% 29th Summer Conference on Topology and its Applications
% July 23-26, 2014
% College of Staten Island, CUNY
% Staten Island, NY, USA
% http://csivc.csi.cuny.edu/SumTopo2014/files/

% Title: Foliation dynamics, shape and classification
% Talk 4 - Embeddings - [July 26, 2014]

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\date[July 26, 2014]


  
\title[Foliation dynamics, shape and classifications]{Foliation dynamics, shape and classification\\IV. Embeddings}


\author{Steve Hurder}

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



\begin{document}

\frame{\titlepage} % # 1


  \section{Introduction}
 
 
  \frame % 1
{
  \frametitle{}
 
 Fix some $r > 0$, usually $r =1$ or $r =2$.
 \bigskip
 
{\bf Problem:} Which matchbox manifolds embed in $C^r$-foliations?

\medskip


{\bf Problem:} Given a finitely generated pseudogroup $\cGX$ modeled on  a Cantor set $\fX$, with generating set
 $$\cGX^0 \equiv \{h_i \colon U_i \to V_i \mid 1 \leq i \leq k\},$$
 when is there an embedding  $\tau \colon \fX \to M$ into a manifold $M$, and smooth $C^r$-diffeomorphisms $f_i \colon N_i \to M_i$ defined on open neighborhoods of the images of the domains and ranges of the generators $h_i$, for $1 \leq i \leq k$, whose restrictions agree with the $h_i$? 
 
 
  \medskip

   \vfill   

}

 
  \frame % 1
{
  \frametitle{}

Recall that a Vietoris solenoid $\cS_{\cP}$ is defined up to homeomorphism by the collection of exponents $P = \{m_1,m_2, \ldots \}$ with $m_{\ell} \geq 2$, the covering multiplicities.

\bigskip

{\bf Theorem:}[Markus \&  Meyer, 1980]  Let $M$ be a compact symplectic manifold of dimension $2m \geq 4$, and let an infinite collection $P$ be fixed.  Then there exists a generic set of flows in the space of Hamiltonian dynamical systems on $M$, so that each flow in this set contains a minimal set homeomorphic to $\cS_{\cP}$. 

\medskip

The proof is based on the   Smale embedding trick in [Smale, 1967]
     
   \vfill   

}

 
  \frame % 1
{
  \frametitle{}


 
   
 \bigskip
 
  \begin{center}
\includegraphics[width=1.0\textwidth]{pix/sw_attractor.png}
\end{center}

     
   \vfill   

}


  \frame % 1
{
  \frametitle{}

 
 The illustration above is for the case where all covering degrees are the same,   $n_{\ell} = p^{\ell}$ for integer $p > 1$, and the resulting solenoid is an attractor for an Axiom A diffeomorphism which is expanding on the leaves of $\cS_{\cP}$ by construction. 
 
 
A variation on this idea was used in [Williams, 1967], [Williams, 1974] to show that a generalized solenoid defined by a stable presentation, 
    $$\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ \sigma \colon M \to M\}  $$
where $M$ is a branched $n$ manifold, and the bonding map $\sigma \colon M \to M$ is a smooth uniformly expanding surjection, admits an embedding into a foliation $\F$ of an open set $U \subset \mR^m$ for some $m > n$. Moreover, $\cS_{\cP}$ is again an attractor for an Axiom A diffeomorphism, which is expanding on the leaves of $\F$.

\medskip

 {\bf Question:} Is there a similar embedding result  for all generalized solenoids?

   
   \vfill   

}


  \frame % 1
{
  \frametitle{}


 

The works by [Gambaudo, Tresser, 1994], [Gambaudo, Sullivan, Tresser, 1994] in the 1990's studied the dynamical properties of $C^r$-diffeomorphisms with solenoidal attractors. 
\medskip

The thesis of [Brown, 2010] studied  solenoidal attractors for partially hyperbolic diffeomorphisms of compact $3$-manifolds. 


 \medskip
     
  Extensions of the results above,  from minimal sets for flows to minimal sets for foliations, is not so immediate.

\medskip
   
Every non-singular  $1$-dimensional $C^1$-distribution  $F \subset TM$ can be integrated to give a foliation $\F$ of $M$, so a perturbation of a flow is always integrable. This is very useful!

     
   \vfill   

}


  \frame % 1
{
  \frametitle{}

 
 
For $n \geq 2$,   a smooth distribution $F \subset TM$ of dimension $n$ can only be integrated to a foliation $\F$ if the \emph{Frobenius integrability conditions} are satisfied. A celebrated result of Bott shows that there are distributions for which no perturbation can be integrated.

