% Geometry and Foliations 2013
% September 9--14, 2013
% slides for talk "Minimal sets for foliations" on September 11
% slides completed September 11, 2013



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\date[September 11, 2013]


  
\title[Minimal Sets]{Minimal sets for foliations}


\author{Steve Hurder}

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



\begin{document}

\frame{\titlepage} % # 1



 
  \section{Classification}
 
  
  \frame % 1
{
  \frametitle{}
 
   
\medskip

 
Given  \emph{one} derivative, 

\begin{center}
\emph{What can you do with one derivative?}
\end{center}

Topological dynamics for flows (and foliations) is a very wild world.

But along with a derivative, sometimes arrives order.

Model example for this:  attractors for \emph{Axiom A} diffeomorphisms, 
and Williams' results on their topological structure.

\begin{center}
\emph{{\color{DarkOrange4} Goal: Classify (exceptional)  minimal sets for $C^1$-foliations.}}
\end{center}

 

Study the ``simplest'' cases.

  
 \medskip
 

   \vfill   

}

\frame % 1
{
  \frametitle{}
 
Themes in  classification of foliations:  
 \begin{center}
 {\LARGE  $2 \leftrightarrow 1 \leftrightarrow 0 \leftrightarrow 1 \leftrightarrow 2$}
 \end{center}

 
 $\bullet$ ``$2$'' ---   $C^2$-invariants, such as characteristic classes
 
 $\bullet$ ``$1$'' ---    $C^1$-invariants, such as entropy \& Lyapunov spectrum 
 
 $\bullet$ ``$0$'' ---    $C^0$-invariants,  topological dynamics of foliations
 
 $\bullet$ ``$1$'' ---    Lipshitz invariants,  tempered cocycles, measure properties 

$\bullet$ ``$2$'' ---   Classification of cycles in $B\G_q^2$

\medskip

 
 \medskip
 
 Discuss some aspects about transitioning from ``0'' to ``1''.  


   \vfill   

}

  
\frame % 1
{
  \frametitle{}
 
  Classification of foliations in the 1970's and 80's
 \medskip
  
 $\bullet$ Haefliger, Thurston $\Longrightarrow$ 
 
 \qquad \emph{Classifying spaces $B\Gamma_q^r$ for $r \geq 1$}
 
 \smallskip
 
  $\bullet$ Bott--Haefliger, Gelfand--Fuks, Kamber--Tondeur $\Longrightarrow$ 
  
 \qquad \emph{Secondary classes and how to calculate them}
 
 \smallskip
 
  $\bullet$ Mizutani, Morita, Tsuboi, Duminy, Sergiescu $\Longrightarrow$ 
  
\qquad  \emph{Decompose foliations, Godbillon-Vey class of  constituents}
   
 \smallskip
 
  $\bullet$ Ghys,  Hector, Langevin, Moussu,  Rosenberg, Roussarie, \\
  Cantwell, Conlon, Plante,  Sullivan, Walczak, Williams, \\
  Inaba,  Matsumoto,  Mizutani,  Nishimori,  Tsuchiya $\Longrightarrow$ 
  
\qquad  \emph{Foliations as dynamical systems}
   
 \medskip
 
    

   \vfill   

}


  \section{Topological Dynamics}
 
 
\frame % 1
{
  \frametitle{}

A minimal set $\cZ \subset M$ is \emph{exceptional} if the intersection   $\cZ \cap \cT$ is totally disconnected for every transversal $\cT$ to $\F$.

Leaves of $\F$ are recurrent in a minimal set $\Longrightarrow$    $\cZ \cap \cT$ is Cantor set.

Holonomy of $\F$ along paths in $\cZ$ defines  pseudogroup   on  $\cZ \cap \cT$.

