% S�minaire G�om�trie et applications
% Institut de Recherche Math�matique Avanc�e, UMR 7501
% http://www-irma.u-strasbg.fr/article286.html
% March 31, 2014
% slides for talk "Foliation dynamics, shape and classification"
 


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\date[March 31, 2014]


  
\title[Shape and Classification]{Foliation dynamics, shape and classification}


\author{Steve Hurder}

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



\begin{document}

\frame{\titlepage} % # 1


 
  \section{Intro}
 
 

\frame % 1
{
  \frametitle{}


{\bf Theorem:} [Denjoy, 1932] There exist a  $C^1$-foliation $\F$ of codimension-1 with an exceptional minimal set $\fM$ with no fixed-points for its holonomy pseudogroup, 
and $\fM$ has the \emph{shape} of wedge of two circles.


 
\medskip

{\bf Theorem:} [Sacksteder, 1965]  An exceptional minimal set $\fM$ for a $C^2$-foliation of codimension-1  always has   hyperbolic fixed-points for its holonomy pseudogroup.

 
  \medskip

 

{\bf Theorem:} [Rosenberg \& Roussarie, 1970] There exists $C^{\infty}$-foliations of codimension-1 with exceptional minimal sets.

\medskip
 

{\bf Theorem:} [Williams, 1974] Let $\Lambda \subset M$ be an expanding attractor for a  diffeomorphism $f \colon M \to M$, so that the \emph{stable foliation} of $f$ is $C^1$ on some open neighborhood of $\Lambda$. Then $f \colon \Lambda \to \Lambda$ is $C^0$-conjugate    to the shift map on a \emph{stationary} generalized  solenoid.



 
   \vfill   

}

 
  
\frame % 1
{
  \frametitle{}

  
  $\F$ is a $C^r$-foliation of codimension-$q$ of a compact manifold $M$.
  
   $\fM \subset M$ is exceptional minimal set - transversally is a Cantor set.




 The classification of exceptional minimal sets for $C^r$-foliations of codimension-$q$ remains an   open problem, for $q \geq 1$ and $r\geq 1$.

\medskip
% \pause

  {\it Problem: classify the possible}
  
\medskip

$\bullet$ ~     topological types of $\fM$, up to homeomorphism; 
 
 
$\bullet$ ~      holonomy pseudogroups for $\fM$, up to Morita equivalence; 

  

$\bullet$ ~      shapes of $\fM$, up to shape equivalence.

\bigskip


We consider this in a more general topological setting, and ask to what extent the Theorems above   admit generalizations.

   \vfill   

}

 
    \section{Matchbox manifolds}
 
\frame % 1
{
  \frametitle{}
 \medskip

 
 {\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 \emph{totally disconnected}, 
and for each $x \in \fM$, the transverse model space $\fX_x \subset \fX$ is a clopen  subset.

\begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{pix/blue_matches.png} 
\caption{Blue tips are points in Cantor set $\fX_x$}
\end{center}
\end{figure}
\vspace{-1cm} 
 
   
   \vfill   

}



 
\frame % 2
{
  \frametitle{}
 
  
    
\bigskip

 {\bf Definition:}     $\fM$ is   an \emph{$n$-dimensional matchbox manifold} if:

$\bullet$  $\fM$  is a continuum, a compact, connected metric space; 

$\bullet$ $\fM$  admits a covering by  foliated coordinate charts $\ds \cU = \left\{ \vp_{i} \colon U_i \to [-1,1]^n \times \fT_i \mid 1 \leq i \leq \kappa \right\}$; 

$\bullet$  each  $\fT_i$ is a     \emph{clopen} subset of  a \emph{totally disconnected}   space   $\fX$. 

\medskip

% \pause
  
Transversal    $\cT = \cT_1 \cup \cdots \cup \cT_{\kappa} \subset \fM$ defined by coordinate charts.


 Holonomy of $\F$ on $\cT$ $\Longrightarrow$  finitely generated pseudogroup $\cG_{\F}$ acting by \underline{homeomorphisms} defined on clopen subsets of $\fX$. 

