% slides completed April 24, 2013
% for Dynamics Seminar at Penn State University




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\date[April 24, 2013]


  
\title[Kuperberg minimal sets]{The dynamics of Kuperberg flows}


\author{Steve Hurder\\(joint with Ana Rechtman)}

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



\begin{document}

\frame{\titlepage} % # 1



 
  \section{Seifert Conjecture}
 
  
\frame % 1
{
  \frametitle{Introduction}
 
 {\bf Theorem:} [K. Kuperberg]  Let  $M$ be  a closed, oriented $3$-manifold. Then $M$ admits a  non-vanishing smooth vector field $\cK$ without periodic orbits.
  
 \bigskip
 
 
{\footnotesize 
$\bullet$     K.~Kuperberg, ``A smooth counterexample to the {S}eifert conjecture'',   Annals of Mathematics,   1994.\\
$\bullet$     S.~Matsumoto, ``{K}uperberg's {$C^\infty$} counterexample to the {S}eifert conjecture'',  S\=ugaku, Mathematical Society of Japan, 1995.\\
$\bullet$     {\'E.}~Ghys, ``Construction de champs de vecteurs sans orbite p\'eriodique (d'apr\`es {K}rystyna {K}uperberg)'',    S{\'e}minaire Bourbaki, Vol. 1993/94, Exp.\ No.\ 785, 1995.
}


   \vfill   

}

\frame % 1
{
  \frametitle{}

   {\bf Theorem:}  [Ghys, Matsumoto, 1994] 
   A  Kuperberg flow $\Phi_t$ has a \emph{unique minimal set} $\Sigma$.

 \medskip
 


{\bf Problem:}  \emph{Describe the topological shape of $\Sigma$, and analyze the dynamics of $\Phi_t$ restricted to   open neighborhoods of $\Sigma$.}  

 
 \bigskip
 
    {\bf Theorem:} [Katok, 1980] If a smooth flow on $M^3$ has positive topological entropy, then it has periodic orbits.
    
Hence,    the  Kuperberg flow $\Phi_t$ has  topological  entropy $0$.
   
\medskip
 


{\bf Problem:} \emph{What type of entropy-zero dynamical system does the restricted flow $\Phi_t | \Sigma$ flow yield? For example, is it an odometer? Does its type depend on the construction of the flow?}
 


  \vfill   

}
 
 
\section{Plugs}


\frame % 2
{
  \frametitle{Plugs}
 
 A plug $\mP \subset \mR^3$ is a $3$-manifold with boundary,   with a non-vanishing vector field that agrees with the vertical field on the boundary of $\mP$:
  
 \begin{center}
\includegraphics[width=0.26\textwidth]{pix/Plug1.png}\qquad
\includegraphics[width=0.26\textwidth]{pix/Plug2.png}
\end{center}

  
 {\bf Mirror Symmetry Property:} An orbit entering a plug (from the bottom) either never leaves the plug  (it is ``trapped''), or exits the plug  at the mirror image point at the top of the plug.

 
  \vfill   

}




\frame % 1a
{
  \frametitle{Wilson Plug}
 
  
 {\bf Theorem:} [Wilson,  1966] A closed oriented $3$-manifold $M$  admits a smooth non-vanishing vector field $\cX$ with two    periodic orbits. 
 
 
{\it Proof:}  Trap orbits so they limit to a periodic orbit contained in the plug.    The two periodic orbits are attractors:
 
 \begin{center}
\includegraphics[width=0.16\textwidth]{pix/Wilson_flow.png}\qquad
\includegraphics[width=0.4\textwidth]{pix/Wilson_merged_orbits.png}
\end{center}


 
  \vfill   

}
 
\frame % 3
{
  \frametitle{Schweitzer Plug}

  
  
  {\bf Theorem:} [Schweitzer, 1974] Every homotopy class of non-vanishing vector fields on a closed $3$-manifold $M$ contains a $C^1$-vector field without closed orbits. 
  