\medskip

{\color{sienna4}  {\bf Problem:} Study the properties of \emph{minimal pseudogroup actions} on Cantor sets.
Look for properties characterizing their ``$C^1$-ness''. }

Or, a little more precisely: 

{\color{sienna4}  {\bf Problem:}  Given a   pseudogroup $\cGX$ acting on a Cantor set $\fX$,   determine if it came from a minimal set in a $C^r$-foliation, $r \geq 1$.}

\medskip

The action must be \emph{minimal} \&   \emph{compactly generated}. 

    \vfill   

}

  
\frame % 1
{
  \frametitle{}
 
 
{\bf Definition:}
 A pseudogroup $\cGX$ acting on a Cantor set $\fX$ is \emph{compactly generated}, if there exists two collections of \emph{clopen} subsets  $\{U_1, \ldots, U_k\}$ and $\{V_1, \ldots, V_k\}$ of $\fX$ and homeomorphisms $\{h_i \colon U_i \to V_i \mid 1 \leq i \leq k\}$ which generate all elements of $\cGX$.
 
 
 \medskip
 
  $\cGX^*$  is defined to be the  compositions of the  generators $\{h_i \colon U_i \to V_i \mid 1 \leq i \leq k\}$ and their inverses, on the maximal domains for which the composition is defined. 
 
 \medskip
 
 
There must be given   a well-defined Lipshitz class of metrics on $\fX$.

 
   \vfill   

}


\frame % 1
{
  \frametitle{}
 
  
 $\dX$ and $\dXp$ are \emph{bi-Lipshitz equivalent},  if they satisfy a Lipshitz condition  for some   $C \geq 1$,
$$
C^{-1} \cdot \dX(x,y) ~ \leq ~ \dXp(x,y) ~ \leq ~ C \cdot \dX(x,y) \quad {\rm for ~ all} ~ x,y \in \fX
$$
  \emph{Lipshitz geometry} of the pair $(\fX, \dX)$  investigates the geometric properties common to all metrics in the Lipshitz class of the given metric $\dX$.  Example: 
Hausdorff dimension.
\medskip

Cantor set has many metrics,  need not be ``locally homogeneous,''  or satisfy a ``doubling property'' of Assouad,   which is necessary if  there is an embedding in some Euclidean space.  
 \vfill   

}



\frame % 1
{
  \frametitle{}
 
  

 
{\bf Definition:}
 The action of a compactly-supported pseudogroup  $\cGX$  is \emph{Lipshitz}  with respect to $d_{\fX}$ if there exists $C \geq 1$ such that for each   $1 \leq i \leq k$ then for all   $w, w' \in U_i = \Dom(h_{i})$ we have
$$
C^{-1} \cdot d_{\fX}(w,w') \leq d_{\fX}(h_{i}(w), h_{i}(w')) \leq C \cdot d_{\fX}(w,w') ~ .
$$
We then say that $\cGX^*$  is $C$-Lipshitz with respect to $\dX$. 
 
 

\bigskip

{\color{DarkGoldenrod} 
{\bf Proposition:} Let $\cGX$ be a compactly-generated pseudogroup acting on a Cantor set $\fX$. If $\cGX$ is the defined by the restriction of the holonomy of a $C^1$-foliation to a transversal $\cZ \cap \cT$, then $\fX$ admits a metric $\dX$ such that the action is Lipshitz.
}


\vfill   

}

 
\frame % 1
{
  \frametitle{}
 
 
 
Let $\cGX$ be a minimal pseudogroup acting on a Cantor space $\fX$, and let $V \subset \fX$ be a clopen subset.   $\cGX | V$ is defined as the restrictions of all maps in $\cGX$ with domain  and range in $V$.   