 \medskip
  
      A ``smooth matchbox manifold'' $\fM$ is analogous to a compact manifold,  where the pseudogroup $\cG_{\F}$   is   the  ``fundamental group''. 


 

  

 
 \vfill   

}
 
     \section{Metrics}
  
\frame % 1
{
  \frametitle{}

 {\it Metric properties:} 
\medskip

   
 Two metrics $\dX$ and $\dXp$ are \emph{Lipschitz equivalent},  if they satisfy a Lipschitz condition  for some   $C \geq 1$,
\begin{equation}
C^{-1} \cdot \dX(x,y) ~ \leq ~ \dXp(x,y) ~ \leq ~ C \cdot \dX(x,y) \quad {\rm for ~ all} ~ x,y \in \fX
\end{equation}

  
\bigskip

{\bf Proposition:} Let   $\cGX$ be a      pseudogroup    defined by the restriction of the holonomy for a minimal set $\cM$ of a $C^1$-foliation  to a transversal $\fX = \cM \cap \cT$, then $\fX$ has a metric $\dX$  for which the action of  $\cGX$ is Lipschitz.
 
    \vfill   

}

    
 
\frame % 1
{
  \frametitle{}
 
For a Cantor set defined as the space of ends of a tree with bounded complexity, there is a natural ultrametric.   
 
  
 For a ``fractal'' $\fC$ defined by an \emph{Iterated Function System}, the tree associated to the pseudogroup action defines an ultrametric on $\fC$.
 
These classes of examples are said to be  \emph{locally homogeneous}.
 
 % \pause


\medskip
    {\bf Theorem:} [Hurder, 2013] 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 Lipschitz condition.    

% \pause
\medskip

Construct pseudogroup adding elements with ``ultra-contraction''.

This construction yields   pseudogroup analogs of the construction of open Riemannian manifolds which are not quasi-isometric to leaves of $C^1$-foliations by Attie \& Hurder.

    \vfill   

}

    
 
\frame % 1
{
  \frametitle{}
 
 \medskip
  
{\bf Definition:}
Let $\cGX$ and    $\cGY$ be a \emph{minimal}   pseudogroup actions on metric  Cantor sets $(\fX, \dX)$ and $(\fY, \dY)$, respectively. 

 

\bigskip

  $(\cGX,  \fX)$ and  $(\cGY,  \fY)$ are \emph{Morita equivalent}   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 $\cGY | W$.   

\bigskip

  $(\cGX, \fX, \dX)$ and  $(\cGY, \fY, \dY)$  are \emph{Lipschitz equivalent} if they are Morita equivalent, 
  and   the conjugation map $h$ can be chosen to be a Lipschitz equivalence of the metric Cantor sets.
   
 \bigskip
% \pause 
 
  {\bf Problem:} Classify the   minimal Lipschitz pseudogroup actions   on a Cantor set $\fX$, up to Lipschitz equivalence.
  

 \vfill   

}

  
\frame % 1
{
  \frametitle{}

  \medskip
 
 {\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 Lipschitz, when is there a Lipschitz equivalence to the pseudogroup   defined by an exceptional minimal set of a $C^1$ foliation?
 
  
   % \pause


   \medskip
   
   Lipschitz invariants of minimal Lipschitz pseudogroup actions:

$\bullet$~ Hausdorff dimension

$\bullet$~ Positive geometric entropy

   \medskip

{\bf Theorem:} [Lukina, 2014] The Ghys-Kenyon construction of a metric Cantor set  from the space of pointed subtrees of the Cayley graph of the free group $\mF_n$ has \emph{infinite} Hausdorff dimension, and admits a minimal Lipschitz pseudogroup action induced by $\mF_n$. 

 
\vfill   

}

 

     
\frame % 13
{

  \frametitle{}

Two dynamical properties of pseudogroup actions:

\medskip 

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

  
 
  

{\bf Definition:} $\fM$ is an $\e$-\emph{expansive matchbox manifold}  if it admits some covering by foliation charts, and  there exists $\epsilon > 0$,  
so that for all $w \ne w' \in  \cT$ with $\ds   d_{\cT}(w , w') < \epsilon$,    there exists  $g \in \cGF$  defined by the holonomy along some path, with $w,w' \in \mathrm{Dom}(g)$ such that   $\ds   d_{\cT}( g(w) , g(w')) \geq \epsilon$.

