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\date[July 31, 2007]
 {Special Session on Real Dynamics \\
Joint International AMS--PMS Meeting}

  
\title[Solenoidal minimal sets in foliations]{Solenoidal minimal sets in foliations}
 
 
\author{Steven Hurder}
 
\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder}

 

\begin{document}

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% \frame{\tableofcontents}

\section{Introduction}
 
  

\frame
{

  \frametitle{Reeb--Thurston--Stowe Stability Theorems}

Let $\F$ be a $C^r$-foliation of a smooth compact manifold $M$, for $r \geq 1$.

 

\medskip


{\bf Theorem:} (Reeb [1952]) Let $L$ be a compact leaf of a codimension one foliation $\F$ such that $\pi_1(L,x) = 0$.  
Then there exists an open saturated neighborhood $L \subset U$ such that $\F \mid U$ is a product foliation.


\bigskip


{\bf Theorem:} (Thurston [1974]) Let $L$ be a compact leaf of a codimension one foliation $\F$ such that $H^1(L,\mR) = 0$.  
Then there exists an open saturated neighborhood $L \subset U$ such that $\F \mid U$ is a product foliation.

 \bigskip


{\bf Theorem:} (Stowe [1983]) Let $L$ be a compact leaf of a codimension $q$  foliation $\F$ such that $H^1(L,\mV) = 0$   
for all flat finite-dimensional vector bundles associated to a representation of $\pi_1(L,x)$. Then there exists an open saturated neighborhood $L \subset U$ such that if $\F'$ is a  sufficiently $C^1$ close to $\F$, then $\F' \mid U$ is a product foliation.


\vfill
}
  
  
  
\frame
{

  \frametitle{Instability of leaves}

Suppose that $L$ is a compact leaf with $H^1(L,\mR) \ne 0$. It is trivial to construct foliations with $L$ as a leaf, such that every leaf $L'$ which intersects an open neighborhood $L \subset U$ is necessarily non-compact.

\medskip

{\bf Problem:} Suppose that $L$ is a compact leaf with $H^1(L,\mR) \ne 0$, and $L \subset U$ is a saturated open neighborhood for which $\F \mid U$ is a product foliation. Then what can be said about the behavior of a foliation $\F'$ which is    $C^1$ close to $\F$?

\bigskip


This question arose from a related problem posed by Alex Clark:

\medskip

{\bf Problem:}  Given a solenoid $\cS$ with $p$-dimensional leaves, when does there exists a smooth foliation $\F$ of a compact manifold such that $\cS$ is a minimal set for $\F$?

\vfill
}
 
\frame
{

  \frametitle{Embedded solenoids}

{\bf Theorem:} (Clark \& Hurder [2006])   $\F$ is a   codimension $q$  $C^1$-foliation. 
Let $L$ be a compact leaf with $H^1(L,\mR) \ne 0$, and $L \subset U$ is a saturated open neighborhood for which $\F \mid U$ is a product foliation. Then there exists $\F'$ arbitrarily $C^1$ close to $\F$ such that $U$ is  saturated for $\F'$, $\F$ and $\F'$ agree outside of $U$, and $\F' \mid U$ has a compact solenoidal minimal set $\K \subset U$, where
$$\K = \lim_{\leftarrow} \left\{ f_n \colon L_{n+1} \to L_n \right\}$$
 where each $L_n$ is a covering of $L$.

\bigskip



{\bf Remark:} The proof uses new constructions in foliation theory, and seems of interest for the questions it raises about perturbations of foliations.

 

\vfill
}
 
\frame
{

  \frametitle{Flat bundles}

Choose a basepoint $x \in L$, and set $\Gamma = \pi_1(L, x)$. 
\medskip


$\G$ acts on the right as deck transformations of the universal cover $\wtL \to L$.

\medskip


Let $\rho \colon \Gamma \to \SOq$ be an orthogonal  representation.  

\medskip

$\G$ acts on the left as isometries of $\mR^q$ by $\gamma \cdot \vec{v} = \rho(\gamma) \vec{v}$.

