in S$. Then $S \subseteq \image
  {\phi}{S} \inverse{W}$, so $\image{\phi}{S}$ is not pre-compact.
  And $\returntimes{V} \inverse{W} \intersection S \subseteq
  \returntimes{U} \intersection S = \emptyset$, so $\returntimes{V}
  \intersection \image{\phi}{S} \subseteq \returntimes{V} \intersection
  SW = \emptyset$, thus $V$ is $\image{\phi}{S}$-transient.
\end{propositionproof}

Given a superequivalence relation $E$ of $\orbitequivalencerelation
{G}{X}$, we say that a set $Y \subseteq X$ is \definedterm
{$E$-locally non-trivially-transient} if its intersection with each
$E$-class is non-trivially transient.

\begin{proposition} \label{transience:minimallytransient:local}
  Suppose that $G$ is a locally compact Po\-lish\xspace group, $X$ is a
  Po\-lish\xspace space, $G \curvearrowright X$ is continuous, $\mathcal{F}$ is the family of
  co-pre-compact sets $S \subseteq G$ containing $\identity{G}$,
  and $\mathcal{S}$ is the family of non-pre-compact sets $S \subseteq G$.
  Then the following are equivalent:
  \begin{enumerate}
    \item The family $\mathcal{F}$ is not in the recurrence spectrum of
      $G \curvearrowright X$.
    \item The family $\principleF{\mathcal{S}}$ is not in the recurrence
      spectrum of $G \curvearrowright X$.
    \item There is a smooth $\aleph_0$-universally Baire\xspace
      superequivalence $F$ of $\orbitequivalencerelation{G}{X}$ for
      which each action $G \curvearrowright \equivalenceclass{x}{F}$ has an
      $\orbitequivalencerelation{G}{X}$-locally non-trivially-transient
      $\sigma$-complete $\aleph_0$-universally-Baire\xspace set.
    \item The action $G \curvearrowright X$ has an $\minimaldecomposition{G}
      {X}$-locally non-trivially-transient $\sigma$-complete $\Sigmaclass
      [0][4]$ set.
  \end{enumerate}
\end{proposition}

\begin{propositionproof}
  As $\principleF{\mathcal{S}}$ is the family of co-pre-compact sets $S
  \subseteq G$, the fact that $\identity{G} \in \returntimes{Y}$ for all
  $Y \subseteq X$ yields $(1) \iff (2)$. As a set $Y \subseteq X$
  is $\orbitequivalencerelation{G}{X}$-locally non-trivially-transient if
  and only if $\returntimes{C \intersection Y} \notin \principleF{\mathcal{S}}$
  for all equivalence classes $C$ of $\orbitequivalencerelation{G}
  {X}$, Proposition \ref{recurrence:characterization:local} yields $(2)
  \iff (3)$. The fact that every $\minimaldecomposition{G}{X}$-locally
  non-trivially-transient set $Y \subseteq X$ is
  $\orbitequivalencerelation{G}{X}$-locally non-trivially-transient
  yields $(4) \implies (2)$. To see $(2) \implies (4)$, fix a countable
  dense set $H \subseteq G$, and let $\mathcal{T}$ denote the family of
  non-pre-compact sets $T \subseteq G$ that are moreover
  contained in $H$. Now observe that if condition (2) holds, then
  Proposition \ref{recurrence:characterization:global} ensures that
  there is no equivalence class $C$ of $\minimaldecomposition{G}
  {X}$ for which $G \curvearrowright C$ is $\principleF{\mathcal{S}}$-recurrent, so
  Proposition \ref{transience:nonprecompact} implies that there is no
  equivalence class $C$ of $\minimaldecomposition{G}{X}$ for
  which $G \curvearrowright C$ is $\principleF{\mathcal{T}}$-recurrent, thus
  $\principleF{\mathcal{T}}$ is not in the recurrence spectrum of $G \curvearrowright
  X$. Fix an increasing sequence $\sequence{K_n}[n \in
  \amsmathbb{N}]$ of compact subsets of $G$ that is \definedterm{cofinal} in the
  sense that every compact set $K \subseteq G$ is contained in
  some $K_n$, and note that if $U \subseteq G \times X$ and $x \in
  X$, then $\horizontalsection{U}{x} \in \principleF{\mathcal{T}} \iff \exists
  n \in \amsmathbb{N} \ H \subseteq K_n \union \horizontalsection{U}{x}$, so
  $\principleF{\mathcal{T}}$ is $\Sigmaclass[0][3]$-on-open. As every set
  $Y \subseteq X$ with the property that $\returntimes{C \intersection
  Y} \notin \principleF{\mathcal{T}}$ for all equivalence classes $C$ of
  $\minimaldecomposition{G}{X}$ is $\minimaldecomposition{G}
  {X}$-locally non-trivially-transient, Proposition \ref
  {recurrence:complete} yields condition (4).
\end{propositionproof}