 

  \vfill

}


 

\frame % 1
{
  \frametitle{}

\medskip

 {\bf Examples:} 
 
 $\bullet$~ The classical $1$-dimensional solenoids are equicontinuous.   
 
  $\bullet$~ Denjoy minimal sets are expansive.

  $\bullet$~ The tiling space for an aperiodic tiling of $\mR^n$ with finite local complexity  has expansive pseudogroup dynamics.

\bigskip

Need property of matchbox manifolds  and their pseudogroup actions, which applies for both the usual examples, as well as the exceptional examples, which are not so exceptional.   

  \vfill

}



 
\section{Shape}


\frame % 1
{
  \frametitle{}
 
 
A \emph{continuum} is a compact connected metric space.

 
{\bf Definition:} [Borsuk, 1968] The \emph{shape} of a continuum $\Omega$ is the homotopy type of the inverse limit of the realizations of the   \v{C}ech complexes associated to finite open covers of  $\Omega$.  
  
 % \pause
 \medskip
  
  {\bf Definition':} The shape of a continuum $\Omega$ is defined by a collection of maps $\{q_{\ell} \colon \Omega  \to Y_{\ell} \mid \ell = 1,2,\ldots\}$ where each $Y_{\ell}$ is a finite simplicial complex, and each $q_{\ell}$ is a continuous onto map whose fibers have diameter bounded by $\e_{\ell}$ where $\e_{\ell} \to 0$. 
  
 % \pause
   \medskip

   {\bf Definition'':} [Alexandroff, 1928] Let $\cY$ be a   collection of finite simplicial complexes. Then the shape of $\Omega$ is $\cY$-like, if for all $\e > 0$, there exists  $Y_\e  \in \cY$ and a map $q_{\e} \colon \Omega \to Y_\e$ such that  the fibers of $q_{\e}$ have diameters uniformly less than $\e$. 
  
    
\vfill   

}

 
\frame % 1
{
  \frametitle{}
 
 
\medskip


 
 
 {\bf Definition:} A continuum $\Omega$ is $Y$-like if it is $\cY$-like for $\cY = \{Y\}$. We say that $\Omega$ has the shape of $Y$.
 
 \medskip
 
  {\bf Example:} A classical $1$-dimensional solenoid has the shape of  $\mS^1$.
  
 \medskip
  {\bf Example:} A Denjoy minimal set has the shape of  $\mS^1  \vee  \mS^1$.
  
 \medskip
 
  {\bf Theorem:} [Anderson \& Putnam, 1999] The tiling space for an aperiodic tiling of $\mR^n$ defined by a substitution rule has the shape of a branched $n$-manifold $Y$.
  
\vfill   

}

 
\frame % 1
{
  \frametitle{}
 
  
 \emph{Presentation}   is a collection $\cP = \{ p_{\ell+1} \colon Y_{\ell+1} \to Y_{\ell} \mid \ell \geq 0\}$, 
 
 $\bullet$  each $Y_{\ell}$ 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
 
  
 Associated to $\cP$ is the  \emph{generalized solenoid}
 $$\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon Y_{\ell +1} \to Y_{\ell}\} ~ \subset \prod_{\ell \geq 0} ~ Y_{\ell} 
$$
where  $\cS_{\cP}$ is given  the product topology.
 
 \medskip
 
$\cP$    is \emph{stationary} if  $Y_{\ell} = Y_0$ for all $\ell \geq 0$, and the bonding maps $p_{\ell} =p_1$ for all $\ell \geq 1$. 
 