{\bf Definition:}
Let $\cGX$ be a minimal pseudogroup action on the Cantor set $\fX$  via Lipshitz homeomorphisms   with respect to the metric $\dX$. 
Likewise, let   $\cGY$ be a minimal pseudogroup action on the Cantor set $\fY$    via Lipshitz homeomorphisms with respect to the  metric $\dY$.
Then 
\begin{enumerate}
\item $(\cGX, \fX, \dX)$ is \emph{Morita equivalent} to $(\cGY, \fY, \dY)$ if there exist clopen subsets $V \subset \fX$ and $W \subset \fY$, and a   homeomorphism $h \colon V \to W$ which conjugates $\cGX | V$ to $\cGY | W$.\\
\item $(\cGX, \fX, \dX)$ is \emph{Lipshitz equivalent} to $(\cGY, \fY, \dY)$ if the conjugation $h$ is Lipshitz.
\end{enumerate}

 \vfill   

}

  
\frame % 1
{
  \frametitle{}
Two very interesting problems:
\medskip
 
{\color{sienna4}  {\bf Problem:} Given a compactly-generated  pseudogroup $\cGX$ acting minimally on a Cantor set $\fX$, and suppose there exists a metric $\dX$ on $\fX$ such that the generators are Lipshitz, when is there a Lipshitz equivalence to the pseudogroup   defined by an exceptional minimal set of a $C^1$ foliation?}
 
  
 \bigskip
 
 
{\color{sienna4}   {\bf Problem:} Classify the compactly-generated Lipshitz pseudogroups acting minimally on a Cantor set $\fX$, up to Lipshitz equivalence.}
  
 
  \bigskip
 
\vfill   

}

  
\frame % 1
{
  \frametitle{}
 
 
   
Classify means, for example, that we are looking for \emph{effective invariants} that distinguish the actions.

\bigskip

There are several known ``types'' of standard examples, of compactly-generated  pseudogroups $\cGX$ acting minimally on a Cantor set $\fX$, which we recall. But first,  there are some ``bad characters'' that the Lipshitz condition rules out.
 

\vfill   

}



  \section{Non-embedding}
 
\frame % 1
{
  \frametitle{}
  \medskip
 
{\color{DarkGoldenrod} 
    {\bf Theorem:} There exist  compactly-generated   pseudogroups $\cGX$ acting minimally on a Cantor set $\fX$, such that there is no metric  on $\fX$ for which  the generators of $\cGX$ satisfy a Lipshitz condition.    
} 
 \medskip
 
This is related to the construction of complete Riemannian manifolds which cannot   be realized as the leaf of a $C^1$-foliation.
 
 \medskip
 
Sketch of   proof -- for details 
see {\it ``Lipshitz matchbox manifolds''}, arXiv:1309.1512.
\vfill   

}

 
\frame % 1
{
  \frametitle{}
 
     
Begin by  constructing the model for the Cantor set $\fX$, with   $\dX$. 
    
Let   $G_{\ell} = \mZ / (2^{\ell} \, \mZ)$ be the cyclic group of order $2^{\ell}$.

 Let  $p_{\ell+1} \colon G_{\ell+1}  \to G_{\ell}$ be the natural quotient map. Set:
$$
\fX ~ = ~   \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon G_{\ell +1} \to G_{\ell}\} ~ \subset \prod_{\ell \geq 1} ~ \mZ / (2^{\ell} \, \mZ) ~ .
$$
Metric on $\fX$:    $\ovx = (x_1, x_2, x_3, \ldots )$ and $\ovy = (y_1, y_2, y_3, \ldots )$, then 
$$
 \dX\left( \ovx , \ovy \right) = \sum_{\ell =1}^{\infty} ~ 3^{-\ell} \delta(x_{\ell} , y_{\ell}), 
$$
where $\delta(x_{\ell} , y_{\ell}) = 0$ if $x_{\ell} = y_{\ell}$, and is equal to $1$ otherwise.    

 \vfill   

}

 
\frame % 1
{
  \frametitle{}
 

Define  action $A \colon \mZ \times \fX \to \fX$, where $\mZ$ acts on each factor $\mZ / (2^{\ell} \, \mZ) $ by translation. 

 Action of $A$ on $\mZ$ on $\fX$ is minimal.


 Let $\sigma \colon \fX \to \fX$ be the shift map,  $\ds  \sigma(x_1,x_2, \ldots ) = (x_2, x_3, \ldots )$.  
 