\section{Generic compressibility}
\label{measures}

Suppose that $E$ is a Bor\-el\xspace equivalence relation on $X$ that is
\definedterm{countable}, in the sense that all of its equivalence
classes are countable. We say that a function $\rho \colon E \to
\openinterval{0}{\infty}$ is a \definedterm{cocycle} if $\rho(x, z) =
\rho(x, y) \rho(y, z)$ whenever $x \mathrel{E} y \mathrel{E} z$. When
$\rho \colon E \to \openinterval{0}{\infty}$ is a Bor\-el\xspace cocycle, we say
that a Bor\-el\xspace measure $\mu$ on $X$ is \definedterm{$\rho$-invariant}
if $\mu(\image{T}{B}) = \int_B \rho(T(x), x) \ d\mu(x)$ for all Bor\-el\xspace
sets $B \subseteq X$ and Bor\-el\xspace automorphisms $T \colon X \to X$
such that $\graph{T} \subseteq E$. We say that $\rho$ is
\definedterm{aperiodic} if $\sum_{y \in \equivalenceclass{x}{E}} \rho
(y, x) = \infty$ for all $x \in X$. Here we generalize the following fact
to orbit equivalence relations induced by Bor\-el\xspace actions of locally
compact Po\-lish\xspace groups, while simultaneously strengthening
comeagerness to a notion under which the recurrence spectrum is
invariant.

\begin{theorem}[Kech\-ris\xspace-Mill\-er\xspace] \label{measures:KM04}
  Suppose that $X$ is a standard Bor\-el\xspace space, $E$ is a countable
  Bor\-el\xspace equivalence relation on $X$, and $\rho \colon E \to
  \openinterval{0}{\infty}$ is an aperiodic Bor\-el\xspace cocycle. Then there
  is an $E$-invariant comeager Bor\-el\xspace set $C \subseteq X$ that is
  null with respect to every $\rho$-invariant Bor\-el\xspace probability measure.
\end{theorem}

A function $\phi \colon X \to Z$ is \definedterm{$E$-invariant} if $\phi
(x) = \phi(y)$ whenever $x \mathrel{E} y$. The \definedterm
{$E$-saturation} of a set $Y \subseteq X$ is the set of $x \in X$ for
which there exists $y \in Y$ such that $x \mathrel{E} y$. We say that
a Bor\-el\xspace probability measure $\mu$ on $X$ is \definedterm
{$E$-quasi-invariant} if the $E$-saturation of every $\mu$-null set
$N \subseteq X$ is $\mu$-null. Let $\probabilitymeasures{X}$
denote the standard Bor\-el\xspace space of Bor\-el\xspace probability measures
on $X$ (see, for example, \cite[\S17.E]{Kec95}). The \definedterm
{push-forward} of a Bor\-el\xspace measure $\mu$ on $X$ through a Bor\-el\xspace
function $\phi \colon X \to Y$ is the Bor\-el\xspace measure $\pushforward
{\phi}{\mu}$ on $Y$ given by $(\pushforward{\phi}{\mu})(B) = \mu
(\preimage{\phi}{B})$ for all Bor\-el\xspace sets $B \subseteq Y$.