  
  \medskip
  
 In [Clark-Hurder, 2011] we replaced the technique of taking nested perturbations of flows near a periodic orbit as above, with the techniques of taking variations of flat bundles over a manifold. This guarantees that integrability issues do not arise. 
 
It is necessary to show that an infinite sequence of  variations of flat bundles yields a $C^r$-foliation in the limit, for $r > 0$.
 
   
 
 
     
   \vfill   

}


  \frame % 1
{
  \frametitle{}

  For $M_0 = \mT^n$, and a presentation   $\cP   =    \{p_{\ell+1} \colon \mT^n \to \mT^n \mid \ell \geq 0\}$, recall that the homeomorphism type of $\cS_{\cP}$ is determined by a sequence of integer matrices $A_{\ell}$ with $\det (A_{\ell}) > 1$ which induce the proper non-trivial covering maps $p_{\ell}$. 
  
  Let $\cA = \{A_1, A_2, \ldots\}$ denote such a sequence, and $\cS_{\cA}$ the corresponding normal solenoid. In   [Clark-Hurder, 2011] we showed:

   
 {\bf Theorem:}   For each $r \geq 1$, there are recursive conditions on the sequence $\cA$ which guarantee that  $\cS_{\cA}$  embeds as a minimal set for a $C^r$-foliation of codimension $q \geq 2n$. Moreover, for fixed $r$ there are uncountably many such $\cA$ for which the spaces $\cS_{\cA}$ are pairwise non-homeomorphic. 
 
 \medskip
  
 The recursive conditions are very technical to state, as they are based on giving uniform estimates to show the convergence of the flat bundle perturbations in the $C^r$-topology on foliations. 
     
   \vfill   

}
 
\section{Lipschitz}

  \frame % 1
{
  \frametitle{}

An alternate approach to proving that a given matchbox manifold does not embed into any foliation, is to look for criteria showing that it does not embed.

\bigskip

The simple observation is that a foliated embedding $h \colon \fM \to M$ where $\F_M$ is a $C^r$-foliation on $M$, induces a type of $C^r$-geometry on the transversal space $\fT$ for $\F$, and the holonomy of the induced pseudogroup $\cGX$ must preserve this ``geometry''. So, we can look for invariants of these geometries which serve as obstructions to the existence of an embedding.

\medskip

There are two notions in the literature related to this: Lipschitz geometry, and quasi-conformal structures. 
We consider the Lipschitz geometries.

   \vfill   

}


 
  \frame % 1
{
  \frametitle{}

 

 {\bf Definition:} Metrics $\dX$ and $\dXp$ on $\fX$ are  \emph{Lipschitz equivalent},   if     for some   $C \geq 1$,
they satisfy the condition:
$$
C^{-1} \cdot \dX(x,y) ~ \leq ~ \dXp(x,y) ~ \leq ~ C \cdot \dX(x,y) \quad {\rm for ~ all} ~ x,y \in \fX
$$
 
 
{\bf Definition:}
 The action of a pseudogroup  $\cGX$,  which is compactly generated by the collection
 $$\{h_i \colon U_i \to V_i \mid 1 \leq i \leq k\}~, ~ U_i , V_i \subset \fX $$
  is \emph{C-Lipschitz}  with respect to $\dX$, if there exists   $C \geq 1$, such that for each   $1 \leq i \leq k$  and for all $w, w' \in U_i$ we have
$$C^{-1} \cdot \dX(w,w') \leq \dX(h_i(w), h_i(w')) \leq C \cdot \dX(w,w')$$
That is, $\cGX$ is generated by \emph{bi-Lipschitz homeomorphisms}.


     
   \vfill   

}

 

 
  \frame % 1
{
  \frametitle{}


 Let $\fM$ be a matchbox manifold, with a choice of  a regular covering by  foliated coordinate charts,
  $$ \cU = \left\{ \vp_{i} \colon U_i \to [-1,1]^n \times \fX_i \mid 1 \leq i \leq k \right\}$$

 Identify   $\fT_i = \vp_{i}^{-1}(0 \times \fX_i) \subset \fM$ with $\fX_i$ and thus $\fX$ with $\fT \subset \fM$.
  