 $\sigma$ is a $2-1$ map, and so  is not invertible.   
 
 $\sigma$ is a $3$-times expanding map. 
 
Partition   $\fX$ into clopen subsets, for $i=0,1$, 
  $$U_1(i) = \{(i, x_2, x_3, \ldots ) \mid 0 \leq x_{j} < 2^{j} ~, ~ p_{j+1}(x_{j +1}) = x_{j} ~, ~  j > 1\} . $$
$\diamX(U_1(0)) = \diamX(U_1(1)) = \dX(U_1(0), U_1(1)) = 1/3$.

Inverse map    $\tau_i = \sigma_i^{-1} \colon \fX \to U_1(i)$ given by the usual formula for the section, $\tau_i(x_1, x_2, x_3, \ldots ) = (i, x_1, x_2, x_3, \ldots )$. 
  
\vfill   

}


\frame % 1
{
  \frametitle{}
 
 For $\ovx \in \fX$,  set $\ovx_{\ell} =  (x_1, \ldots , x_{\ell})$. 
 
 For $\ell \geq 1$, define the clopen neighborhood of $\ovx$, 
\begin{eqnarray*}
U_{\ell}(\ovx) & = &  \{(x_1, \ldots , x_{\ell},\xi_{\ell+1}, \xi_{\ell+2}, \ldots ) \\
& & \mid 0 \leq \xi_{j} < 2^{j} ~, ~ p_{j+1}(\xi_{j +1}) = \xi_{j} ~, ~  j >  \ell\} .
\end{eqnarray*}
The restriction $\sigma^{\ell} \colon U_{\ell}(\ovx) \to \fX$ is 
$1-1$ and onto,
 $3^{\ell}$-expansive.

  $\diamX(U_{\ell}(\ovx)) = 3^{-\ell}/2$. 

So far, this is just the standard shift model.

A standard example of a Cantor minimal action by affine group.

 
\vfill   

}


\frame % 1
{
  \frametitle{}
 
 
The key point:   construct hypercontraction $\vp  \colon \mZ \times \fX \to \fX$.  

Choose two distinct points $\ovy, \ovz \in \fX$, and choose a sequence $\{\ovx_{k} \mid -\infty < k < \infty\} \subset \fX - \{\ovy,\ovz\}$ of distinct points with $\ds \lim_{k \to \infty}  \ovx_{k} = \ovy$ and  $\ds \lim_{k \to -\infty}  \ovx_{k} = \ovz$.


Choose disjoint clopen neighborhoods $V_{k} \subset \fX$ of the points $\ovx_k$ recursively. 

 $\diamX(V_k) = \diamX(V_{-k})   < \rho_k/(3 \, \ell_k!)$
 
 $\rho_k$ is distance between all previous choices.

Homeomorphism $\vp \colon \fX \to \fX$ such that for all $-\infty < k < \infty$, 

the restriction $\vp_{k} \colon V_{k} \to V_{k +1}$ is a homeomorphism onto, and 

 $\vp$ is defined to be  the identity on the complement of the union  $\ds V = \cup \{ V_{k} \mid -\infty < k < \infty \}$. 
 
 The map $\vp$ is a homeomorphism. 
\vfill   

}


\frame % 1
{
  \frametitle{}
 
 Let $\cGX = \langle A, \tau_1, \tau_2, \vp \rangle$ be   pseudogroup they generate. 

\medskip

{\bf Claim:}
There does    not exists a metric $\dXp$ on $\fX$ such that  the generators $\{A, \tau_1, \tau_2, \vp\}$ of $\cGX$ satisfy a Lipshitz condition.

{\it Proof:} If such a metric $\dXp$ exists, then some power of the contractions $\tau_i$ are  contractions for the new metric $\dXp$.

But then the Lipshitz condition on $\vp$ becomes impossible.


 \bigskip
   
What is going on? 


This is a ``toy problem'' for a broader geometric problem.

Return to minimal sets, and construct  a graph which cannot be embedded quasi-isometrically in a $C^1$-foliation.
 