  

\vfill   

}


\frame % 1
{
  \frametitle{}
 
     
 
 {\bf Theorem:} [Marde\v{s}i\'{c} and Segal, 1963] Let $\Omega$ be a continuum which is $\cY$-like, then there exists $Y_{\ell} \in \cY$ and continuous surjections $p_{\ell +1} \colon Y_{\ell +1} \to Y_{\ell}$ for $\ell \geq 1$, such that $\Omega$ is homeomorphic to a generalized solenoid with these maps defining the presentation. 
 
  
  \medskip
% \pause


  {\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 $m_{\ell+1} > 1$.
  
\medskip
  
    {\bf Theorem:} [McCord, 1965] A weak solenoid is a  {matchbox manifold}.
 
        {\bf Remark:} A generalized solenoid may be a matchbox manifold, such as for Williams solenoids which are stationary, and the Anderson--Putnam and G\"{a}hler--Sadun construction of finite approximations to tiling spaces. 


  

  \medskip
   
    
\vfill   

}

 
\frame % 1
{
  \frametitle{}

Let $\fM$ be a minimal matchbox manifold $\fM$. 

Consider the properties of $\fM$: 

\medskip

 

$\bullet$~ Morita   class of pseudogroup $\cGF$ acting on Cantor set $\fX$

\medskip

$\bullet$~ Dynamical properties of $\cGF$ acting on Cantor set $\fX$

\medskip

$\bullet$~ Shape of $\fM$

\medskip

$\bullet$~ $\fM$ is realized as exceptional minimal set of $C^1$-foliation

\medskip

 $\bullet$~ Homeomorphism type  of $\fM$

 \medskip
 
 {\bf Program:} Combine properties to obtain conclusions about others.


  
   
\vfill   

}

 
\section{Results}


\frame % 1
{
  \frametitle{}
  


{\bf Theorem:} [Hagopian, 1977] A matchbox manifold  $\fM$ is $\mS^1$-like if and only if it is homeomorphic to the inverse limit $\cS_{\cP}$ of a 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 $n_{\ell} > 1$. These spaces were introduced by     Vietoris [1927] and  van Dantzig [1930]. 

\medskip

 If all $n_{\ell} = p > 1$ are constant, then $\cS_{\cP}$ can be realized as a Smale-Williams hyperbolic attractor for a smooth map of $\mS^3$.
 
 
 \begin{figure}[H]
\begin{center}
\includegraphics[width=0.3\textwidth]{pix/solenoid-wiki.png} 
% \caption{Blue tips are points in Cantor set $\fX_x$}
\end{center}
\end{figure}
\vspace{-1cm} 


   
\vfill   

}


\frame % 1
{
  \frametitle{}
 A continuum $\Omega$ is \emph{homogeneous} if the group of homeomorphisms of $\Omega$  is \emph{point-transitive}.
  
 \medskip
 
{\bf Theorem:} [Clark \& Hurder, 2010] 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   $\fM$ has equicontinuous pseudogroup, and is moreover, homeomorphic to a McCord (normal) solenoid.
 
 \medskip
 
 This last result is a higher-dimensional version of the \emph{Bing Conjecture} for 1-dimensional matchbox manifolds. 
 
 The proof uses coding functions for the pseudogroup action on a transversal. Inspired by work of 
 Coornaert \&  Papadopoulos in   {\bf Symbolic dynamics and hyperbolic groups},  1993.

 
 \vfill   

}
  

\frame % 1
{
  \frametitle{}
   
   Let $\cM_n =\{ M \mid M ~ {\rm is ~  closed ~ oriented ~ manifold ~ of ~ dimension} ~  n\}$
 \medskip
 
{\bf Theorem:} [Clark, Hurder, Lukina, 2014] Let $\fM$ be a matchbox manifold which is  $\cM_n$-like.  Then $\cGF$ has equicontinuous dynamics, hence is homeomorphic to a weak solenoid. 

