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\title[Dynamical invariants of foliations]{Dynamical Invariants of Foliations\footnote{Colloquium GT3,  IRMA, Universit\'{e} de Strasbourg}
}
\author{Steven Hurder}
\institute{University of Illinois at Chicago}
\date{October 19, 2012}
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\section{Introduction}
 
\begin{frame}{Foliations}

 A foliation $\F$ of   a compact manifold $M$ is \ldots
 
 \bigskip
 
 $\bullet$ a ``uniform partition'' of $M$ into submanifolds of constant dimension $p$ and codimension $q$.

 
  
 \medskip
{\bf Definition:}  $M$   a smooth manifold of dimension $n$  is \emph{foliated}  if there is a covering of $M$ by coordinate charts whose change of coordinate functions preserve leaves
 \begin{center}
\includegraphics[width=0.35\textwidth]{strasbourg_pix/chart.pdf}
\end{center}
 
 \vfill
 
\end{frame}

\begin{frame}{Reeb,   Vincensini  \&   Ehresmann (1949)}

 
   
   \begin{center}
\includegraphics[width=.6 \textwidth]{strasbourg_pix/RVE.jpg}
% \caption{Georges Reeb, Paul Vincensini and Charles Ehresmann, from left to right, at a topology conference in Oberwolfach, 1949.}
\end{center}
 
 Andr\'{e}~Haefliger,
 {\it Naissance des feuilletages, d'{Ehresmann}-Reeb \`a {Novikov}},
  {\bf G\'eom\'etrie au vingti\`eme si\`ecle : 1930-2000}, {Hermann, \'Editeurs des Sciences et des Arts}, Paris.
http://www.unige.ch/math/folks/haefliger/Feuilletages.pdf 
 \vfill
 
\end{frame}

\begin{frame}{Foliations}

 A foliation $\F$ of   a compact manifold $M$ is also \ldots
 
 \bigskip


$\bullet$ a local geometric structure on $M$, given by a $\G_{\mR^q}$-cocycle for a ``good covering''. (Ehresmann, Haefliger)

 
 \medskip

$\bullet$ a  dynamical system on $M$ with multi-dimensional time.

\medskip

$\bullet$ a groupoid   $ \GF \to M$ with fibers complete manifolds, the holonomy covers of leaves.  
 
 \bigskip
  % \pause

 Each point of view has  advantages and disadvantages.
 
 \medskip
 
 {\bf Problem:}  How to distinguish and classify foliations?
 
 \vfill
 
\end{frame}

 
\begin{frame}{Foliation invariants}
Each viewpoint yields its own classes of invariants to distinguish foliations.

\bigskip

 $\bullet$ a ``uniform partition'' of $M$ into submanifolds of constant dimension $p$
 \medskip
 
 $\Longrightarrow$ Growth rates of leaves, other  quasi-isometry invariants.

  % \pause

 \bigskip


 $\bullet$   local geometric structure on $M$, given by a $\G_{\mR^q}$-cocycle  
 
 \medskip
 $\Longrightarrow$ Secondary classes, other cohomology invariants derived from sheaves.

  % \pause

 \bigskip

$\bullet$   dynamical system on $M$ with multi-dimensional time 
 
 \medskip
 $\Longrightarrow$ Geometric entropy,  Lyapunov spectrum,    invariant \& harmonic measures.

 
 \bigskip

 % \pause


$\bullet$   groupoid   $ \GF \to M$ with fibers   the holonomy covers of leaves  
 
 \medskip
 $\Longrightarrow$ $C^*$-algebras $C_r^*(M,\F)$ and its K-Theory invariants, von~Neumann algebra $W^*(M, \F)$ and its flow of weights.