\vfill   

}


 \section{Matchbox manifolds}

\frame % 1
{
  \frametitle{}
 
 {\bf Definition:} A \emph{matchbox manifold} is a continuum with the structure of a
smooth foliated space $\fM$, such that  the transverse model space $\fX$ is totally disconnected, and for each $x \in \fM$, the transverse model space $\fX_x \subset \fX$ is a clopen  subset, hence is homeomorphic to a Cantor set.

 All matchbox manifolds  are assumed to be smooth with a given leafwise Riemannian metric. 
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{blue_matches.png} 
\caption{Blue tips are points in Cantor set $\fX_x$}
\end{center}
\end{figure}
 
\vfill   

}


\frame % 1
{
  \frametitle{Examples}
 
$\bullet$  Exceptional minimal set  for   foliation $\F$ on $M$, with   metric induced from Riemannian metric  on $M$. \\
  
 \medskip
 
 $\bullet$ Given a   repetitive, aperiodic tiling of the Euclidean space $\mR^n$ with finite local complexity, the associated tiling space $\Omega$ is   the   closure of the set of translations by $\mR^n$ of the given tiling,   in an appropriate  {Gromov-Hausdorff topology} on the  space of tilings. Metric is induced from ``tiling matching'', so a type of ultra-metric. 
 
   
\vfill   

}


\frame % 1
{
  \frametitle{}
 
  $\bullet$   Ghys-Kenyon construction of laminations, for Cayley graphs of finitely-generated groups, and for foliated Cayley graphs. 

\medskip
{\small
[Ghys, 1999] {\it Laminations par surfaces de Riemann.}

[Blanc, 2001] {\it Propri\'et\'es g\'en\'eriques des laminations.}

[Lozano-Rojo, 2006] {\it {C}ayley foliated space of a graphed pseudogroup.}


[Lukina, 2012] {\it Hierarchy of graph matchbox manifolds.}

[Lozano-Rojo and Lukina, 2013] {\it Suspensions of {B}ernoulli shifts.}
 
}

\medskip

{\bf Remark:} $L \subset \cZ$ a leaf of minimal set for foliation, $\G_L$ the graph of the pseudogroup restricted to $L$, then there is an associated graph matchbox manifold $\fM(\G_L)$ which captures the dynamics of $L$ in $\cZ$, and is transversally Cantor set. Plus, can keep all leaves as sticks, no need to fatten them up, so really does look like   matchboxes. 
 \begin{center}
{\it Matchbox dynamics captures  dynamics of foliation minimal sets.}
\end{center}

\vfill   

}

\section{Solenoids}

\frame % 1
{
  \frametitle{}
 
 Example   above,     tilings, trees, etc:
$$  {\rm expansive} ~  \longleftrightarrow ~ {\rm shifts ~ on ~ trees} ~  \longleftrightarrow ~ {\rm parabolic~ groups}$$  
  Need to also  also look for rotational part of geometry, as in the decomposition of a semi-simple Lie group
  $G = P \times K$:
$$  {\rm equicontinuous} ~  \longleftrightarrow ~ {\rm Cantor ~ rotations} ~  \longleftrightarrow ~ {\rm compact ~ groups}$$  

 
\medskip

The \emph{weak solenoids} correspond to the Cantor rotations, or maximal compact factors in Bruhat decomposition.

\vfill   

}


\frame % 1
{
  \frametitle{}
 
  
 \emph{Presentation}   is a collection $\cP = \{ p_{\ell+1} \colon M_{\ell+1} \to M_{\ell} \mid \ell \geq 0\}$, 
 
 $\bullet$  each $M_{\ell}$ is a connected compact simplicial complex, dimension $n$,
 
 $\bullet$ each ``bonding map'' $p_{\ell +1}$  is a proper surjective map of   simplicial complexes with discrete fibers.
 
 \medskip
 
 The \emph{generalized solenoid}
 $$\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} 
$$
 $\cS_{\cP}$ is given  the product topology.
 