\medskip

    
{\it Proof:}  Replace the circular orbits of the Wilson Plug with Denjoy minimal sets, embedded as pictured, so trapped orbits limit to Denjoy minimal set.
  
   
 \begin{center}
\includegraphics[width=0.45\textwidth]{pix/Schweitzer_plug.png}
\end{center}


  
  \vfill   

}



\frame % 6
{
 \frametitle{Handel's Theorem}
  
A minimal set $K$ for a flow $\cX$ on a $3$-manifold $M$ is said to be ``surface-like'' if there is a tamely embedded surface $\Sigma \hookrightarrow M$ whose image contains $K$.

 
\medskip

{\bf Theorem:} [Handel, 1980] Let $\cX$ be a flow on a $3$-manifold $M$ such that its minimal sets are surface-like, then $\cX$ cannot be $C^2$. 
 
  
\medskip

Handel's analysis implies any construction of counter-examples  to the Seifert Conjecture requires ``$3$-dimensional dynamics'', in that its minimal sets cannot be planar.
  \vfill   

}

 
 \section{Kuperberg   Plug}

\frame % 5
{
  \frametitle{Modified Wilson Plug}

 Define a radially symmetric  vector field  on the plug $\mW$
 $$\cW = g(r,\theta, z) \partial /\partial z + f(r,\theta, z) \partial /\partial \theta$$
 $(r,\theta, z) \in [1,3] \times \mS^1 \times [-2,2] = W$ is radially symmetric, with $f(r,0) = 0$ and $g(r,z) = 0$ only near the boundary, and $g(r,z) = 1$ away from the boundary $\partial W$. 


   \begin{center}
\includegraphics[width=0.18\textwidth]{pix/Wilson_modified.png}\qquad
\includegraphics[width=0.55\textwidth]{pix/Kuperberg_cylinder22.png}
\end{center}


  \vfill   

}

  
\frame % 5
{
  \frametitle{}

% \vspace{1cm}
 The flattened orbits of the modified Wilson flow $\cW$ appear like:
\begin{center}
\includegraphics[width=0.7\textwidth]{pix/Figure2a.png} % modified wilson
\end{center}


  \vfill   

}

\frame % 5
{
  \frametitle{}

% \vspace{1cm}
3d-orbits of  $\cW$ appear   for $r > 2$ appear like:
\begin{center}
\includegraphics[width=0.6\textwidth]{pix/figure3bfinal.png} % modified wilson
\end{center}

 
  \vfill   

}


\frame % 6
{
 \frametitle{Embed modified Wilson as a double cover}
  
   \begin{center}
\includegraphics[width= .58\textwidth]{pix/Figure4.png}
\end{center}
  
  \vfill   

}


\frame % 6
{
 \frametitle{Grow two horns}
  
   \begin{center}
\includegraphics[width= .6\textwidth]{pix/Kuperberg_ingredients1.png}
\end{center}
  
  \vfill   

}

 

\frame % 6
{
 \frametitle{Twist the horns}
  
   \begin{center}
\includegraphics[width= .6\textwidth]{pix/Kuperberg_ingredients2.png}
\end{center}
  
  \vfill   

}


\frame % 6
{
 \frametitle{Insert the horns}
  
 The vector field $\cW
$ induces a field $\cK$ on the surgered manifold. Then   the Kuperberg Plug is   pictured as:
  
   \begin{center}
\includegraphics[width=.8\textwidth]{pix/figure7final.png}
\end{center}
  
  \vfill   

}

 
\frame % 6
{
 \frametitle{Wilson dynamics + insertions = Kuperberg dynamics}
  
   
   \begin{center}
\includegraphics[width=.5\textwidth]{pix/figure8final.png}
\end{center}

This is an aperiodic plug, as only chance for periodic orbit is via the circular Wilson orbits, and they get broken up.
    