 
 \bigskip

 \vfill
 
\end{frame}

\section{Three examples}

\begin{frame}{Reeb foliation}

Here is maybe the prettiest picture of the Reeb foliation of $\mS^3$, from the Wikimedia Commons:

 \begin{center}
\includegraphics[width=.5\textwidth]{strasbourg_pix/Reeb_foliation_half-torus.png}
\end{center}
 
 \vfill
 
\end{frame}



\begin{frame}{}

$\bullet$  It's Godbillon-Vey class is zero. 
\smallskip

Mizutani, Morita \& Tsuboi: {\it The Godbillon-Vey classes of codimension one foliations which are almost without holonomy}, [Annals of Math., 1981]

\medskip

This was the model for   studies of how the Godbillon-Vey class is related to dynamics, culminating in the celebrated result of Gerard Duminy in 1982 (discussed later).

 

 % \pause

\bigskip

$\bullet$  The K-Theory of its $C^*$-algebra can be explicitly calculated:
\smallskip


 Anne-Marie Torpe: {\it K-theory for the leaf space of foliations by Reeb components},  [J. Funct. Anal., 1985]

\medskip

This was a model example for the geometric approach to foliation index theory:

\smallskip

Paul Baum \& Alain Connes: {\it Geometric K-theory for Lie groups and foliations}, [Enseign. Math., 2000: preprint, 1985]

\bigskip
 % \pause

\emph{Not bad},  for such a ``modest'' example.  The status of the classification problem today - 30 years later - has seen great progress.

 \vfill
 
\end{frame}


 

  
 \begin{frame}{Riemannian foliations}

 

 
 $\F$ is a \emph{Riemannian foliation} if there is a Riemannian metric on $TM$ so that its restriction to 
 $Q = TM/T\F$ is invariant under the leafwise parallelism. 
 
 \medskip
 
 {\bf Theorem:} [Molino, 1984] Let $\F$ be a smooth Riemannian foliation on a compact manifold $M$. Then for each leaf $L$, its closure $\overline{L}$ is a manifold, and: 
 
 \begin{enumerate}
\item the  restricted foliation $\F \mid \overline{L}$ is Riemannian, even homogeneous;
\item all leaves of $\F$ in $\overline{L}$ are dense in $\overline{L}$;
\item the closures of the leaves form a singular Riemannian foliation of $M$.
\end{enumerate}
  
If the group of foliated homeomorphisms of $M$ is \emph{transitive}, then the foliation by leaf closures is defined by a submersion to the quotient manifold $W = M / \overline{T\F}$.
 
 \bigskip
 
 {\bf Remark:} Molino's Theorem and related works by Carri\`{e}re, Ghys, and others give (almost)  a complete classification of Riemannian foliations  in low dimensions. 
 
\vfill

\end{frame}


 \begin{frame}{Anosov foliations}

 
\medskip

\emph{Anosov foliations} are at the opposite extreme from Riemannian foliations:

 

$$TM = E^+ \oplus \langle \vec{X} \rangle \oplus E^- $$

where the flow $\varphi_t$ of $\vec{X}$ uniformly expands $E^+$, and uniformly contracts $E^-$. 

\medskip

 $\F^{\pm}$ is the foliation given by the integral manifolds of the distribution $\vec{X} \oplus E^{\pm}$.
 
 \medskip
 
The restriction of a Riemannian metric on $TM$   to 
 $Q^{\pm} \cong E^{\mp}$ is either uniformly contracted/expanded  under the leafwise parallelism along $\vec{X}$. 
 
 \bigskip
   

 Anosov foliations are extremely well-studied, and though not classified, there are algebraic models for ``what they should be'', and proofs that they are algebraic in important cases. 
 
 \medskip
 
 Concept extends to actions of Lie groups, and suspensions of countable groups acting smoothly on compact manifolds, where there are many open questions. 
 
 
 
 \vfill

\end{frame}

 \begin{frame}{Intermediate classes of foliation dynamics}

 
\medskip

The two cases above have exceptional uniformity in their structure:

\medskip

$\bullet$ For a Riemannian foliation $\F$, all leaves have quasi-isometric \& diffeomorphic   holonomy coverings, given by a ``typical'' leaf $L \subset M$.