Presentation    is \emph{stationary} if  $M_{\ell} = M_0$ for all $\ell \geq 0$, and the bonding maps $p_{\ell} =p_1$ for all $\ell \geq 1$. 


\vfill   

}


\frame % 1
{
  \frametitle{}
 {\bf Definition:}  $\cS_{\cP}$  is  a \emph{weak solenoid} if  for each $\ell \geq 0$,  
  $M_{\ell}$ is a compact manifold without boundary,  and  the map $p_{\ell+1}$ is a proper covering map of degree $m_{\ell+1} > 1$.
  
   Classic example: Vietoris solenoid, defined by tower of coverings:
  
  $$ \longrightarrow \mS^1  \stackrel{n_{\ell+1}}{\longrightarrow} \mS^1    \stackrel{n_{\ell}}{\longrightarrow} \cdots  \stackrel{n_{2}}{\longrightarrow}\mS^1   \stackrel{n_{1}}{\longrightarrow} \mS^1$$
where all covering degrees $n_{\ell} > 1$.
  
  Weak solenoids are the most general form of this construction.
 
 \medskip
  
{\color{DarkGoldenrod} 
    {\bf Proposition:} A weak solenoid is a matchbox manifold.
  }
  
{\bf Remark:} A generalized solenoid may be a matchbox manifold, such as for Williams solenoids, and Anderson-Putnam construction of finite approximations to tiling spaces. 
Or, it may not be. 

  \medskip
   
    
\vfill   

}


\frame % 1
{
  \frametitle{}
  
 
 Associated to a presentation: sequence of proper surjective maps 
$$q_{\ell} = p_{1} \circ \cdots \circ p_{\ell -1} \circ p_{\ell} \colon M_{\ell} \to M_0 .$$
and a fibration map $\Pi_{\ell} \colon \cS_{\cP}  \to M_{\ell}$ obtained by projection onto the $\ell$-th factor. 
$\Pi_0 = \Pi_{\ell} \circ q_{\ell} \colon \cS_{\cP} \to M_0$ for all $\ell \geq 1$.

 
Choice of a basepoint $x \in \cS_{\cP}$ gives basepoints  $x_{\ell} = \Pi_{\ell}(x) \in M_{\ell}$.

$\cH_{\ell} = image(q_{\ell} \colon  \pi_1(M_{\ell}, x_{\ell})) \subset \cH_0$.

\medskip

 
 {\bf Definition:}  $\cS_{\cP}$  is  a \emph{McCord (or normal) solenoid} if  for each $\ell \geq 1$,  $\cH_{\ell}$ is a normal subgroup of $\cH_0$.
 
 
 
 \medskip
 
 $\cP$ normal presentation $\Longrightarrow$    fiber $\fX_x = (\Pi_0)^{-1}(x)$ of $\Pi_0 \colon \cS_{\cP} \to M_0$ is a Cantor group, and monodromy action of $\cH_0$   on $\fX_x$ is minimal. 
 
\vfill   

}


\frame % 1
{
  \frametitle{}
 A continuum $\Omega$ is \emph{homogeneous} if its group of homeomorphisms is point-transitive.
 Alex Clark and I proved the following   in 2010. 
 
 \medskip
 
{\color{DarkGoldenrod} 
{\bf Theorem:} Let $\fM$ be a matchbox manifold. 

$\bullet$ If $\fM$ has equicontinuous pseudogroup, then  $\fM$ is homeomorphic to a weak solenoid   as foliated spaces.

$\bullet$ If  $\fM$ is homogeneous, then   $\fM$ is homeomorphic to a McCord solenoid   as foliated spaces.
 }
\medskip

This looks almost like the Molino Theorem for TP foliations! 

Though one point was troubling...
\vfill   

}

\frame % 1
{
  \frametitle{}
  Solenoids have many possible Lipshitz classes of metrics. 
  
  For a weak solenoid, choose a metric $d_{\ell}$ on each $X_{\ell}$. 