 Let $\cGX$ be the pseudogroup for $\fM$ generated by the collection of transition maps
  $$ \cGX^0 \equiv \{h_{i,j} \mid U_i \cap U_j \ne \emptyset\}$$
 
  The inclusion $\fT \subset \fM$ induces a metric $d_{\fT}$ on the transversal, and hence a metric $\dX$ on $\fX$. 
 
   \vfill   

}
  
 
  \frame % 1
{
  \frametitle{}

{\bf Proposition:}   Let $\fM$ be a minimal matchbox manifold, and  $M$ a smooth Riemannian manifold with a $C^1$-foliation $\F_M$.
If there exists  a homeomorphism $f \colon \fM \to \cZ \subset M$, then there exists a metric $\dX$ on $\fX$ 
such that the action of  the holonomy pseudogroup  $\cGX$ on $\fX$    is Lipschitz.

\medskip

     
     It is immediate that the holonomy maps associated to a $C^1$-foliation of a compact manifold satisfy a bi-Lipschitz condition. 
     
     This    implies the induced transition maps $h_{i,j}$ for the  clopen subsets of $\fX$ are Lipschitz, though with some technical considerations.
     
     
   \vfill   

}

\section{Entropy}

  \frame % 1
{
  \frametitle{}

   
The  \emph{geometric entropy}   for  pseudogroup actions was introduced in [Ghys, Langevin \& Walczak,  1988],  to give a  measure    of the ``exponential complexity'' of the orbits of the holonomy pseudogroup for a $C^1$-foliation. Conveniently, their work formulates the main ideas and results for pseudogroups of local homeomorphisms of topological spaces. We recall some details. 
 
 \bigskip
 
 Let $\{h_i \colon U_i \to V_i \mid 1 \leq i \leq k\}$ be a generating set.
    
  $\cGX^*$  is defined to be all  compositions of elements of $\cGX^0$ on the maximal domains for which the composition is defined. 

\medskip

Recall that  the \emph{word length} of   $g \in \cG^*$ satisfies     $\| g \| \leq m$ 
if $g$ can be expressed as the composition of at most $m$ elements of $\cGX^0$. 

That is, $\| g \| \leq m$ implies that $g = h_{i_\ell} \circ \cdots h_{i_1}$ for $\ell \leq m$. 

   \vfill   

}



  \frame % 1
{
  \frametitle{}


   Let $\e > 0$ and $\ell > 0$. 
   
   A subset $\cE \subset \fX$ is said to be $(\dX, \e, \ell)$-separated if   for all $w,w' \in \cE \cap \fX_i$   there exists $g \in \cGX^*$ with $w,w' \in \mathrm{Dom}(g) \subset \fX_i$, and  $\|g\|_w \leq \ell $  so that  $\dX(g(w), g(w')) \geq \e$. 

If$w  \in \fX_i$ and $w' \in \fX_j$ for $i \ne j$ then   they are   $(\e, \ell)$-separated.   

The  ``expansion growth function'' is:
  $$h(\cGX, \dX, \e, \ell) =     \max \{ \# \cE \mid \cE \subset \fX ~ \text{is} ~ (\dX, \e,\ell) \text{-separated} \}   $$
The (geometric) entropy  is   the \emph{asymptotic exponential growth type} of the expansion growth function:
\begin{eqnarray*}
 h(\cGX, \dX, \e)  & = &  \limsup_{\ell \to  \infty} ~  \ln \left\{ h(\cGX, \dX, \e, \ell) \right\}/ \ell  \\
 h(\cGX,\dX)   & = & \lim_{\e \to 0} ~  h(\cGX, \dX, \e)
\end{eqnarray*}
Then  $0 \leq h(\cGX, \dX) \leq \infty$.

   
   \vfill   

}



  \frame % 1
{
  \frametitle{}
We recall some properties of the entropy which were shown in  [Ghys, Langevin \& Walczak,  1988].


{\bf Proposition:}  
Let    $\cGX$ be a compactly generated pseudogroup, acting on the compact space  $\fX$ with the metric $\dX$. Then   $h(\cGX,\dX)$ is independent of the choice of metric $\dX$. 

 \bigskip
 
 {\bf Proposition:}  
Let    $\cGX$ be a compactly generated pseudogroup, defined as the induced pseudogroup for a compact leafwise saturated subset $\cZ \subset M$ for a $C^1$-foliation $\F_M$ on a compact manifold $M$.   Then   $h(\cGX,\dX) < \infty$.