\medskip


Define a   flat $\mR^q$-bundle with holonomy $\rho$ by
$$\mE_{\rho}^q =  (\wtL \times \mR^q)/(\wty \cdot \gamma, \vec{v}) \sim (\wty, \gamma \cdot \vec{v}) ~ \longrightarrow ~ L$$
 



The most familiar example is for $L = \mS^1$ and $\G = \pi_1(\mS^1 , x) = \mZ \to {\bf SO(2)}$. Then $\mE^2_{\rho}$ is the flat vector bundle over $\mS^1$ with the foliation by lines of slope $\rho(1) = \exp(2\pi \sqrt{-1} \, \alpha)$.

\medskip

In general, the bundle $\mE_{\rho}^q \to L$ need not be a product vector bundle. 


\vfill
}

\frame
{

  \frametitle{Trivializing flat bundles}


{\bf Proposition:} Suppose that there exists a 1-parameter family of representations $\rho_t \colon \G \to \SOq$ such that $\rho_0$ is the trivial map, and $\rho_1 = \rho$, then $\rho_t$ canonically defines a vector bundle map $\mE_{\rho}^q \cong L \times \mR^q$.

\bigskip

{\bf Proof:} The family of representations defines a family of flat bundles $\mE^q_{\rho_t}$ over the product space $L \times [0,1]$.  This defines an isotopy between the bundles $\mE^q_{\rho_0}$ and $\mE^q_{\rho_1}$, which induces a bundle isomorphism  between them.   The initial bundle $\mE^q_{\rho_0}$ is a product, hence the same holds for $\mE^q_{\rho_1}$.

\medskip

In the case of the example above over $\mS^1$ the product structure can be written down explicitly.  

\medskip

The key point is that the bundle isomorphism between  $\mE^q_{\rho_0}$ and $\mE^q_{\rho_1}$ depends smoothly on the path $\rho_t$.


\vfill
}
 
\frame
{

  \frametitle{Abelian representations}

$k =$ the greatest integer such that $2k\leq q$.
\medskip

  $\mT^k \subset \SOq$   a maximal embedded $k$-torus. 

\medskip

  $\xi = (\xi_1, \ldots, \xi_k) \colon \G \to \mR^k$   a representation. Define  
\begin{eqnarray*}
 \rho_t^{\xi} \colon  \G & \to &  \SOq\\
  \rho_t^{\xi}(\gamma) & = & [\exp(2 \pi t \sqrt{-1} \xi_1(\gamma), \ldots , \exp(2 \pi t \sqrt{-1} \xi_k(\gamma))]
  \end{eqnarray*}
  
\begin{eqnarray*}
\mD^q_{\epsilon} & = &  \{ (z_1, \ldots , z_q) \mid z_1^2 + \cdots z_q^2 < \epsilon\} \subset \mR^q \\
\mB^q_{\epsilon} & = &  \{ (z_1, \ldots , z_q) \mid z_1^2 + \cdots z_q^2 \leq \epsilon\} \subset \mR^q \\
\mS^{q-1}_{\epsilon} & = & \{ (z_1, \ldots , z_q) \mid z_1^2 + \cdots z_q^2 = \epsilon\} \subset \mR^q
\end{eqnarray*}

\vfill
}

 
\frame
{

  \frametitle{Realizing abelian representations}


{\bf Proposition:} $\xi  \colon \G \to \mR^k$ defines a flat bundle foliation $\F_{\xi}$ of $L \times \mS^{q-1}$ whose leaves cover $L$. Moreover, if the image of $\xi$ is contained in the rational points $\mQ^k \subset \mR^k$, then all leaves of $\F_{\xi}$ are compact. 

 \bigskip

{\bf Proof:} $\rho_t^{\xi}$ is an isotopy from $\xi$ to the trivial representation. $\Box$

\bigskip

The idea is that if we take a path $\lambda \colon [0,\epsilon] \to {\bf Rep}(\G, \SOq)$ of such representations, then this will yield a    foliation $\F_{\lambda}$ of $L \times \mD^q_{\epsilon}$ whose restriction to the spherical fiber 
$L \times \mS^{q-1}_{s}$ is $\F_{\lambda(s)}$, for $0 \leq s < \epsilon$.
\vfill
}
 
\frame
{

  \frametitle{The basic plug}

In the main theorem, we can assume that $U = L \times \mD^q_{\epsilon}$.