\begin{proposition} \label{measures:cocycle}
  Suppose that $X$ is a standard Bor\-el\xspace space, $E$ is a countable
  Bor\-el\xspace equivalence relation on $X$, and $\phi \colon X \to
  \probabilitymeasures{X}$ is an $E$-invariant Bor\-el\xspace function such
  that $\mu$ is $E$-quasi-invariant and $\preimage{\phi}{\mu}$ is
  $\mu$-conull for all $\mu \in \image{\phi}{X}$. Then there is a
  Bor\-el\xspace cocycle $\rho \colon E \to \openinterval{0}{\infty}$ such that
  $\mu$ is $\rho$-invariant for all $\mu \in \image{\phi}{X}$.
\end{proposition}

\begin{propositionproof}
  By standard change of topology results (see, for example, \cite
  [\S13]{Kec95}), we can assume that $X$ is a zero-dimensional
  Po\-lish\xspace space. Fix a compatible complete ultrametric on $X$. By
  \cite[Theorem 1]{FM77}, there is a countable group
  $\Gamma$ of Bor\-el\xspace automorphisms of $X$ whose induced
  orbit equivalence relation is $E$. For all $\gamma \in \Gamma$,
  define $\rho_\gamma \colon X_\gamma \to \openinterval{0}{\infty}$
  by $\rho_\gamma(x) = \lim_{\epsilon \rightarrow 0} (\pushforward
  {(\inverse{\gamma})}{\phi(x)})(\ball{x}{\epsilon}) / \phi(x)(\ball{x}
  {\epsilon})$, where $X_\gamma$ is the set of $x \in X$ for which
  this limit exists and lies in $\openinterval{0}{\infty}$.

  Note that if $\gamma \in \Gamma$, $\mu \in \image{\phi}{X}$,
  and $\psi \colon X \to \openinterval{0}{\infty}$ is a Ra\-don\xspace-Nik\-o\-d\'ym\xspace
  derivative of $\pushforward{(\inverse{\gamma})}{\mu}$ with respect
  to $\mu$ (see, for example, \cite[\S17.A]{Kec95}), then the
  straightforward generalization of the Leb\-es\-gue\xspace density theorem
  for Po\-lish\xspace ultrametric spaces (see, for example, \cite[Proposition
  2.10]{Mil08a}) to integrable functions ensures that $\psi(x)
  = \lim_{\epsilon \rightarrow 0} \int_{\ball{x}{\epsilon}} \psi \ d\mu / \mu
  (\ball{x}{\epsilon}) = \rho_\gamma(x)$ for $\mu$-almost all $x \in X$.

  It immediately follows that for all $\gamma \in \Gamma$, the
  complement of $X_\gamma$ is null with respect to every $\mu
  \in \image{\phi}{X}$. Moreover, if $B \subseteq X$ is Bor\-el\xspace,
  $\gamma, \delta \in \Gamma$, and $\mu \in \image{\phi}{X}$, then
  \begin{align*}
    \pushforward{\inverse{(\gamma \delta)}}{\mu}(B)
      & = \int_{\image{\delta}{B}} \rho_\gamma(x) \ d\mu(x) \\
      & = \int_B \rho_\gamma(\delta \cdot x) \ d(\pushforward{(\inverse
        {\delta})}{\mu})(x) \\
      & = \int_B \rho_\gamma(\delta \cdot x) \rho_\delta(x) \ d\mu(x),
  \end{align*}
  so the almost-everywhere uniqueness of Ra\-don\xspace-Nik\-o\-d\'ym\xspace
  derivatives ensures that the set of $x \in X$ for which there exist
  $\gamma, \delta \in \Gamma$ such that $\rho_{\gamma \delta}(x)
  \neq \rho_\gamma(\delta \cdot x) \rho_\delta(x)$ is null with respect
  to every $\mu \in \image{\phi}{X}$.