\bigskip

$\bullet$ For an Anosov foliation $\F$, all leaves are  quasi-isometric \& diffeomorphic, to either $\mR^n$  or   to a simply-connected nil-manifold $\cN$.  
 \bigskip
 
 
 For the general foliation $\F$, the closure of a leaf $L$ can contain a variety of diffeomorphism types of manifolds as leaves  (Reeb foliation, and beyond).
 
  
 

\medskip

$\bullet$  Foliation dynamics is a ``mash-up'' of these two extremes of behavior.
\medskip

$\bullet$  Foliation dynamics  defined ``germinally'', as     no uniform notion of ``time''.

 \vfill

\end{frame}

\section{Local to global}

 
  \begin{frame}{}

   
  The problem in the study of foliation theory:
  \medskip
  
  ``How to obtain global conclusions from local assumptions?''
  \bigskip
  \vfill

\end{frame}
 

  \begin{frame}{Pseudogroups}

A section $\cT \subset M$ for $\F$ is an embedded submanifold of dimension $q$ which intersects each leaf of $\F$ at least once, and always transversally.  The   holonomy of $\F$ on $\cT$ yields a compactly generated pseudogroup $\cGF$.

 
\medskip

 
{\bf Definition:}
A     pseudogroup of transformations $\cGF$ of $\cT$ is \emph{compactly generated}     if there is 
\begin{itemize}
\item  relatively compact open subset $\cT_0 \subset \cT$ meeting all orbits of $\cGF$;
\item  a finite set $\G = \{g_1, \ldots , g_k\} \subset    \cGF$  which generates $\cGF | \cT_0$; 
\item   $g_i \colon D({g_i}) \to R({g_i})$ is the restriction of  $\widetilde{g}_i \in   \cGF$ with   
$\overline{D(g)} \subset D({\widetilde{g}_i})$.
\end{itemize}

  \medskip
  
 


  \vfill

\end{frame}


  \begin{frame}{Groupoid  word length}

    
{\bf Definition:} The groupoid of $\cGF$ is the space of germs 
$$\GF = \{[g]_x \mid g \in \cGF ~ \& ~ x \in D(g)\} ~ , ~ \GF$$
with source map $s[g]_x = x$ and range map $r[g]_x = g(x) = y$.

% % \pause 

 
\medskip


The \emph{word length} $\| [g] \|_x$ of the germ $[g]_x$ of g at $x$  is the least $k$ such that 
$$[g]_x = [g_{i_1}^{\pm 1} \circ \cdots \circ g_{i_k}^{\pm 1}]_x$$

Word length is a measure of the ``time'' required to get  from one point on an orbit to another along an orbit or leaf, while preserving the germinal dynamics.
  
 \vfill
 
\end{frame}


  \begin{frame}{Derivative cocycle}

    There is really no good ideas for classifying foliations which are just $C^0$.
    \medskip
   % \pause
  
    Assume that the holonomy maps in $\cGF$ are uniformly $C^1$.
    
    \bigskip
    
  Assume $(\cGF, \cT)$ is a compactly generated pseudogroup, and $\cT$ has a uniform Riemannian metric. Choose a uniformly bounded, Borel trivialization, ~ $T\cT \cong \cT \times \mR^q$,  ~ $T_x\cT \cong_x  \mR^q$ ~ for all $x \in \cT$.
  
  \medskip
  
  
 {\bf Definition:} The \emph{normal derivative cocycle} $D  \colon \GF \times \cT \to {\bf GL(\mR^q)}$ is defined by
 $$D ([g]_x) =  D_x g   \colon T_x \cT \cong_x \mR^q  \to T_{y}\cT \cong_y \mR^q$$

 which satisfies the  cocycle law 
 $$ D( [h]_y \circ [g]_x ) = D([h]_y) \cdot D ([g]_x) $$
 
  
  % \pause

  {\bf Problem:} Extract from the normal derivative cocycle   classification data: dynamical invariants for $C^1$-foliations,  secondary classes for $C^2$-foliations.
 