    \vfill   

}



  \frame % 1
{
  \frametitle{}

{\bf Definition:} The action of the pseudogroup $\cGX$ is equicontinuous, if   for all $\e > 0$, there exists $\delta > 0$ so that for all $h \in \cGF$ we have
$$x,y \in D(h) ~ {\rm with} ~ \dX(x,y) < \delta \quad \Longrightarrow  \quad \dX(h(x), h(y)) < \e$$

\medskip
 
{\bf Proposition:}  
Let    $\cGX$ be a compactly generated pseudogroup, acting on the compact space  $\fX$ with the metric $\dX$. If the action of $\cGX$ is equicontinuous, then $h(\cGX,\dX) =0$.
    
   \vfill   

}



  \frame % 1
{
  \frametitle{}


{\bf Proposition:} Let    $\cGX$ be a compactly generated pseudogroup associated to a regular covering of $\fM$, acting on the compact space  $\fX$ with the metric $\dX$. 
Then the property that $h(\cGX, \dX)  = 0$, $0 < h(\cGX, \dX)  < \infty$, $h(\cGX, \dX)  = \infty$ is independent of the choice of the regular covering of $\fM$.

\medskip

We can thus speak of a matchbox manifold $\fM$ as having zero, positive, or infinite entropy.


\bigskip


{\bf Proposition:}  
Let    $\cGX$ be a compactly generated pseudogroup associated to a regular covering of $\fM$, acting on the compact space  $\fX$ with the metric $\dX$. 
If $\fM$ admits a foliated embedding into a $C^1$-foliation of a compact manifold $M$,  then  $h(\cGX, \dX)  < \infty$.

 \bigskip
 
 {\bf Question:} Which   matchbox manifolds have infinite entropy?
   
   \vfill   

}


 
\section{Examples}

  \frame % 1
{
  \frametitle{}

{\bf Proposition:} Let $\fM$ be a matchbox manifold which is homeomorphic to a weak solenoid $\cS_{\cP}$. Then for any regular covering of $\fM$, and metric $\dX$ on the transversal space $\fX$, the associated pseudogroup $\cGX$ satisfies  $h(\cGX, \dX)  =0$.

\bigskip

Thus, it is reasonable to ask which weak solenoids admit embeddings into $C^1$-foliations.
   \vfill   

}


 
\section{Examples}

  \frame % 1
{
  \frametitle{}

 
 {\bf Proposition:} Let $\fM$ be a matchbox manifold which is homeomorphic to a generalized solenoid $\cS_{\cP}$, associated to the pseudogroup $\cGX$ acting on a Cantor set $\fX$. If $\fX$ has finite Hausdorff dimension, hence finite box dimension, then  $h(\cGX, \dX)  < \infty$.

\bigskip

The proof uses that for any $\e > 0$, there is a finite stage $M_{\ell}$ of the   presentation for $\fM$ for which the fibers of the projection $\Pi \colon \fM \to M_{\ell}$ have diameters less than $\e$. Then for any $\ell > 0$ the holonomy of words of length at most $\ell$ in $\cGX$ is defined by a leafwise path in $\fM$ whose length is bounded above by   $k \cdot \ell$ for some constant $k$. These paths project to paths in the branched manifold $M_{\ell}$ with length at most $k' \cdot \ell$, for some $k'$. Such a path crosses branch submanifolds in $M_{\ell}$ at most an exponential function of $\ell$. As $\fX$ has finite box dimension, this implies that $h(\cGX, \dX)  < \infty$.

   \vfill   

}


  \frame % 1
{
  \frametitle{}

Recall the space $\fM_2$ obtained from the  Ghys-Kenyon construction, using the Cayley tree for the free group $\mF_2 = \mZ * \mZ$ on $2$-generators. 

From the talk yesterday by Lukina, and in [Lukina, 2014], one can show that :

{\bf Theorem:} The matchbox manifold $\fM_2$ is the suspension of a Lipschitz pseudogroup action on a Cantor set $\fX_2$ with Hausdorff dimension infinity, and with  $h(\cGX, \dX)  = \infty$.

\bigskip

{\bf Corollary:} $\fM_2$ does not embed into any $C^1$-foliation of a  manifold.

Note: there is no requirement in the corollary, that the embedding is Lipschitz. This big space just doesn't embed!
     