  \vfill   

}

 

\frame % 4
{
  \frametitle{Kuperberg Plug}

 
   
    
Shigenori Matsumoto's summary: 
 \begin{center}
\includegraphics[width=\textwidth]{pix/Matsumoto_quote.png}
\end{center}

\medskip




\begin{quotation}
{\footnotesize  We therefore must demolish the two closed orbits in the Wilson Plug beforehand. But producing a new plug will take us back to the starting line. The idea of Kuperberg is to \emph{let closed orbits demolish themselves}. We set up a trap within enemy lines and watch them settle their dispute while we take no active part. \\
\hfill (transl. by Kiki Hudson Arai)}
\end{quotation}

    \vfill   

}


 
    
 
 \section{Generic dynamics}



 
  
\frame % 7
{
  \frametitle{The transverse section}
  
  Consider a rectangular section 
  $$\bRt = \{(r,\pi, z) \mid 1 \leq r \leq 3 ~ \& ~ -2 \leq z \leq 2\}$$ 
  as pictured here, which is disjoint from insertions. The periodic orbits in $\mW$ intersect this in two special points $\omega_1 = (2,\pi, -1)$ and $\omega_2 = (2,\pi , 1)$.
  
 \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(20,3)
  \put(2,0.1){\includegraphics[width=60mm]{pix/figure13final.png}}
 \end{picture}
\label{fig:KR1}
 \vspace{-20pt}
\end{figure}



   \vfill   

}
  
\frame % 7
{
  \frametitle{The minimal set}
 
 $\Phi_t \colon \mK \to \mK$ is Kuperberg flow.
 \medskip
  
$\bullet$  Every trapped orbit in $\mK$ contains either $\omega_1$ or $\omega_2$ in its closure.

\medskip


Define the orbit closures 
$\ds \Sigma_i = \overline{\{ \Phi_t(\omega_i) \mid t \in \mR\} }$ for $i =1,2$.

\medskip

{\bf Theorem:} $\Sigma_1 = \Sigma_2$ and    $\Sigma\equiv \Sigma_1$ is the unique minimal set for $\Phi_t$. 

\medskip
 
 What is the topological shape of $\Sigma$?  
 
 What is the Hausdorff dimension of the   intersection $\Sigma \cap \bRt$?
 
    \vfill   

}

 

\frame % 7
{
  \frametitle{}
  
Suggestive computer illustration of $\Sigma \cap \bRt$,   by B. S\'{e}vennec (1994).
    
\begin{center}
\includegraphics[width=.45\textwidth]{pix/MinimalSet_model.png}
\end{center}

 \vfill   

}

  

\frame % 7
{
  \frametitle{Quadratic hypotheses}

A more detailed study of the Kuperberg dynamics requires some type of regularity hypotheses on the construction.

{\bf Definition:} A Kuperberg flow $\cK$ is said to be \emph{generic} if   the singularities for the vanishing of the
vertical part $\ds g(r,\theta,z) \frac{\partial}{\partial z}$ of the Wilson vector field $\cW$ are of quadratic type, and each insertion map  $\sigma_i$ for $i=1,2$ yields a
quadratic radius function.

  \setlength{\unitlength}{1cm}
\begin{figure}[H]
\begin{picture}(20,4)
 \put(1,0.1){\includegraphics[width=44mm]{pix/figure6final0.png}}
  \put(6,.6){\includegraphics[width=44mm]{pix/RadiusInequality.png}}
\end{picture}
\label{fig:twisted}
%\vspace{-20pt}
\end{figure}
 
     \vfill   

}

  
 \section{Laminates}

\frame % 7
{
  \frametitle{The twisted band and the propellers}
 
 Reeb cylinder $\cR = \{(2,\theta,z) \mid 0 \leq \theta \leq 2 \pi ~ \& ~ -1 \leq z \leq 1\} \subset \mW$ 
 
  $\cR'$ is Reeb cylinder minus insertions. Introduce $\Phi_t$-invariant sets
 $$\fM_0 = \bigcup_{t \in \mR}~  \Phi_t(\cR')  \quad , \quad  \fM \equiv \overline{\fM_0} \subset \mK$$

  
  
  \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(20,3)
  \put(2.4,0.1){\includegraphics[width=66mm]{pix/figure19final0.png}}
 \end{picture}
\label{fig:KR2}
 \vspace{-20pt}
\end{figure}