  Let $N$ denote the $E$-saturation of the union of these sets, and
  let $\rho \colon E \to \openinterval{0}{\infty}$ be the extension of the
  constant cocycle on $\restriction{E}{N}$ given by $\rho(\gamma
  \cdot x, x) = \rho_\gamma(x)$ for all $\gamma \in \Gamma$ and $x
  \in X$.
\end{propositionproof}

As a consequence, we obtain the following.

\begin{theorem} \label{measures:comeagernullset}
  Suppose that $X$ is a Po\-lish\xspace space, $E$ is a Bor\-el\xspace equivalence
  relation on $X$ admitting a Bor\-el\xspace complete set $B \subseteq X$
  on which $E$ is countable, $F$ is a superequivalence relation of
  $E$ for which every $F$-class is $G_\delta$\xspace and the $F$-saturation
  of every open set is Bor\-el\xspace, and $\phi \colon X \to \probabilitymeasures
  {X}$ is an $E$-invariant Bor\-el\xspace function for which every measure
  $\mu \in \image{\phi}{X}$ has $\mu$-conull $\phi$-preimage and
  concentrates off of Bor\-el\xspace sets on which $E$ is smooth. Then there
  is an $E$-invariant Bor\-el\xspace set $C \subseteq X$ that is comeager in
  every $F$-class, but null with respect to every measure in $\image
  {\phi}{X}$.
\end{theorem}

\begin{theoremproof}
  By the Lu\-sin\xspace-No\-vik\-ov\xspace uniformization theorem (see, for example,
  \cite[Theorem 18.10]{Kec95}), there is a Bor\-el\xspace extension $\pi
  \colon X \to B$ of the identity function on $B$ whose graph is
  contained in $E$. Fix a sequence $\sequence{\epsilon_n}[n \in
  \amsmathbb{N}]$ of positive real numbers whose sum is $1$, in addition to a
  countable group $\set{\gamma_n}[n \in \amsmathbb{N}]$ of Bor\-el\xspace
  automorphisms of $B$ whose induced orbit equivalence relation is
  $\restriction{E}{B}$, and define $\psi \colon B \to
  \probabilitymeasures{B}$ by $\psi(x) = \sum_{n \in \amsmathbb{N}} \pushforward
  {(\gamma_n \circ \pi)}{\phi(x)} / \epsilon_n$. As each $\nu
  \in \image{\psi}{B}$ is $(\restriction{E}{B})$-quasi-invariant,
  Proposition \ref{measures:cocycle} yields a Bor\-el\xspace cocycle $\rho
  \colon \restriction{E}{B} \to \openinterval{0}{\infty}$ such that every
  $\nu \in \image{\psi}{B}$ is $\rho$-invariant.

  Given $\nu \in \image{\psi}{B}$, fix $x \in B$ for which $\nu =
  \psi(x)$, set $\mu = \phi(x)$, and observe that $\nu(\preimage{\psi}{\nu})
  \ge \mu(\preimage{\phi}{\mu}) = 1$. Moreover, as $E$ is smooth on
  the periodic part $P = \set{x \in B}[\sum_{y \in \equivalenceclass
  {x}{\restriction{E}{B}}} \rho(y, x) < \infty]$ of $\rho$ (see, for example,
  \cite[Proposition 2.1.1]{Mil08b}), and therefore on
  its $E$-saturation, it follows that $\saturation{P}{E}$ is null with
  respect to every measure in $\image{\phi}{X}$, thus $P$ is null with
  respect to every measure in $\image{\psi}{B}$.

  By the proof of Theorem \ref{measures:KM04} (see \cite
  [Theorem 13.1]{KM04}), there is a Bor\-el\xspace set $R \subseteq
  \Bairespace \times B$, whose vertical sections are $(\restriction{E}
  {B})$-invariant and null with respect to every $\rho$-invariant Bor\-el\xspace
  probability measure, such that every $x \in B$ is contained
  in comeagerly-many vertical sections of $R$. It follows that the
  vertical sections of the set $S = \preimage{(\mathrm{id} \times \pi)}{R}$ are
  $E$-invariant and null with respect to every measure in $\image
  {\phi}{X}$, and every $x \in X$ is contained in comeagerly-many
  vertical sections of $S$. The Kur\-at\-ow\-ski\xspace-U\-lam\xspace theorem therefore
  ensures that for all $x \in X$, comeagerly-many vertical sections of
  $S$ are comeager in $\equivalenceclass{x}{F}$.

