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\date[December 18, 2010]
 {\footnotesize   First Joint Meeting of the \\Sociedad de Matem\'{a}tica de Chile\\
  American Mathematical Society\\
 Special Session on Group Actions: Probability and Dynamics\\
}

  
\title{Dynamics of Group Actions and   Minimal Sets}


\author{Steven Hurder}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder}



\begin{document}

\frame{\titlepage} % # 1



\section{Introduction}
 
  
\frame % 0
{
  \frametitle{Abstract:}
 
 {\footnotesize  Ongoing joint works with Alex Clark \& Olga Lukina,  Leicester University, UK. 


The study of foliation dynamics aims to understand the asymptotic properties of its leaves, and identify geometric and topological ``structures'' which are associated to them; e.g.,  the minimal sets of the foliation. 
 
 The dynamics of a foliation partitions the ambient manifold into three disjoint saturated Borel sets: the Elliptic, Parabolic and Hyperbolic regions. A fundamental open problem is to describe the properties of minimal sets contained in each of these regions.
 
 Alex Clark and the author showed that there exists  smooth   actions of $\mZ^n$ with a continuum of distinct   minimal sets, all contained in the union of elliptic and parabolic sets, and no two of which are homeomorphic. These minimal sets are ``weak solenoids'',  and  give rise to a continuum of secondary invariants. 
  
  The study of these examples leads to the more general study of properties and classification of matchbox manifolds, a particular class of continua that arise in dynamical systems.
   }
 
   
  
    
  
   
  \vfill   

}

 
\frame % 1
{
  \frametitle{Smooth dynamical systems}
 
   
  Smale [1967, Bulletin AMS] -- differentiable dynamics for a $C^r$-diffeomorphism $f \colon N \to N$ of a closed manifold $N$, $r \geq 1$:
  
  \pause
  
$\bullet$ Classify dynamics as hyperbolic, or otherwise.
  
  \pause
  
$\bullet$ Describe the minimal/transitive closed invariant sets \& attractors.
  
  \pause
  
$\bullet$  Is the system structurally stable under $C^r$-perturbations, $r \geq 1$?
   
   \pause
   
$\bullet$  Find cohomology invariants of the system which characterize it.
  
  \pause
  \bigskip
  
 
Also consider non-singular vector field $\vec{X}$ on a closed manifold $M$ which defines a $1$-dimensional foliation $\F$ on $M$.  

  Smale also suggested to study these points for large group actions.
  
    Students of Godbillon,  Smale, Tamura studied foliation dynamics.

    
  
   
  \vfill   

}
  
\frame % 2
{
  \frametitle{Foliation dynamics}
 
  
 A foliation $\mathcal F$ of dimension $n$ on a smooth manifold $M^m$ decomposes the space  into ``uniform layers'' -- the leaves.



  $M$  is a \emph{$C^r$ foliated manifold}   if the    transition functions for the foliation charts
   $\vp_{i} \colon U_i \to [-1,1]^n \times T_i$ (where $T_i \subset \mR^q$ is   open)  are  $C^{\infty}$ leafwise, 
   and vary $C^r$ with the transverse parameter in the leafwise $C^{\infty}$-topology.

 \pause
For a continuous dynamical system on a compact manifold $M$ defined by  a flow $\varphi \colon M \times \mR \to M$,   the orbit $L_x = \{\varphi_t(x) = \varphi(x,t) \mid t \in \mR\}$ is thought of as the time trajectory of the point $x \in M$.   
  
 
\pause
 
Foliation dynamics: replace the concept of time-ordered trajectories with multi-dimensional futures for points; then study   the    aggregate and statistical behavior of the collection of its leaves.

 
 

  \vfill   

}

\frame % 3
{
  \frametitle{Group actions}
 
   $\G = \langle \gamma_1 , \ldots , \gamma_d \rangle$ is a finitely generated group.
   
   $\varphi \colon \G \times N \to N$ is $C^r$-action on closed manifold of dimension $q$, $r \geq 1$.

If $\G \cong \pi_1(B, b_0)$ and $\widetilde{B} \to B$ is the universal covering, then 
$$M = (\widetilde{B} \times N)/\G  \to B$$ 
is a foliated bundle, where the transverse holonomy of $\F_{\varphi}$ determines the action $\varphi$ up to conjugacy.
$$ \text{Dynamics of action} ~ \varphi \quad \Longleftrightarrow \quad \text{Dynamics of leaves of} ~ \F_{\varphi}$$

Each point of view has advantages, limitations. 


   \vfill   

}




\frame
{

  \frametitle{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 $\cG_{\F}$.