   
 \vfill

\end{frame}


\section{Dynamics}

 \begin{frame}{Linear algebra}


$A \in GL(\mR^q)$ acts on $\mR^q$ via $L_A(\vec{v}) = A \cdot \vec{v}$.
\bigskip

There is well-known trichotomy for dynamics of  linear actions:

\medskip

$\bullet$~ Elliptic, or isometric $\Leftrightarrow$ all orbits of $L_{A}$ action are bounded $\Leftrightarrow$ $\|A^{\ell}\|$ is bounded  as $\ell \to \infty$.

\medskip

$\bullet$~ Parabolic, or shear $\Leftrightarrow$   orbits of $L_{A}$ action not bounded, but do not grow exponentially fast in norm $\Leftrightarrow$ $\|A^{\ell}\|$ grows at subexponential rate   as $\ell \to \infty$.


\medskip

$\bullet$~ Hyperbolic, or partially hyperbolic $\Leftrightarrow$   some orbits of $L_{A}$ action   grow exponentially fast in norm
$\Leftrightarrow$ $\|A^{\ell}\|$ grows at exponential rate   as $\ell \to \infty$.


\medskip

 



 \vfill

\end{frame}


  \begin{frame}{Asymptotic exponent}

    
   
    {\bf Definition:} The   \emph{transverse expansion rate}  function at $x$ is
   $$\lambda(\cGF, k, x) =  
   \max_{\| [g] \|_x \leq k} ~  \frac{ \ln \left(  \max \{ \| D_x g  \| ,  \| (D_x g)^{-1}  \|\} \right) }{k }~ \geq ~ 0$$

\bigskip

    \begin{center}
\includegraphics[width=0.6\textwidth]{strasbourg_pix/Figure3.png}
\end{center}

 \medskip

 \vfill

\end{frame}


  \begin{frame}{}
  
  \bigskip
  \bigskip
   
    {\bf Definition:} The \emph{asymptotic transverse growth rate} at $x$ is
   $$\lambda(\cGF, x) = \limsup_{k \to \infty} ~ 
 \lambda(\cGF, k, x) ~ \geq ~ 0$$
This is essentially the ``maximum Lyapunov exponent'' for $\cGF$ at $x$.

\medskip

 $\lambda(\cGF, x)$ is a Borel function of $x \in \cT$, as each norm function $\| D_{w'} h_{\sigma_{w,z}} \|$ is continuous for $w' \in D(h_{\sigma_{w,z}})$ and the maximum of Borel functions is Borel.

\bigskip


{\bf Lemma:}   $\lambda_{\F}(z)$ is constant along leaves of   $\F$. 

\medskip

$\Longrightarrow$ $\F$ decomposes $M$ into saturated Borel sets.


 \vfill

\end{frame}


  
  \begin{frame}{Expansion classification}

  
{\bf Theorem:} (Hurder, 2000, 2005)  Let $\F$ be a $C^1$-foliation of compact manifold $M$. Then there is a disjoint decomposition 
$$M = \cE_{\F} \cup \cP_{\F} \cup \cH_{\F}$$
consisting of  $\F$-saturated, Borel subsets of $M$, defined by:

 

\begin{enumerate}

% \pause

\item Elliptic points:  $\cE_{\F} \cap \cT  = \{ x \in \cT \mid \forall ~ k \geq 0, ~ \lambda(\cGF, k, x) \leq \kappa(x) \}$\\
i.e., ``points of bounded expansion'' ~ (example: Riemannian foliations)

 \medskip
% % \pause

\item Parabolic  points:  $\cP_{\F} \cap \cT  = \{ x \in \cT - ( \cE_{\F} \cap \cT)\mid  \lambda(\cGF, x) = 0 \}$ \\
i.e., ``points of slow-growth expansion'' ~  (example: Distal foliations)
 