 Choose a    series $\{a_{\ell} \mid a_{\ell} > 0\}$ with total sum  $< \infty$.
 
 Define a metric on $\fX_x$ by setting, for $u,v \in \fX_x$ so 
 
$u = (x_0, u_1, u_2, \ldots)$ and $v = (x_0, v_1, v_2, \ldots)$, 
$$
\dX(u,v) = a_1 d_1(u_1, v_1) + a_2 d_1(u_2 , v_2) + \cdots 
$$

{\color{sienna4} 
{\bf Problem:} What is the classification of weak (or McCord) solenoids, up to Lipshitz equivalence?
}

\medskip

This problem seems to be wide open.

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}

 
 
\frame % 1
{
  \frametitle{}
 
 We give a simple example, in the case of Vietoris solenoids. 
 
 \medskip
 
  Let   $m_{\ell}$ be the covering degrees for a presentation $\cP$ with base $M_0 = \mS^1$ ,  given by $m_{\ell} = 2$ for $\ell$ odd, and $m_{\ell} = 3$ for $\ell$ even. 
  
 \medskip
 
 Let   $n_{\ell}$ be the covering degrees for a presentation $\cQ$ with base $M_0 = \mS^1$ ,  given by  $\{n_1, n_2, n_3, \ldots\} = \{2,3,2,2,3,2,2,2,2,3, \ldots\}$. 
 The $\ell$-th cover of degree $3$ is followed by $2^{\ell}$ covers of degree $2$. 
 
 \medskip
Sequences are   equivalent for   \emph{Baer classification} of solenoids, 

$\Longrightarrow$  $\cS_{\cP} \cong \cS_{\cQ}$.

\medskip
But for the metrics they define, the solenoids $\cS_{\cP}$ and $\cS_{\cQ}$   are not Lipshitz equivalent as matchbox manifolds.

\vfill   

}

 
\frame % 1
{
  \frametitle{}
Recent result by Alex Clark, Olga Lukina and myself.
 
 \medskip
 
 {\color{DarkGoldenrod} 
 {\bf Theorem:}  
Let $\fM$ be a minimal  matchbox manifold without holonomy. Then there exists a presentation $\cP$ by  simplicial maps   between compact branched manifolds   such that $\fM$ is homeomorphic to $\cS_{\cP}$ as foliated spaces.  
 }
 
 \medskip
 
 This implies a sort of generalization of the Denjoy/Sacksteder:
 
  

{\color{DarkGoldenrod}  
 {\bf Corollary:} Let $\fM$ be an exceptional  minimal  set for a $C^1$-foliation $\F$ of a compact manifold $M$.  If all leaves of $\F | \fM$ are simply connected, then  there is a  Lipshitz homeomorphism of $\fM$ with the inverse  limit space $\cS_{\cP}$ defined by   a   presentation $\cP$, given by  simplicial maps   between compact branched manifolds.
}

\medskip

{\color{sienna4} 
{\bf Problem:} How to ``classify''  $C^1$-minimal sets which are inverse limit spaces of branched manifolds. 
}

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}
 
 \section{Conclusion}
 
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  Amidst the vast wasteland formed by the class of all minimal Cantor actions by compactly generated-pseudogroups:
  
  \medskip
  
$\bullet$   Study the \emph{Lipshitz subclass}, and its classification by invariants such as \emph{Bratteli diagrams} and o\emph{rdered K-Theory}, \emph{dimension properties}, and possibly other invariants such as secondary classes associated to  their embeddings into $C^1$-foliations, which reflect their ``inner $C^1$-ness''.

\medskip
  
 $\bullet$  Study  the  \emph{Zygmund subclass}, such as $C^2$-embedded normal solenoids, and   their geometric and cohomological invariants, such as secondary classes associated to  their embeddings into $C^2$-foliations, which reflect their ``inner  $C^2$-ness''.
  
   
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}

  
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 \begin{center}
 {\Large Thank you for your attention!}
 
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