 

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

  
 


  \vfill

}



\frame
{

  \frametitle{Groupoid  word length}
  
{\bf Definition:} The groupoid of $\cG$ is the space of germs 
$$\Gamma_{\cG} = \{[g]_x \mid g \in \cG ~ \& ~ x \in D(g)\} ~ , ~ \G_{\F} = \G_{\cG_{\F}}  $$
with source map $s[g]_x = x$ and range map $r[g]_x = g(x) = y$.

\pause 

 


 For $g \in \G_{\cG}$,    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

}


\section{Smooth Dynamics}



\frame
{

  \frametitle{Derivative cocycle}
  
  Assume $(\cG, \cT)$ 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 $x \in \cT$.
  
  \bigskip
  
  
 {\bf Definition:} The normal cocycle $D\varphi \colon \G_{\cG} \times \cT \to {\bf GL(\mR^q)}$ is defined by
 $$D\varphi [g]_x =  D_x g   \colon T_x \cT \cong_x \mR^q  \to T_{y}\cT \cong_y \mR^q$$

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

}



\frame
{

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


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

 $\lambda(\cG, x)$ 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.

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

\medskip

 \vfill

}


\frame
{

  \frametitle{Asymptotic exponent -- group actions}
  
   
Let  $\varphi \colon \G \times N \to N$ be a $C^1$-action.

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


  
 
   
   
    {\bf Definition:} The asymptotic transverse growth rate at $x$ is
   $$\lambda(\varphi, x) = \limsup_{k \to \infty} ~  \lambda(\varphi, k, x) ~ \geq ~ 0$$
This is essentially the ``maximum Lyapunov exponent'' for $\cG$ at $x$ --
for all $x \in N$, $\epsilon > 0$, there exists a sequence $\{\gamma_{\ell} \in \G \mid \|\gamma \| \to \infty\}$, 
 $$\max \left\{ \| D_x \varphi(\gamma_{\ell})  \| , \| (D_y \varphi(\gamma_{\ell}) ^{-1} \| \right\} \geq \exp \left\{ \ell \cdot (\lambda(\varphi, k, x) -\epsilon) \right\}$$

 
 \vfill

}


\frame
{
  \frametitle{Expansion classification}

 

$$M = \cE \cup \cP \cup \cH$$

where each are $\F$--saturated, Borel subsets of $M$, defined by:

 

\begin{enumerate}

\pause

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

 \medskip
\pause

\item Parabolic  points:  $\cP \cap \cT  = \{ x \in \cT - ( \cE \cap \cT)\mid  \lambda(\cG, x) = 0 \}$ \\
i.e., ``points of slow-growth expansion'' --  distal foliations
 
\medskip
\pause

\item Partially Hyperbolic  points:  $\cH \cap \cT  = \{ x \in \cT \mid  \lambda(\cG, x) > 0 \}$ \\
 i.e., ``points of exponential-growth expansion'' --  non-uniformly, partially   hyperbolic foliations
 
\end{enumerate}

\vfill
}

\section{Minimal Sets}

\frame % 4
{
  \frametitle{Minimal and transitive sets}
 
  $M$ compact foliated, $\F$ foliation of codimension-$n$.  $\fM \subset M$ is 
  
 $\bullet$  \underline{minimal} if it is closed, $\F$-saturated, and every leaf in $\fM$ is dense.  
   
 $\bullet$  \underline{transitive} if it is closed, $\F$-saturated, and there exists a dense leaf in $\fM$.  

  {\bf Remark:}  A minimal set $\fM$ is an example of a \emph{continuum}; that is, a compact and connected metrizable space. In fact, it is  an \emph{indecomposable continuum}, which  is a continuum that is not the union of   two   proper subcontinua. 

\pause
\medskip

For a group action $\varphi$, a minimal set $\fM$ for $\F_{\varphi}$ is the compactification   of  $\G$ associated to the  sub $C^*$-algebra $\varphi_x^* \colon C^0(N) \to C_b(\G)$.

  \vfill   

}


\frame % 5
{
  \frametitle{Shape dynamics of  minimal sets}
 

The \emph{shape} of a minimal set $\fM$ is defined by a co-final descending chain $\{U_{\ell} \mid \ell \geq 1\}$ of open neighborhoods
$$ U_1 \supset U_2 \supset \cdots \supset U_{\ell} \supset \cdots \supset \fM \quad ; \quad \bigcap_{\ell = 1}^{\infty} ~ U_{\ell} = \fM$$
Such a tower is called a shape approximation to $\fM$.