   \vfill   

}


\section{Classification}


  \frame % 1
{
  \frametitle{}

Recall the definition:

 {\bf Definition:}  Matchbox manifolds  $\fM^1$ and $\fM^2$ are \emph{return equivalent} if  there exists 
 
 $\bullet$~ regular coverings of $\fM^1$ and $\fM^2$ by foliated coordinate charts, with  transversals $\fT^1$ and $\fT^2$, 
 
  $\bullet$~ pseudogroups $\cG_{\fX^1}$ and $\cG_{\fX2}$, respectively,  
  
   $\bullet$~ sections $C^1 \subset \fT^1$ and $C^2 \subset \fT^2$, 
   
    $\bullet$~ a homeomorphism $h \colon C^1 \to C^2$ which conjugates the restricted pseudogroups $\cG_{C^1}$ to $\cG_{C^2}$.

 

     
   \vfill   

}


  \frame % 1
{
  \frametitle{}

 
 {\bf Definition:}  Lipschitz  pseudogroups  $(\cG_{\fX^1}, d_{\fX^1})$ and $(\cG_{\fX^2}, d_{\fX^2})$ are \emph{Lipshitz return equivalent} if  there exists 
 
   
   $\bullet$~ sections $C^1 \subset \fT^1$ and $C^2 \subset \fT^2$, 
   
    $\bullet$~ a bi-Lipschitz homeomorphism $h \colon C^1 \to C^2$ which conjugates $\cG_{C^1}$ to $\cG_{C^2}$.

 \bigskip
 
 {\bf Proposition:} Lipschitz return equivalence is an equivalence relation on minimal Lipschitz pseudogroups. 

 \bigskip
 
 
   {\bf Meta-problems:}  
   
   $\bullet$~ Find invariants of Lipschitz return equivalence.
   
   $\bullet$~ For a group action $\G \times \fX \to \fX$,   does  Lipschitz return equivalence become tractable for particular classes of groups  $\G$? Examples to consider include $\G = \mZ^n$, and $\G$ a higher rank lattice for example.
   
   
   
   \vfill   

}

  
 
  
\frame % 1
{
  \frametitle{}
  
   \begin{center}
{\it Thank you for your continued attention and interest, \\ throughout these four lectures!}
\end{center}

 

\vfill   

}


  
\frame % 1
{
  \frametitle{References}
 {\tiny   

  {H.~Bell and K.R.~Meyer},
  {\it Limit periodic functions, adding machines, and solenoids},
 {\bf J. Dynam. Differential Equations}, 7:409--422, 1995.
  
{L.~Block, J.~Keesling},
{\it Iterated function systems and the code space},
{\bf Topology Appl.}, 2265--75, 2002.

{R.~Bott},
  {\it On topological obstructions to integrability},
   In {\bf Actes du Congr\`es International des Math\'ematiciens (Nice, 1970), Tome 1},
  {Gauthier-Villars, Paris},  1971:27--36.


{A.~Brown}, 
{\it Nonexpanding attractors: conjugacy to algebraic models and    classification in 3-manifolds},
{\bf J. Mod. Dyn.}, 4(3):517--548, {2010}.
   


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

{J.-M.~Gambaudo and C.~Tresser},
  {\it Diffeomorphisms with infinitely many strange attractors},
  {\bf J. Complexity}, 6:409--416, 1990.
 
{\'E.~Ghys, R.~Langevin, and P.~Walczak},
  {\it Entropie g\'eom\'etrique des feuilletages},
  {\bf Acta Math.}, 160:105--142, 1988.

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


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


{\'A.~Lozano-Rojo and O.~Lukina},
 {\it  Suspensions of {B}ernoulli shifts},
  {\bf Dynamical Systems. An International Journal},to appear 2013; arXiv:1204.5376.


 
{O.~Lukina},
 {\it Hausdorff dimension of graph matchbox manifolds},
{preprint}, July 2014; {arXiv:1407.0693}.

  {L.~Markus and K.~Meyer},
  {\it Periodic orbits and solenoids in generic {H}amiltonian dynamical systems},
  {\bf Amer. J. Math.}, 102:25--92, 1980.

{S.~Smale}, 
  {\it Differentiable Dynamical Systems}, 
  {\bf Bull. Amer. Math. Soc.}, 73:747-817, 1967.

{P.~Walczak},
   {\bf Dynamics of foliations, groups and pseudogroups},
  {Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series)   [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)]}, Vol. 64.
   Birkh\"auser Verlag, Basel, 2004.


{R.F.~Williams},
 {\it One-dimensional non-wandering sets},
 {\bf Topology}, 6:473--487, 1967.

{R.F.~Williams},
 {\it Expanding attractors},
 {\bf Inst. Hautes \'Etudes Sci. Publ. Math.}, 43:169--203, 1974.



  

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\end{document}

 