\medskip
% % \pause

\item   Hyperbolic  points:  $\cH_{\F} \cap \cT  = \{ x \in \cT \mid  \lambda(\cGF, x) > 0 \}$ \\
 i.e., ``points of exponential-growth expansion''   ~ 
 (example: Anosov foliations, or more generally, non-uniformly, partially   hyperbolic foliations) 
\end{enumerate}


\vfill
\end{frame}

  \begin{frame}{Classification via transverse growth}
  
  \medskip
  {\bf Proposition:}   $\F$   a foliation with all leaves compact, then $ \lambda(\cGF, x) = 0$,  $\forall ~ x \in \cT$.
  
  \medskip
   The derivative cocycle is not trivial, but has ``slow growth'' as must be parabolic (upper triangular). 

  \bigskip
   % \pause


  {\bf Theorem:} If there exists $\kappa > 0$ so that $\lambda(\cGF, k, x) \leq \kappa$ ~ $\forall ~ x \in \cT$,  then $\F$ is equicontinuous (and so Riemannian?)
  
  \medskip
  
  The case where  there is a bound $\kappa(x)$ depending on $x$, but there is no uniform upper bound,  this remains a mystery!
  
  \medskip
  
 The parabolic case  of $ \lambda(\cGF, x) = 0$ for all  $x \in \cT$, but no pointwise upper bound, should be  some sort of nil-geometry.
 
 \bigskip
  % \pause


 The case when the set $\cH_{\F} $ of partially hyperbolic points is   most interesting. 
 
\vfill
\end{frame}



\section{Secondary classes}

  \begin{frame}{Godbillon-Vey classes}
  
  \medskip
  
  Assume that $\F$ is a $C^2$-foliation and the normal bundle $Q = TM/T\F$ is oriented. 
  \medskip
  
  Let $\omega$ be a $q$-form defining $T\F$, so then $d \omega = \eta \wedge \omega$ for a $1$-form $\eta$.
  
  \medskip
  
  The form $\eta \wedge (d \eta)^q$ is then a closed $2q+1$-form on $M$, and its cohomology class is independent of choices.
  
  $$GV(\F) = [\eta \wedge (d \eta)^q] \in H^{2q+1}_{deR}(M)$$
  
  is the \emph{Godbillon-Vey class} of $\F$. 
  
  \bigskip
   % \pause


  {\bf Question:} (Moussu-Pelletier 1974, Sullivan 1975) If $GV(\F) \ne 0$,  what does this imply about the dynamical properties of $\F$? Must there be leaves with exponential growth?
  
  \medskip
  
  This question motivated many developments in the study of foliations.
  
  
\vfill
\end{frame}


  \begin{frame}{Godbillon-Vey and exponent}
  
  \medskip
 
 {\bf Theorem:} (Hurder-Langevin 2004) Let $\F$ be a $C^2$-foliation of codimension-$q$ with $GV(\F) \ne 0$. Then the hyperbolic set $\cH_{\F}$ has \emph{positive Lebesgue measure}.
 
 \bigskip
  % \pause


 {\bf Theorem:} (Hurder-Langevin 2004) Let $\F$ be a $C^1$-foliation of codimension-$1$ such that the hyperbolic set $\cH_{\F}$ has positive Lebesgue measure. Then $\F$ has a resilient leaf, and hence has positive geometric entropy. 
 
 \medskip
 
 {\bf Corollary:} (Duminy 1982) Let $\F$ be a $C^2$-foliation of codimension-$1$ with $GV(\F) \ne 0$. Then $\F$ has a resilient leaf.  

\bigskip

Combining results, we have the stronger statement: 
  \medskip
   % \pause


  {\bf Theorem:} (Hurder-Langevin 2004) Let $\F$ be a $C^1$-foliation of codimension-$1$ such that the hyperbolic set $\cH_{\F}$ has positive Lebesgue measure. Then $\F$ has positive geometric entropy, in the sense of Ghys-Langevin-Walczak, so that $GV(\F) \ne 0$ implies positive geometric entropy. 