% \pause

 \bigskip
 
 {\bf Theorem:} [Clark, Hurder, Lukina, 2013] Let $\fM$ be a matchbox manifold which is  $\mT^n$-like.   Then $\fM$ is homeomorphic to an inverse limit space defined by a presentation  

  $$ \longrightarrow \mT^n  \stackrel{p_{\ell+1}}{\longrightarrow} \mT^n  \stackrel{p_{\ell}}{\longrightarrow} \cdots  \stackrel{p_{2}}{\longrightarrow}\mT^n   \stackrel{p_{1}}{\longrightarrow} \mT^n$$

 {\bf Corollary:} Let $\fM$ be a $\mT^n$-like minimal matchbox manifold, then $\fM$ is homeomorphic to a minimal set of a $C^1$-foliation. 
  \vfill   

}
  
\section{Classification}
  
\frame % 1
{
  \frametitle{}
 
Recall the  algebraic classification of $\mS^1$-like solenoids.
 
Two sequences of primes ${\cP} = (p_1,p_2,\ldots)$ and ${\cQ} = (q_1,q_2,\ldots)$ are equivalent, ${\cP} \sim {\cQ}$, if and only if one can delete a finite number of entries from each sequence so that every prime occurs the same number of times in deleted sequences.


\medskip
{\bf Example:} ${\cP} = (2,3,3,\ldots)$, ${\cQ} = (3,3,3,\ldots)$, ${\cS} = (2,3,2,3,2,3,\ldots)$. Then ${\cP} \sim {\cQ}$, and ${\cP} \nsim {\cS}$, and ${\cQ} \nsim {\cS}$.

\bigskip
 % \pause

 
{\bf Theorem:} [Bing, 1960] ${\cP} \sim {\cQ}$ $\Rightarrow$ $\cS_{\cP}$ is homeomorphic to $\cS_{\cQ}$.

\medskip
{\bf Theorem:} [McCord, 1965] ${\cP} \sim {\cQ}$ $\Leftarrow$ $\cS_{\cP}$ is homeomorphic to $\cS_{\cQ}$.
 
 \bigskip
 % \pause

{\bf Remark:} Results of   Hjorth  \&  Thomas imply   there is no \emph{algebraic} classification for weak solenoids, when   dimension $n > 1$.  
 
 
\vfill   

}
 
  
\frame % 1
{
  \frametitle{}
 
 A  manifold $Y$ is \emph{aspherical}, if it is connected and higher homotopy groups $\pi_n(X)$, $n \geq 2$ are trivial.

\medskip
Equivalently, $Y$ is \emph{aspherical} if it is connected and has a contractible universal cover.

\medskip
{\bf Example:} $\mathbb{T}^n$, $n \geq 1$, surfaces of genus $g \geq 1$ are aspherical.

\medskip
A sphere $\mathbb{S}^n$, $n \geq 2$, is not aspherical.


 
\vfill   

}
 
  
\frame % 1
{
  \frametitle{}
A collection $\mathcal{A}_B$ of closed manifolds is called \emph{Borel} if it satisfies the following.

\begin{enumerate}
\item Each $Y \in \mathcal{A}_B$ is aspherical,
\item Any closed manifold $X$ which is homotopy equivalent to a $Y \in \mathcal{A}_B$ is homeomorphic to $Y$,
\end{enumerate}

\medskip
A closed manifold $Y$ is \emph{strongly Borel} if the set of its finite covers forms a Borel collection.

\medskip
{\bf Example:} $\mathbb{T}^n$ is strongly Borel, as is  every closed nilmanifold.  

  $n=3$,    Thurston Geometrization Conjecture, proved by Perleman. 

  $n=4$,  Freedman's proof of Poincar\'e Conjecture. 

  $n\geq 5$,  Surgery Theory plus Poincar\'e Conjecture. 

\vfill   

}
 
  
\frame % 1
{
  \frametitle{}
{\bf Theorem:} [Clark, Hurder, Lukina, 2013]
Suppose that $\mathfrak{M}_1$ and $\mathfrak{M}_2$ are  $\mT^n$-like matchbox manifolds.   Then $\mathfrak{M}_1$ and $\mathfrak{M}_2$  are homeomorphic    if and only if $\cG_{\F_1}$ and $\cG_{\F_2}$ are Morita equivalent. 
 