\pause
\medskip


The \emph{shape dynamics} of $\fM$ is the germ of the dynamical system $\F$ defined by a shape approximation to $\fM$.
This is equivalent to specifying the \emph{foliated microbundle} of $\F$ defined by  $\fM \subset M$.

 \pause
\medskip

 {\bf Definition:} The shape of $\fM$   is \emph{stable} if there exists $\ell_0$ such that for $\ell \geq \ell_0$ the inclusion
 $\fM \subset U_{\ell+1} \subset U_{\ell}$ is a homotopy equivalence. 

  \vfill   

}

 

\frame % 6
{
  \frametitle{Remarks and   questions -- codimension-one}
 
 
  
 The first question is very old:
 
 {\bf Question:}  What are the minimal sets in a codimension-$1$,  $C^r$-foliation?

\medskip
\pause

 {\bf Example:} Denjoy minimal sets  for  $C^1$-foliations are stable.
 
  
\medskip

\pause

 {\bf Example:} Markov minimal sets for  $C^2$-foliations are stable, but this need not be true for $C^1$.


  \vfill   

}

 

\frame % 7
{
  \frametitle{Remarks and   questions -- codimension-$q > 1$}
 
 
  {\bf Question:} What indecomposable  continua can arise as  minimal sets in codimension-$q$? 
   Are there restrictions on their   shape types for $C^r$-dynamics, depending on $r > 0$?

  
\medskip

\pause

The Sierpinsky torus $\mT^q$ and its generalizations can be realized as hyperbolic minimal sets, $q > 1$. 

  
\medskip

\pause

 
 {\bf Question:} Are there   conditions on the shape dynamics of $\fM$ which force $\fM$ to have stable shape? (e.g., hyperbolicity)


  \vfill   

}

 
 
 \section{Matchbox Manifolds}
  

 
\frame % 8
{
  \frametitle{Matchbox manifolds}
 
 

 {\bf Definition:}  An $n$-dimensional  \emph{matchbox manifold} is a   continuum  $\fM$   which is a foliated space with codimension zero and leaf   dimension $n$. Essentially, same concept as laminations.
 \medskip
 
 $\fM$ is a foliated space if it admits a covering $\cU = \{\vp_{i} \mid 1 \leq i \leq \nu\}$ with foliated coordinate charts $\vp_{i} \colon U_i \to [-1,1]^n \times \fT_i$. The compact metric spaces $\fT_i$ are totally disconnected $\Longleftrightarrow$  $\fM$ is a matchbox manifold. 
\pause
$$\text{Leaves of} ~ \F ~ \Longleftrightarrow  ~\text{path components of} ~ \fM ~  \Longleftrightarrow ~ \text{proper subcontinua} $$
  \pause
 A ``smooth matchbox manifold'' $\fM$ is analogous to a compact manifold,  with the transverse dynamics of the foliation $\F$ on the Cantor-like fibers $\fT_i$ representing fundamental groupoid data. 
 
  \vfill   

}

   

 
\frame % 9
{
  \frametitle{Embedding matchbox manifolds}
 
 
{\bf Problem:}
Let $\fM$ be a minimal matchbox manifold of dimension $n$. When does there exists a $C^r$-foliation $\F_M$ of a compact manifold $M$ and a foliated topological embedding  $\iota \colon \fM \to M$ realizing $\fM$ as a   minimal set?
 

 \pause
 
The space of tilings associated to a given quasi-periodic tiling of
$\mR^n$ is a matchbox manifold.
 For a few classes of quasi-periodic
tilings of $\mR^n$, the codimension one canonical cut and project
tiling spaces, it is known that the associated
matchbox manifold is a minimal set for a $C^1$-foliation of a torus
$\mT^{n+1}$, where the foliation is a generalized Denjoy example.

\pause

The ``Williams  solenoids'',   introduced by 
Bob Williams in 1967, 1974 as   the attractors of
certain Axiom A systems, are matchbox manifolds. 
It  is unknown which of the Williams solenoids can be
embedded as minimal sets for foliations of   closed manifolds.

  
  \vfill   

}

 
\frame % 10
{

  \frametitle{Topological dynamics}

{\bf Definition:} $\fM$ is an \emph{equicontinuous matchbox manifold}  if it admits some covering by foliation charts as above, such that for all $\e > 0$, there exists $\delta > 0$ so that for all $h_{\cI} \in \cGF$ we have
$$x,x' \in D(h_{\cI}) ~ {\rm with} ~ d_{\cT}(x,x') < \delta \quad \Longrightarrow  \quad d_{\cT}(h_{\cI}(x), h_{\cI}(c')) < \e$$

  
\pause 

 {\bf Theorem:} [Clark-Hurder 2010] Let $\fM$ be an equicontinuous matchbox manifold. Then $\fM$ is   minimal.  