   \vfill   

}


\frame % 7
{
  \frametitle{Levels}
  
$$\fM_0 ~ =  ~ \cR'  ~ \cup  ~ \fM_0^1  ~ \cup  ~ \fM_0^2  ~ \cup  ~ \cdots $$
    
  \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(50,4.6)
  \put(-.8,2.6){\includegraphics[width=28mm]{pix/figure18final0.png}}
  \put(1.2,-.4){\includegraphics[width=32mm]{pix/figure15afinal0.png}}
  \put(4.4,0.4){\includegraphics[width=60mm]{pix/figure20final0.png}}
 \end{picture}
\label{fig:KR3}
 \vspace{-20pt}
\end{figure}

 
   
  
     \vfill   

}


\frame % 7
{
  \frametitle{}
  
   
    {\bf Theorem:} [H \& R]
   For the generic Kuperberg flow, the minimal set $\cZ = \fM$, and thus is $2$-dimensional. 
   
   
   $$\cR' \subset \cZ =  \overline{\{ \Phi_t(\omega_i) \mid t \in \mR\} } \subset \overline{\bigcup_{t \in \mR}~  \Phi_t(\cR')} = \fM .$$
   
      
 {\it Proof:} Show that the orbit $\Phi_t(\omega_1)$ forms a spiral about $\cR'$ which is uniformly  spaced tending to zero as $r \to 2$.
 
 
   \bigskip
   
  The two papers below discuss the existence of open disks in $\Sigma$.
  
  $\bullet$  {\footnotesize   {\'E.}~Ghys, ``Construction de champs de vecteurs sans orbite p\'eriodique (d'apr\`es {K}rystyna {K}uperberg)'',    S{\'e}minaire Bourbaki, Vol. 1993/94, Exp.\ No.\ 785, 1995.}\\

    $\bullet$  {\footnotesize   G.~Kuperberg and K.~Kuperberg, ``Generalized counterexamples to the {S}eifert conjecture'' ,   Annals of Math,   1996.}
      
    \vfill   

}
 

\frame % 7
{
  \frametitle{Wandering points}
  
   
    {\bf Theorem:}  For the generic Kuperberg flow, all points in the complement $\mK - \fM$ are wandering. Moreover, any orbit which is entirely contained in  either the region for $r > 2$, or the region  $r < 2$ of $\mK$, cannot be infinite.
   
   \medskip
   
 {\it Proof:} Combine results of Ghys and  Matsumoto for orbits with $r \leq 2$, and the authors for the case where $r > 2$. 
   
   \medskip
   
   {\bf Conclusion:} $\fM$ is where   the dynamical properties  of the Kuperberg flow are determined.
          
    \vfill   

}


\frame % 7
{
  \frametitle{Zippered laminations}
  
   
    {\bf Theorem:}[H \& R]
   For the generic Kuperberg flow:
   
   
   $\bullet$ There is a lamination $\cL \subset \fM$ with open leaves;
   
   $\bullet$ Each leaf of $\cL$ has a tree structure, with branching number at most $4$;
   
   $\bullet$  $\partial_z \fM = \fM - \cL$ is the union of the boundaries of the leaves of $\cL$;
      
  $\bullet$ $\partial_z \fM$ is dense in $\fM$.
  \medskip
  
 So $\fM$ is like a lamination with boundary, except that its ``boundary'' $\partial_z \fM$ is dense in $\fM$.   
   To describe the topological properties of $\fM$ we must recall how $\fM_0$ was defined. 
      
    \vfill   

}

 
\frame % 7
{
  \frametitle{Cantor transversal}
  
  
  $\cT = \{(r,\pi,0) \mid  1 \leq r \leq 3\} \subset \bRt$ is   transverse to $\fM_0$.
  \medskip
  
  
    {\bf Theorem:} $\fC = \fM \cap \cT$ is a Cantor set, which is a complete transversal for the open leaves $\cL \subset \fM$.
    