\bigskip
% \pause

{\bf Theorem:} [Clark, Hurder, Lukina, 2013]
Suppose that $\mathfrak{M}_1$ and $\mathfrak{M}_2$ are  $Y$-like matchbox manifolds, where $Y$ is strongly Borel. Assume that each of $\mathfrak{M}_1$ and $\mathfrak{M}_2$ has a leaf which is simply connected. Then $\mathfrak{M}_1$ and $\mathfrak{M}_2$  are homeomorphic   if and only if $\cG_{\F_1}$ and $\cG_{\F_2}$ are Morita equivalent. 
 
\medskip

The proofs of these two results are in our most recent paper,

 \emph{Classifying matchbox manifolds}, {arXiv:1311.0226} 

and uses techniques from all previous works. 

\vfill   

}
 
 \section{Problems}
  
\frame % 1
{
  \frametitle{Some Problems}

 

 $\bullet$~   Develop effective invariants for  (Lipschitz)   equicontinuous  minimal pseudogroup actions on (metric) Cantor sets. 

 
\emph{This is work in progress with Jessica Dyer and Olga Lukina.}
\bigskip
   
$\bullet$~  Let   $Y$ be a closed   branched $n$-manifold.
Give a ``combinatorial'' structure theorem for $Y$-like matchbox manifolds.

 
\emph{The case where the $Y$-like matchbox manifold $\fM$ has leaves with non-trivial holonomy is not well-understood.}

\bigskip

$\bullet$~   Given a $Y$-like matchbox manifold $\fM$, when does there exists a (Lipschitz) embedding into a $C^r$-foliation, for some $r \geq 1$?

 
\emph{Extend the results of [Williams, 1974] and [Clark-Hurder, 2011].}


 
 
  
  
\vfill   

}
 
 
\frame % 1
{
  \frametitle{References}
 {\tiny   
{O.~Attie and S.~Hurder}, 
 {\it Manifolds which cannot be leaves of foliations}, 
 {\bf Topology}, 35(2):335--353, 1996.


{A.~Clark, R.~Fokkink and O.~Lukina},
 {\it The Schreier continuum and ends},
 {\bf Houston J. Math.}  to appear, {arXiv:1007.0746v1}.

{A.~Clark and S.~Hurder},
 {\it Embedding solenoids in foliations},
 {\bf Topology Appl.}, 158:1249--1270, 2011; {arXiv:0812.1085}.


{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Trans. Amer. Math. Soc.}, 365:3151--3191, 2013;  {arXiv:1006.5482}.


{A.~Clark, S.~Hurder and O.~Lukina},
 {\it Voronoi tessellations for matchbox manifolds},
 {\bf Topology Proceedings}, 41:167--259, 2013;  {arXiv:1107.1910}.


{A.~Clark, S.~Hurder and O.~Lukina},
 {\it Shape of matchbox manifolds},
accepted to {\bf Indigaciones Mathematicae}; {arXiv:1308.3535}.
 

{A.~Clark, S.~Hurder and O.~Lukina},
 {\it Classifying matchbox manifolds},
 {preprint}, October 2013; {arXiv:1311.0226}.

{A.~Clark, S.~Hurder and O.~Lukina},
 {\it $Y$-like   matchbox manifolds}, {in preparation}, 2014.


{S.~Hurder},
 {\it Lectures on Foliation Dynamics: Barcelona 2010},
 {\bf Foliations: Dynamics, Geometry and Topology}, Advanced Courses in Mathematics CRM Barcelona,  to appear  2014; {arXiv:1104.4852}.

 
{S.~Hurder},
{\it Lipschitz matchbox manifolds},  {preprint}, September 2013; {arXiv:1309.1512}.

{S.~Hurder and O.~Lukina},
 {\it Entropy and dimension for graph matchbox manifolds}, {in preparation}, 2014.

 
{O.~Lukina},
 {\it Hierarchy of graph matchbox manifolds},
 {\bf Topology Appl.}, 159:3461--3485, 2012;  {arXiv:1107.5303v3}.

{O.~Lukina},
 {\it Hausdorff dimension of graph matchbox manifolds},  {in preparation}, 2014.
 
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