\pause 

{\bf Definition:} $\fM$ is an \emph{expansive matchbox manifold}  if it admits some covering by foliation charts as above, such that there exists $\e > 0$,  so that for all $x \ne x' \in \cT$ with $d_{\cT}(x,x') < \e$, there exists 
$h_{\cI} \in \cGF$ such that 
$$  d_{\cT}(h_{\cI}(x), h_{\cI}(x')) \geq \e$$

\bigskip

  \vfill

}

\frame % 11
{
  \frametitle{Weak solenoids}
  
  Let $B_{\ell}$ be compact, orientable manifolds of dimension $n \geq 1$ for $\ell \geq 0$, with   orientation-preserving covering maps
  $$ \stackrel{p_{\ell +1}}{\longrightarrow} B_{\ell} \stackrel{p_{\ell}}{\longrightarrow} B_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  B_1 \stackrel{p_1}{\longrightarrow}  B_0 $$
  \pause
The $p_{\ell}$ are called the \emph{bonding maps} for the weak solenoid
$$\cS = \lim_{\leftarrow} ~ \{p_{\ell} \colon B_{\ell} \to B_{\ell -1}\} ~ \subset ~ \prod_{\ell =0}^{\infty} ~ B_{\ell} $$ 
\pause
Choose basepoints $x_{\ell} \in B_{\ell}$ with $p_{\ell}(x_{\ell}) = x_{\ell -1}$. 

Set $G_{\ell} = \pi_1(B_{\ell} , x_{\ell})$. 

 
  \vfill   

}


\frame % 12
{
  \frametitle{McCord solenoids}
  
There is  a descending chain of groups and injective maps
  $$ \stackrel{p_{\ell +1}}{\longrightarrow} G_{\ell} \stackrel{p_{\ell}}{\longrightarrow} G_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  G_1 \stackrel{p_1}{\longrightarrow}  G_0 $$

Set $q_{\ell} = p_{\ell} \circ \cdots \circ p_1 \colon B_{\ell} \longrightarrow  B_0$.

{\bf Definition:} $\cS$ is a \emph{McCord solenoid}  for some fixed $\ell_0 \geq 0$,    for all $\ell \geq \ell_0$ the image 
$G_{\ell} \to H_{\ell} \subset G_{\ell_0}$ is a normal subgroup of $G_{\ell_0}$. 

 \medskip
 
 {\bf Theorem} [McCord 1965] Let $B_0$ be an oriented  smooth closed manifold. Then a  McCord solenoid $\cS$ is an orientable,  homogeneous, equicontinuous   smooth matchbox manifold.
  
 
   \vfill   

}
 

\frame % 13
{
  \frametitle{Classifying weak solenoids}
  
A weak solenoid is determined by the base manifold $B_0$ and the tower equivalence of the descending chain

 $$\cP \equiv \left\{  \stackrel{p_{\ell +1}}{\longrightarrow} G_{\ell} \stackrel{p_{\ell}}{\longrightarrow} G_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  G_1 \stackrel{p_1}{\longrightarrow}  G_0 \right\}$$
 \pause

{\bf Theorem:}[Pontragin 1934; Baer 1937] For $G_0 \cong \mZ$, the homeomorphism types of McCord solenoids is uncountable. 
 \pause

{\bf Theorem:}[Kechris 2000; Thomas2001] For $G_0 \cong \mZ^k$ with $k \geq 2$, the homeomorphism types of McCord solenoids is not classifiable, \emph{in the sense of Descriptive Set Theory}.

The number of such is not just huge, but indescribably large.
  \vfill   

}
 

\frame % 14
{
  \frametitle{Homogeneous matchbox manifolds}
  
 {\bf Definition:} A matchbox manifold $\fM$ is \emph{homogeneous} if the group of Homeomorphisms of $\fM$ acts transitively.
 
 
  \bigskip
  \pause
  
 {\bf Theorem:} [Clark-Hurder 2010] Let $\fM$ be a homogeneous matchbox manifold. Then $\fM$ is equicontinuous, minimal,  and without holonomy. Moreover, $\fM$ is homeomorphic to a McCord solenoid. 
 
    \medskip
  \pause
  
 {\bf Corollary:}  Let $\fM$ be a homogeneous matchbox manifold. Then $\fM$ is homeomorphic to the suspension of an minimal action of a countable group  on a Cantor group $\mK$.