  By \cite{Sri79}, there is a Bor\-el\xspace set $D \subseteq X$
  intersecting every $F$-class in a single point. As the $F$-saturation
  of every open set is Bor\-el\xspace, the usual proof of the Mont\-gom\-ery\xspace-No\-vik\-ov\xspace
  theorem that the pointclass of Bor\-el\xspace sets is closed under category
  quantifiers (see, for example, \cite[Theorem 16.1]{Kec95}) shows that
  $\set{\pair{b}{x} \in \Bairespace \times X}[\verticalsection{S}{b} \text{ is
  comeager in } \equivalenceclass{x}{F}]$ and $\set{\pair{b}{x} \in
  \Bairespace \times X}[\verticalsection{S}{b} \text{ is non-meager in }
  \equivalenceclass{x}{F}]$ are Bor\-el\xspace, so \cite[Theorem 18.6]{Kec95}
  yields a Bor\-el\xspace function $\beta \colon D \to \Bairespace$ such that
  $\verticalsection{S}{\beta(x)}$ is comeager in $\equivalenceclass
  {x}{F}$ for all $x \in D$. Then the set $C = \union[x \in D][\verticalsection
  {S}{\beta(x)} \intersection \equivalenceclass{x}{F}]$ is as desired.
\end{theoremproof}

We say that a function $\rho \colon G \times X \to \openinterval{0}
{\infty}$ is a \definedterm{cocycle} if $\rho(gh, x) = \rho(g, h \cdot x)
\rho(h, x)$ for all $g, h \in G$ and $x \in X$. When $\rho \colon G
\times X \to \openinterval{0}{\infty}$ is a Bor\-el\xspace cocycle, we say that
a Bor\-el\xspace measure $\mu$ on $X$ is \definedterm{$\rho$-invariant} if
$\mu(gB) = \int_B \rho(g, x) \ d\mu(x)$ for all Bor\-el\xspace sets $B
\subseteq X$ and group elements $g \in G$. The following fact is the
desired generalization of Theorem \ref{measures:KM04}.

\begin{theorem} \label{measures:genericcompressibility}
  Suppose that $G$ is a locally compact Po\-lish\xspace group, $X$ is a Po\-lish\xspace
  space, $G \curvearrowright X$ is a continuous action, $F$ is a superequivalence
  relation of $\orbitequivalencerelation{G}{X}$ for which every $F$-class
  is $G_\delta$\xspace and the $F$-saturation of every open set is Bor\-el\xspace, and $\rho
  \colon G \times X \to \openinterval{0}{\infty}$ is a Bor\-el\xspace cocycle with
  the property that every $G$-orbit is null with respect to every
  $\rho$-invariant Bor\-el\xspace probability measure. Then there is a
  $G$-invariant Bor\-el\xspace set $C \subseteq X$ that is comeager in every
  $F$-class, but null with respect to
  every $\rho$-invariant Bor\-el\xspace probability measure.
\end{theorem}

\begin{theoremproof}
  By \cite[Theorem 1.1]{Kec92}, there is a complete
  Bor\-el\xspace set $B \subseteq X$ on which $\orbitequivalencerelation{G}{X}$
  is countable. Fix a $\rho$-invariant uniform ergodic decomposition
  $\phi \colon X \to \probabilitymeasures{X}$ of $G \curvearrowright X$ (see \cite
  [Theorem 5.2]{GS00}), and appeal to Theorem \ref
  {measures:comeagernullset}.
\end{theoremproof}

We next check that the special case of Theorem \ref
{measures:genericcompressibility} for $F = \minimaldecomposition
{G}{X}$ provides a proper strengthening of Theorem \ref
{measures:KM04}. While this can be seen as a consequence
of the Kur\-at\-ow\-ski\xspace-Ulam theorem, we will show that the usual proof
of the latter easily adapts to yield a generalization to a natural class of
equivalence relations containing $\minimaldecomposition{G}{X}$.