 
   
\vfill
\end{frame}


 
  \begin{frame}{Positive entropy and hyperbolicity}

Questions for codimension $q > 1$:
\medskip

 {\bf Question:} Suppose $\cGF$ has positive geometric entropy. What can be said about the dynamics of $\F$, beyond the easy conclusion that $\lambda(\cGF, x) > 0$ for some   $x$?
 
  % \pause

 \bigskip
 
 {\bf Question:} Suppose $\cH_{\F}$ has positive Lebesgue. What can be said about the dynamics of $\F$? Must   $\cGF$ have positive entropy?

\medskip
 % \pause

{\bf Question':} Suppose $\cH_{\F}$ has positive Lebesgue, and the normal derivative cocycle has \emph{algebraic hull} $G \subset GL(\mR^q)$ where $G$ acts transitively on $\mR^q$.  Must   $\cGF$ have positive entropy? 
 

 \bigskip
  % \pause
 

These are all part of the general program to understand the structure of partially hyperbolic foliations. 

 
       
\vfill
\end{frame}


 \section{Matchbox manifolds}

  \begin{frame}{Closed invariant sets}

  As an alternative to asking for the ``big picture'' of foliation dynamics,  study   the dynamics near closed invariant sets.   This follows Smale's plan for the study of the dynamics of flows, where attractors are  a central concept.   
 \bigskip
 
 $\cZ \subset M$   \emph{minimal} $\Longleftrightarrow$ closed and every leaf $L \subset \cZ$ is dense. 
 
 \medskip
 
$\cW \subset M$ is \emph{transitive} $\Longleftrightarrow$  closed and there exists a dense leaf $L \subset \cW$

% % \pause
\medskip

$M$ compact, then minimal sets for foliations always exist.
 
 \medskip
 
Transitive sets are most important for flows -- Axiom A attractors  are transitive sets, while the minimal sets include the  periodic orbits in the domain of attraction.
  
  \bigskip
  
  {\bf Problem:} Describe the minimal sets for $\F$ in each type of dynamic. 
   
   \medskip
   
   Restrict attention to \emph{exceptional minimal sets} $\cZ$, where $\cZ \cap \cT$ is a Cantor set.
     
     \medskip 

$\fM \subset M$ has the structure of a \emph{minimal matchbox manifold}; 
that is, it is a foliated space which is \emph{transversally Cantor-like}. 

\vfill
\end{frame}


 
 \begin{frame}{Structure of exceptional minimal sets}

{\bf Conjecture:} If $\F$ is a $C^2$-foliation of codimension-one, and $\fM \subset M$ is an exceptional minimal set, then  $\fM$ has the structure of a \emph{Markov Minimal Set}. 

\medskip
That is, there is a covering of $\fX = \fM \cap \cT$ by clopen sets, such that the restricted holonomy pseudogroup $\cGF^{\fX}$ is generated by a free semigroup.

    \bigskip
  % \pause
   
{\bf Theorem:} [Clark--H--Lukina, 2012]  Let  $\fM$ be an exceptional minimal set of dimension $p$ for a $C^r$-foliation, $r \geq 0$. Then  there exists a presentation  $\ds \cP = \{ p_{\ell} \colon M_{\ell} \to M_{\ell -1}  \mid \ell \geq \ell_0\}$ consisting of   branched $p$-manifolds $M_\ell$ and local    covering $C^{r-1}$-maps $p_\ell$, such that $\fM$   is   homeomorphic to the inverse system, 
$$
  \fM ~  \cong  ~  \cS_{\cP} = \varprojlim ~ \{ p_{\ell} \colon M_{\ell} \to M_{\ell -1} \mid \ell \geq \ell_0\}. 
$$
  
 \medskip
 
 This generalizes a well-known result for the case where $\fM$ is the tiling space associated to a  
   repetitive, aperiodic tiling of $\mR^q$ with      finite local complexity. 
  