    \vfill   

}
 
  
 

\section{Embeddings}

  
\frame % 15
{
  \frametitle{Embedding results}
  
   
   
 {\bf Problem:} Let $r \geq 0$.
What types of towers of finitely-generated groups
   $$\cP \equiv \left\{  \stackrel{p_{\ell +1}}{\longrightarrow} G_{\ell} \stackrel{p_{\ell}}{\longrightarrow} G_{\ell -1} \stackrel{p_{\ell -1}}{\longrightarrow} \cdots \stackrel{p_{2}}{\longrightarrow}  G_1 \stackrel{p_1}{\longrightarrow}  G_0 \right\}$$
arise from equicontinuous   minimal sets of $C^r$-foliations?
  
  $\cP$ is called a presentation of the inverse limit $\cS$. 
  
  The foliated homeomorphism type of $\cS$ is completely determined by $\cP$.
  
     \medskip
  \pause
  
  It seems that extremely little is known about such questions.  
  
  We present some results for the case $\G = \mZ^k$.
  
  Note that   solenoids do not have stable shape.
  
  \vfill   

}

  
\frame % 16
{
  \frametitle{Topological embeddings}
  
   
Our strongest results are for $C^0$-embedding problem -- every presentation of a solenoid with base $\mT^k$ admits an embedding into a $C^0$-foliation.

{\bf Theorem:} [Clark \& Hurder] 
Let $\cP$ be a presentation of the solenoid $\cS$ over the base space $\mT^k$, and let $q \geq 2k$. Then   there exists a $C^0$-foliation $\whF$ of $\mT^k \times \mD^q$  such that:
\begin{enumerate}
\item  $\whF$  is a distal foliation, with   smooth transverse invariant volume form;
\item $L_0 = \mT^k \times \{ \vec0\}$ is a leaf of $\whF$, and  $\whF = \F_0$ near the boundary of $M$;
\item there is an embedding of $\cP$ into the foliation $\whF$;
\item the solenoid $\cS$ embeds as  a minimal set $\whF$.
\end{enumerate}
 


    \vfill   

}


  
\frame % 17
{
  \frametitle{Smooth embeddings}
  
  
  The embedding problem for solenoids into $C^1$-foliations  is the next most general case. 
  
{\bf Theorem:} [Clark \& Hurder] 
Let $\cP$ be a presentation of the solenoid $\cS$ over the base space $\mT^k$, and let $q \geq 2k$. Suppose that $\cP$ admits a sub-presentation $\cP'$ which satisfies condition  (**).
Then   there exists a   $C^1$-foliation $\whF$ of $\mT^k \times \mD^q$   such that:
\begin{enumerate}
\item  $\whF$  is a distal foliation, with    smooth transverse invariant volume form;
\item $L_0 = \mT^k \times \{ \vec0\}$ is a leaf of $\whF$, and  $\whF = \F_0$ near the boundary of $M$;
\item there is an embedding of $\cP'$ into the foliation $\whF$;
\item the solenoid $\cS$ embeds as  a minimal set $\whF$.
\end{enumerate}
 

  
  
    \vfill   

}


 
 
\frame % 18
{
  \frametitle{anti-Reeb-Thurston Stability}
  
  

{\bf Theorem:} [Clark \& Hurder] 
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 $L_0$. 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 $\{\cS_{\alpha} \mid \alpha \in \cA\}$, all contained in $U$, and \emph{pairwise non-homeomorphic}. 

If $\F_0$ is a distal foliation with a smooth transverse invariant volume form, then the same holds for $\F_M$.
 
   
  
    \vfill   

}



 
\frame % 19
{
  \frametitle{Get a grip - some open problems}
  
  
  
   
  {\bf Problem:} What is going on for the dynamics and minimal sets contained in the elliptic and parabolic regions for $C^r$-foliations, $r \geq 1$.
  
  
  \pause
   

  {\bf Problem:} Suppose that $\fM$ is a weak solenoid, homeomorphic to a minimal set in a $C^2$-foliation. Must the fibers of the solenoid be virtually abelian? In particular, can inverse limits of nilpotent, non-abelian countable groups be realized as minimal sets of $C^r$-foliations, $r \geq 2$?
  
  \pause
   

  {\bf Problem:} Suppose that $\fM$ is   homeomorphic to a minimal set in a $C^2$-foliation, and the transversals are $k$-connected, for $0 \leq k < q$. Are there examples besides Sierpinski manifolds, suspensions of minimal actions on Cantor groups, and various products of these?
   Is there a possible structure theory ?
  
  
  



    \vfill   

}


 


 

 
 

        
    
 
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