\medskip

Fix a non-trivial representation  $\xi_1  \colon \G \to \mQ^k$ which exists as $H^1(L,\mR) \ne 0$.

\medskip

Let $0 < \epsilon/2 < \epsilon_1 < \epsilon$, and set $\epsilon_1' = (\epsilon_1 + \epsilon)/2$. Choose a monotone decreasing smooth function $\mu_1 \colon [0,\epsilon] \to [0,1]$ such that 
$$
\mu_1(s) = \begin{cases}
  1    & \text{if }  ~   0 \leq s \leq \epsilon_1, \\
    0  &  \text{if }  ~ \epsilon_1' \leq s \leq \epsilon
\end{cases}
$$

Set $\rho^{\xi_1}_{1,s}   = \rho^{\mu_1(s) \xi_1} \colon \G \to \SOq$. Use this family of representations to define a foliation $\F_1$ of $N_1 = L \times \mD^q_{\epsilon}$.

\medskip

 Note that $\F_1$ is the product foliation outside of $L \times \mS^{q-1}_{\epsilon'_1}$, and has all leaves compact in  
 $L\times \mB^{q}_{\epsilon_1}$.



\vfill
}
 
\frame
{

  \frametitle{Iterating the plug}

Let $L_1$ be a generic  leaf of $\F_1$ contained in $L \times \mS^{q-1}_{\epsilon_1/2}$.

\medskip

By construction, $L_1 \to L$ is the compact covering associated to the kernel $\G_1 \subset \G$ of the homomorphism 
$\rho^{\xi_1} \colon \G \to \SOq$.

\bigskip


Next choose $0 < \epsilon_2 < \epsilon$ sufficiently small so that $\F_1$ restricted to the  $\epsilon_2$-disk bundle $N_2$ about $L_1$ is a product foliation. 

\bigskip

We now repeat the construction: choose a non-trivial map  $\xi_2 \colon \G_1 \to   \mQ^k$ and maps $\mu_2$ as before.

\vfill
}

\frame
{

  \frametitle{Iterating the plug 2}


Iterate for all $n \geq 2$. This yields:

\medskip

A descending sequence of subgroups 
$\G \supset \G_1 \supset \G_2 \supset \cdots$

\medskip

An increasing of open saturated subsets   
$$V_n = L \times \mD^q_{\epsilon} - L_{n+1} \times \mB^q_{\epsilon_{n+1}}$$
 with a foliation $\F_n'$ which has leaves of increasingly high order as coverings, corresponding to the subgroups $\G_n$

\bigskip
Note that $V_{n} \subset V_{n+1}$, hence 
  $\K_n = L \times \mD^q_{\epsilon} - V_n$, forms a nested sequence of   compact sets,  $\K_{n+1} \subset \K_n$.
\vfill
}

\frame
{

  \frametitle{The perturbation $\F'$}



{\bf Proposition:} If the maps $\xi_n$ are suitably chosen (ie the images of the generators of $\G$ are approach   0 in $\mQ^k$ sufficiently rapidly)  then:

\begin{enumerate}
\item  the foliations $\F_n'$ converge to a $C^r$-foliation $\F'$ of $L \times \mD^q_{\epsilon}$. 

\medskip

 \item  $\displaystyle \K = \cap_{n=1}^{\infty} ~  \K_n$ is a saturated compact set.

\medskip

\item $\F' \mid \K$ is a solenoid.

\end{enumerate}

\bigskip


{\bf Problem:}  Is there a classification for the solenoids which arise in this way?

\medskip

Look for an answer to more general formulation.
\vfill
}


\frame
{

  \frametitle{Pseudogroups \& Groupoids}
  
  

Every foliation admits a discrete model by choosing a section $\cT \subset M$, an embedded submanifold of dimension $q$ which intersects each leaf of $\F$ at least once, and always transversally.  The holonomy of $\F$ yields a compactly generated pseudogroup $\cG_{\F}$ acting on $\cT$.