\begin{theorem}
  Suppose that $X$ is a second-countable Baire\xspace space, $E$ is an
  equivalence relation on $X$ such that every $E$-class is a Baire\xspace
  space and the $E$-saturation of every open subset of $X$ is open,
  and $B \subseteq X$ has the Baire\xspace property. Then:
  \begin{enumerate}
    \item $\forall^* x \in X \ B$ has the Baire\xspace property in
      $\equivalenceclass{x}{E}$.
    \item $B$ is comeager $\iff \forall^* x \in X \ B$ is
      comeager in $\equivalenceclass{x}{E}$.
  \end{enumerate}
\end{theorem}

\begin{theoremproof}
  We begin with a simple observation.

  \begin{lemma} \label{measures:densesaturation}
    Suppose that $U \subseteq X$ is a non-empty open set and
    $V \subseteq U$ is a dense open set. Then $\saturation{V}{E}$
    is dense in $\saturation{U}{E}$.
  \end{lemma}

  \begin{lemmaproof}
    If $W \subseteq X$ is open and $\saturation{V}{E} \intersection W
    = \emptyset$, then $V \intersection \saturation{W}{E} = \emptyset$,
    so the openness of $\saturation{W}{E}$ ensures that $\closure{V}
    \intersection \saturation{W}{E} = \emptyset$, thus the density of
    $V$ implies that $U \intersection \saturation{W}{E} = \emptyset$,
    hence $\saturation{U}{E} \intersection W = \emptyset$.
  \end{lemmaproof}

  To see the special case of ($\implies$) of (2) when $B \subseteq X$
  is open, note that if $U \subseteq X$ is non-empty and open, then
  Lemma \ref{measures:densesaturation} yields that $\saturation{B
  \intersection U}{E}$ is dense in $\saturation{U}{E}$, and therefore
  $\forall^* x \in X \ (x \in \saturation{U}{E} \implies x \in
  \saturation{B \intersection U}{E})$, or equivalently,
  $\forall^* x \in X \ (U \intersection \equivalenceclass{x}{E}
  \neq \emptyset \implies B \intersection U \intersection
  \equivalenceclass{x}{E} \neq \emptyset)$. As $X$ is second countable,
  it follows that $\forall^* x \in X \ B$ is dense in
  $\equivalenceclass{x}{E}$.

  To see ($\implies$) of (2), suppose that $B \subseteq X$
  is comeager, fix dense open sets $B_n \subseteq X$ for which
  $\intersection[n \in \amsmathbb{N}][B_n] \subseteq B$, and appeal to the
  special case for open sets to obtain that $\forall^* x \in
  X \ \intersection[n \in \amsmathbb{N}][B_n]$ is comeager in $\equivalenceclass
  {x}{E}$.

  To see (1), fix an open set $U \subseteq X$ for which $B
  \mathrel{\triangle} U$ is meager, and note that
  $\forall^* x \in X \ B \mathrel{\triangle} U$ is meager
  in $\equivalenceclass{x}{E}$, by $(\implies)$ of (2).

  To see ($\impliedby$) of (2), suppose that $B$ is not
  comeager, fix a non-empty open set $V \subseteq X$ in which
  $B$ is meager, note that $\forall x \in V \ V \intersection
  \equivalenceclass{x}{E} \neq \emptyset$, and appeal to
  ($\implies$) of (2) to obtain that $\forall^*
  x \in X \ B \intersection V$ is meager in $\equivalenceclass{x}{E}$,
  thus $\forall^* x \in V \ B$ is not comeager in
  $\equivalenceclass{x}{E}$.
\end{theoremproof}

Finally, we check that no condition on the recurrence spectrum
can yield the existence of an invariant Bor\-el\xspace probability measure.

\begin{theorem} \label{recurrence:compressibility}
  Suppose that $G$ is a locally compact Po\-lish\xspace group, $X$ is a
  standard Bor\-el\xspace space, $G \curvearrowright X$ is Bor\-el\xspace, and $\rho \colon
  G \times X \to \openinterval{0}{\infty}$ is a Bor\-el\xspace cocycle for which
  every $G$-orbit is null with respect to every $\rho$-invariant Bor\-el\xspace
  probability measure. Then there is a $G$-invariant Bor\-el\xspace set
  $B \subseteq X$ that is null with respect to every $\rho$-invariant
  Bor\-el\xspace probability measure but for which the recurrence spectra of
  $G \curvearrowright B$ and $G \curvearrowright X$ coincide.
\end{theorem}