\vfill
\end{frame}


\begin{frame}{Williams solenoids}

Williams (1970, 1974) introduced a class of inverse limit spaces, called generalized solenoids, which model Smale's Axiom A attractors.

\medskip
Let $K$ be  a branched manifold, i.e. each $x \in K$ has a neighborhood homeomorphic to the disjoint union of a finite number of Euclidean disks modulo some identifications.

\medskip
$f: K \to K$ is an expansive immersion of branched manifolds satisfying a flattening condition. Then  
 $$\fM = \lim_{\longleftarrow}\left\{ f: K \to K \right\}$$
each point $x$ has a neighborhood homeomorphic to $[-1,1]^n \times \textrm{Cantor set}$. Moreover, $f$ extends to a hyperbolic map defined on some open neighborhood of $\fM \subset V \subset \mR^m$ for $m$ sufficiently large. Thus, $\fM$ is an exceptional minimal set for the expanding foliation $\F_u$ of $V$ for $f$, and $\fM \subset \cH_{\F_u}$. 

\end{frame}


    



 \begin{frame}{Parabolic minimal sets}

  
 {\bf Definition:} A minimal set $\cZ$ is said to be \emph{parabolic} if $\cZ \cap  \cH_{\F}  = \emptyset$.
 
% % \pause
 \bigskip
 
 {\bf Proposition:} Let $\F$ be a   $C^1$-foliation of a compact manifold $M$, with all leaves of $\F$ compact. Then every leaf of $\F$ is a parabolic minimal set.
 \medskip
 
 {\it Proof:}  If some holonomy transformation along $L_w$  has a non-unitary eigenvalue, then it has a stable manifold.


 % % \pause
 \medskip



{\bf Theorem:} A parabolic minimal set has zero geometric entropy. 

 

\bigskip

{\bf Question:} What are the \emph{zero entropy} exceptional minimal sets for $C^r$-foliations?


\vfill
\end{frame}
   

\begin{frame}{Solenoids  and weak solenoids}

Weak solenoids are generalizations to higher dimensions of $1$-dimensional $p$-adic (Vietoris) solenoids, i.e. inverse limit of finite-to-one coverings of a circle
  \begin{align*} \mathbb{S}_\infty & = \lim_{\longleftarrow} \left\{ p_i: \mathbb{S}^1 \to \mathbb{S}^1, ~ i \geq 0 \right\}. \end{align*}

 


 
 An \emph{$n$-dimensional solenoid}, as studied by   McCord (1965) and  Fokkink and Oversteegen (2002)  is an inverse limit space
$$\ds \cS = \lim_{\leftarrow} ~ \{ p_{\ell+1} \colon L_{\ell+1} \to L_{\ell}\}$$
where for $\ell \geq 0$, $L_{\ell}$ is a closed, oriented, $n$-dimensional manifold, and     $p_{\ell+1} \colon L_{\ell +1} \to L_{\ell}$ are smooth, orientation-preserving proper covering maps.


 \smallskip

$\cS$ is a fibre bundle with Cantor set fibre, and profinite structure group.  
  
\bigskip

If all compositions   $L_{\ell+1} \to L_{0}$ are normal (Galois) coverings, then $\cS$ is called a McCord solenoid, and otherwise is a (weak) solenoid.


% % \pause
\bigskip

 
\vfill

\end{frame}
  
\frame % 4
{
  \frametitle{Embeddings}

  For these abstract solenoidal spaces, the basic question is whether they are homeomorphic to minimal sets of parabolic or hyperbolic type?
   