\bigskip

 
 {\bf Definition:}
A     pseudogroup of transformations $\cG$ of $\cT$ is \emph{compactly generated}     if there is 
\begin{itemize}
\item   a 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}


 
\bigskip

{\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 $r[g]_x = g(x) = y$.



  \vfill

}


\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 
 $$ D\varphi ( [h]_y \circ [g]_x ) = D\varphi [h]_y \cdot D\varphi [g]_x $$
 
\vfill
}

 
\frame
{

  \frametitle{Twist invariant}

 {\bf Proposition:} Given a closed saturated subset $\K \subset M$,  
 $$[D\varphi \mid \K] \in H^1(\cG_{\F} \mid \K ; {\bf GL(\mR^q)}) \cong H_{\F}^1(\K; {\bf GL(\mR^q)})$$
  is an invariant of $\F \mid \K$.
 
 \bigskip
 
 
 {\bf Theorem:} Let $\F'$ be a foliation with solenoidal minimal set $\cS$ as above. Then 
 $[D\varphi \mid \cS] \in   H_{\F'}^1(\cS; {\bf GL(\mR^q)})$ is non-trivial, and measures the ``asymptotic twisting of the lamination''. 
 
 \bigskip
 
 
 {\bf Problem:} Calculate $H_{\F'}^1(\cS; {\bf GL(\mR^q)})$ for a solenoid.
 \vfill

}




\frame
{

  \frametitle{Asymptotic expansion}
  

 {\bf Definition:} For $g \in \G_{\cG}$,    the word length $\| [g] \|_x$ of the germ $[g]_x$ of g at $x$  is the least $n$ such that 

$$[g]_x = [g_{i_1}^{\pm 1} \circ \cdots \circ g_{i_n}^{\pm 1}]_x$$

Word length is a measure of the ``time'' required to get  from one point on an orbit to another.
  
  
  \bigskip
   
    {\bf Definition:} The   transverse expansion rate  function at $x$ is
   $$\lambda(\cG, n, x) =  
   \max_{\| [g] \|_x \leq n} ~  \frac{ \ln \left( \max \{ \| D_x g  \| , \| D_y g^{-1} \|\}\right) }{\| [g] \|_x }~ \geq ~ 0$$

  \vfill
  }
  
  
\frame
{

  \frametitle{Invariant measures}
  

    {\bf Definition:} The asymptotic transverse growth rate at $x$ is
   $$\lambda(\cG, x) = \limsup_{n \to \infty} ~   \lambda(\cG, n, x) ~ \geq ~ 0$$

This is essentially the maximum Lyapunov exponent for $\cG$ at $x$.

\bigskip


{\bf Theorem:} Let $\K \subset M$ be a compact saturated subset such that $\lambda(\cG, x) = 0$ for all $x \in \K \cap \cT$. Then $\F \mid \K$ has a holonomy invariant transverse measure supported on $\K$.


\vfill
}


\frame
{

  \frametitle{Distal foliations}
  

  
{\bf Definition:} A foliation $\F$ is distal if its pseudogroup  $(\cG_{\F}, \cT)$ is distal: that is, for all $x\ne y \in \cT$ there exists $\epsilon_{x,y} >0$ such that 
$$d_{\cT}(g(x),g(y)) \geq \epsilon_{x,y} ~ \mbox{for ~all} ~ g \in \cG_{\F}$$

{\bf Definition:} A foliation is said to be \emph{compact} if all leaves of $\F$ are compact submanifolds. 
\medskip

{\bf Remark:} All compact foliations are distal. 


\bigskip

 

{\bf Theorem:} If $\F$ is distal and transversally  $\C^{1+\alpha}$ for some $\alpha > 0$, then $\lambda(\cG, x) = 0$ for all $x \in \cT$.
 
\medskip

As an application, this gives another proof that a minimal set $\K \subset M$ has a holonomy invariant transverse measure supported on $\K$.

\medskip


{\bf Problem:} Is there a smooth structure theory for solenoids in distal foliations?

 \vfill

}

 

\end{document}
       
 