\begin{theoremproof}
  We can assume that $X$ is a Po\-lish\xspace space and $G \curvearrowright X$ is
  continuous. By Theorem \ref{measures:genericcompressibility},
  there is an $\orbitequivalencerelation{G}{X}$-invariant Bor\-el\xspace set
  $B \subseteq X$ that is comeager in every $\minimaldecomposition
  {G}{X}$-class, but null with respect to every $\rho$-invariant Bor\-el\xspace
  probability measure. Proposition \ref
  {recurrence:characterization:global} then ensures that $G \curvearrowright
  B$ and $G \curvearrowright X$ have the same recurrence spectra.
\end{theoremproof}

\subsection*{Acknowledgements}
We would like to thank the anonymous referee for her suggestions.

\bibliographystyle{asl}

redpajama_set_name: RedPajamaArXiv


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text:
Kristi Krueger, Anchor/Health Reporter
Kathleen Corso, Special Projects Producer
Tags: Health, Broward County, Fort Lauderdale
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Mentoring programs help disadvantaged youth
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FORT LAUDERDALE, Fla. – An estimated 35,000 people in the United States are affected by cystic fibrosis, or CF, a genetic disorder that causes problems with breath and digestion and often leads to the need for a double lung transplant.
Fort Lauderdale resident Kati Gregory was born with CF and wasn't sure she would ever achieve the dream of being a mom.
The disease causes a thickly, sticky mucus to build up in the lungs and other organs.
By 2015, her lungs were failing.
"I was wheelchair bound, I couldn't walk across from me to you without just being so out of breath. That was not something I was used to prior to that," Gregory said.
That same year she underwent a life-saving double lung transplant.
"Kati was a really unique experience in that when I met her I don't think she really knew how sick and how far along her disease had gotten," said Dr. Jason Turowski, a pulmonologist with Cleveland Clinic Ohio.
He performed Gregory's surgery in September 2015.
"I think it's, so far, from the standpoint of transplants, Kati has been an example of a real success story," Turowski said.
Living with a fresh set of lungs meant Gregory would need to take anti-rejection medications for life, making the chances of a future pregnancy risky.
"I kind of just wrote it off that it wasn't in my cards," she said.
Thanks to in vitro fertilization, of IVF, and the support of two surrogates, Gregory and her husband Gerome now have a toddler and twins.
"Having gone on and lived a successful life, having a really great job, getting married and having three babies, that's incredibly unique. I've never seen that before until Kati," said Turowski.
When Gregory and her husband count their blessings, the donor who saved her life is at the top of the list.
"It's just something I will cherish forever and the fact that she made that choice to be an organ donor is why I'm here," she said.
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Kathleen Corso
Veteran journalist Kathleen Corso is the special projects producer for Local 10 News.