 \bigskip
   % \pause
   

 {\bf Embedding Property:}  Given a (generalized) solenoid   $\cS$, does there exists a $C^r$-foliation $\F_M$ of a compact manifold $M$ and an embedding of $\iota \colon \cS \hookrightarrow M$ as a foliated subspace? ($r \geq 0$)
 
   
  \medskip
      % \pause
   

     {\bf Germinal Extension Property:}   Given a (generalized)  solenoid $\cS$, does there exists a $C^r$-foliation $\F_U$ of an open manifold $U$ and an embedding of $\iota \colon \cS \hookrightarrow U$ as a foliated subspace? ($r \geq 0$)

 \bigskip
   
     Solutions to the embedding problem for solenoids modeled on $\mS^1$  were given by  Gambaudo, Tressier, et al in 1990's. 
 
  

  
  
\vfill   


}


 

\frame % 4
{
  \frametitle{Embeddings of toral solenoids}


 \medskip
 
  {\bf Theorem:} [Clark--H, 2008] 
 Let $\F_0$ be a $C^r$-foliation of codimension $q \geq 2$ on a manifold $M$. Let $L_0$ be a compact leaf with
$H^1(L_0 ; \mR) \ne 0$, and suppose that $\F_0$ is a product foliation in some open neighborhood $U$ of $L_0$. Then there exists a foliation $\F$ on $M$ which is $C^r$-close to $\F_0$, and $\F$ has a solenoidal minimal set contained in $U$ with base $L_0$. If $\F_0$ is a distal foliation, then $\F$ is also distal.



  
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 The criteria for embedding depends on the degree of smoothness required, and the tower of subgroups of the fundamental group. 

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The study of minimal foliated spaces in the abstract, leads to very interesting (and seemingly difficult) problems concerning when these can be realized as minimal  sets for $C^r$ foliations, especially for $r \geq 2$. 


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  \frametitle{Instability}

 
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One application is a type of ``Reeb Instability'' result:
 
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  {\bf Theorem:}   [Clark \& H 2010]  Let $\F_0$ be a $C^{\infty}$-foliation of codimension $q \geq 2$ on a manifold $M$. Let $L_0$ be a compact leaf with
$H^1(L_0 ; \mR) \ne 0$, and suppose that $\F_0$ is a product foliation in some saturated open neighborhood $U$ of $L_0$. 

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Then there exists a foliation $\F_M$ on $M$ which is $C^{\infty}$-close to $\F_0$, and $\F_M$ has an uncountable set of  solenoidal minimal sets $\{\cS_{\alpha} \mid \alpha \in \cA\}$, which are  \emph{pairwise non-homeomorphic}.  

  
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Solenoid-type objects are ``typical'' for perturbations of dynamical systems formed by flows,  and possibly also for foliations.


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  \frametitle{Expansive minimal sets}
 
 {\bf Theorem:} [Auslander--Glasner--Weiss, 2007] Let $\G$ be a finitely generated group acting minimally on a Cantor set $\fX$. If the action is distal, then it must be equicontinuous. 
 
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 This result suggests: 
 
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 {\bf Conjecture:} Let $\fM \subset M$ be an exceptional minimal set for a foliation $\F$. Then either   the holonomy action of $\cGF^{\fX}$ on $\fX$ equicontinuous, or it is expansive.
 
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 This brings the study of foliation dynamics full circle:
 
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 {\bf Problem:} Let $\fM \subset M$ be an exceptional minimal set for a $C^r$-foliation $\F$ with $r \geq 1$, for which the   holonomy action of $\cGF^{\fX}$ on $\fX$  is expansive. Show that $\fM \cap \cH_{\F} \ne \emptyset$, and give some sort of structure theory for these examples. 
 
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There exist examples of exceptional minimal sets for $C^2$-foliations with codimension-one, which contain parabolic leaves, so one cannot expect  $\fM \subset \cH_{\F}$. 

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The conclusion is that classifying foliations remains as elusive a goal as ever.

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Though, the study of their dynamical properties provides the best method to date, for the passage from local definitions to global properties. 


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Thank you for your attention!
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