redpajama_set_name: RedPajamaCommonCrawl


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Life Together in Brooklyn
In Ecclesiology, Theologians
Tomorrow, our Theology Circle gathers to start discussing Dietrich Bonhoeffer's Life Together (Gemeinsames Leben). If you're in NYC, come join us. Details are below. I picked LT because it is a very modern, positive exploration of what intentional Christian community can look like and an exploration of what fellowship is. It is a small, accessible book born out of experience. It also serves as a nice communal counterbalance to the radically subjectivist individualism that we discussed in Kierkegaard's Fear and Trembling. FT is about how no one can understand one's relationship with God. LT is about living together in light of God.
It's worth mentioning that LT was written over four weeks in late 1938 after a wealth of intense experience with the good and the bad of Christian community. Bonhoeffer was 32 and had recently finished his most well-known work, Nachfolge (The Cost of Discipleship), immediately prior. He had already experienced a great variety of religious community from the marginal church life of his youth to his theological training at Tübingen to his influential trip to Rome in 1924's Holy Week to his year pastoring an ex-pat church in Barcelona to his year at New York City's historic Union Theological Seminary in 1930/31 where he also taught youth Sunday school at Abyssinian Baptist Church in Harlem to lecturing at the University of Berlin to working as a student minister at a German trade university to pastoring two ex-pat churches in London to all throughout his work with the European ecumenical and youth movements to his sharpening battles as part of the Confessing Church with the German Christians and the Nazi's Reich Church. By the time 1938 arrives, he has already directed an illegal, underground seminary for pastors of the Confessing Church. Bonhoeffer knew life together.
He's on the Nazi's radar screen. He has gone virtually toe to toe with Reich Bishop Müller and keeping one step ahead of him as the incompetent bishop seeks to solidify Hitler's power among the suspicious churches. Bonhoeffer knows what is happening. He has seen (not all, but growing number of) Christians in his country duped and slowly seduced to solidify Hitler's increasing power grab since 1933. He has resisted the "Aryan Paragraph" that prevented all non-Aryans from becoming ministers or religious teachers. The Nazi State sought to turn every aspect of German life together towards and for Adolf Hitler. Bonhoeffer sought to resist to ensure life together in and toward and for Christ.
It is important to not take Christian community for granted. In the United States and United Kingdom, we do. We have churches on almost every corner. Christendom, in its many forms, is alive and well. There are those who would claim that the US is a "Christian nation", whatever that means. However, anyone who compares a perceived persecution of Christians in the US to that of the situation in Hitler's Germany is a fool and has no knowledge or understanding of Europe's first half of the twentieth century. It is fear-mongering and naive. It is important to say that up front, because there are those who will twist Bonhoeffer to make him their own instead of letting Bonhoeffer be the very nuanced and complex and brilliant Dietrich Bonhoeffer.
For Bonhoeffer, the idea of decision was critical. In his many encounters with ecclesiastic, ecumenical, international, and state bureaucracies, he realized that Christian community must be decisive about its identity and its place in the World. It should not mince words and only say things when it needs to. But when it does, it must be decisive. It must be concrete. It must take a stand, if only for itself. Life together is to live decisively. It is a decision to live together. It is intentional. It is loving. It is where Christianity encounters itself. This is fellowship.
We decide to gather at 61 Local at 61 Bergen Street, Brooklyn, on Wednesday nights at 7pm to explore what life together looks like. We discuss and drink together. Join us.
Dietrich Bonhoeffer Life Together St. Lydia's Theology Circle
A Theological Take on Good Omens: A Prologue
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Bonhoeffer at Charlottesville
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ADVENT IS NOW. ADVENT IS NOW. ADVENT IS NOW.
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By Burke March 26, 2013
Church Strippers
EV'RY DAY I'M PASTORIN'… – EASTER MUSICAL ON PALM SUNDAY


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«Аланга» — бывший узбекистанский футбольный клуб из города Касан Кашкадарьинской области.

Названия
 2000-2001 — «Алтын Тола».
 2009-2011 — «Еркурган».
 2012 — «Ёшлик».
 с 2013 — «Аланга».

История
Основан в 2000 году и возрождён в 2009 году. 2 сезона (2009 и 2010) провёл во Второй лиге чемпионата Узбекистана, ещё 7 сезонов (2000-2001 и 2011-2015) выступал в Первой лиге.

Достижения
4-е место в Первой лиге (2011).

Ссылки
 Клуб в базе footballfacts

Футбольные клубы Кашкадарьинской области

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Volver – polski zespół grający muzykę pop założony w 2009 przez Tomasza Luberta.

W 2010 zespół zdobył trzy festiwalowe nagrody. Na festiwalu TOPtrendy 2010 w Sopocie otrzymał główną nagrodę w konkursie Trendy. Podczas Bydgoszcz Hit Festiwal utwór "Volveremos" został uznany za Polski Hit Lata roku 2010. Na XLVII Krajowym Festiwalu Piosenki Polskiej w Opolu zespół zdobył Superjedynkę w kategorii debiut roku. 4 sierpnia 2012 zespół zdobył Złoty Samowar na Festiwalu Piosenki Rosyjskiej za utwór "Million ałych roz".

Dyskografia
 Albumy

 Single

Nagrody i wyróżnienia

Przypisy

Polskie zespoły popowe
Zdobywcy złotych płyt