  \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(30,4.8)
  \put(2,0.4){\includegraphics[width=60mm]{pix/figure28final.png}}
 \end{picture}
\label{fig:KR4}
 \vspace{-20pt}
\end{figure}

 

     \vfill   

}

\frame % 7
{
  \frametitle{Transverse pseudogroup}
  
  The flow $\Phi_t$ induces a pseudogroup $\cGK$ on $\bRt$
  
Five generators $\{\psi, \phi_1^\pm, \phi_2^\pm\}$ -- these 
  act on curves in $\fM_0 \cap \bRt$ (illustrated below) to obtain $\cGM $ acting on $\fC$.
  
 \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(30,4)
  \put(2.4,0){\includegraphics[width=24mm]{pix/figure16a.png}}
  \put(5.6,0){\includegraphics[width=24mm]{pix/figure16b.png}}
 \end{picture}
\label{fig:KR5}
 \vspace{-20pt}
\end{figure}

  
 

     \vfill   

}

\frame % 7
{
  \frametitle{Lamination entropy}
  

For $\e > 0$,  say that    $\xi_1, \xi_2 \in \fC$ are $(n,\e)$-separated if there exists $\vp \in \cGM^{(n)}$ with $\xi_1, \xi_2 \in Dom(\vp)$, and 
$\ds  d_{\fC} (\vp(\xi_1) , \vp(\xi_2)) ~ \geq ~ \e$. 
\medskip

 A finite set $\cS \subset \fC$ is said to be   $(n,\e)$-separated if   every distinct pair $\xi_1 , \xi_2 \in \cS$  are $(n,\e)$-separated.
 \medskip
 
 Let $s(\cGM,n,\e)$ be the maximal cardinality of an $(n,\e)$-separated subset of $\fM$.   
  
   \vfill   

}


\frame % 7
{
  \frametitle{}
  
  The \emph{lamination entropy} of $\cGM$ is   defined by: 
$$
h(\cGM) ~ = ~ \lim_{\e \to 0} \left\{ \limsup_{n\to\infty}\frac{1}{n}\ln(s(\cGM,n,\e)) \right\} .
$$
The limit $h(\cGM)$   depends on the generating set, but the fact of being non-zero does not.

\medskip

For $0 < \alpha < 1$,  define the \emph{slow entropy} for $\cGM$
$$
h_{\alpha}(\cGM) ~ = ~ \lim_{\e \to 0} \left\{ \limsup_{n\to\infty}\frac{1}{n^{\alpha}}\ln(s(\cGM,n,\e)) \right\} .
$$
 
  {\bf Theorem:} (H and R) For $\alpha = 1/2$, the pseudogroup $\cGM$ has positive slow entropy, $h_{1/2}(\cGM) > 0$. 

   \vfill   

}


\frame % 7
{
  \frametitle{}
  
 
  {\it Proof:} The parabolic arcs are one-half of a stretched ellipse, and the action of $\cGK$ maps ellipses into ellipses, so can get a good game of ping-pong up and running. The only catch is that the game runs slow. The $n^{th}$--volley takes approximately $n^2$ steps.
    
 \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(30,4)
  \put(0.6,0){\includegraphics[width=44mm]{pix/arcs_double.png}}
  \put(5.6,0){\includegraphics[width=44mm]{pix/transform2.png}}
 \end{picture}
\label{fig:KR6}
 \vspace{-20pt}
\end{figure}

    
  
  
   \vfill   

}


\frame % 7
{
  \frametitle{Sullivan dictionary}
  
    
  {\bf Remark:} The entropy $h(\Phi_t)$ of the flow $\Phi_t$ is calculated by following the path around the perimeter of the leaf $\fM_0$, while for the entropy $h(\cGM)$ we are allowed to follow straight paths, along the Cayley graph of $\cGM$ in $\fM_0$. 
 
 \medskip
  
  The same points get separated, but just at different rates.
  
  \medskip
  
  Analogous to behavior of horocycle flow (Kuperberg flow $\Phi_t$) verses geodesic flow (Wilson flow $\Psi_t$), even including ``nested ellipses at infinity'' in $\bRt$.
  
  
  
   \vfill   

}


\frame % 7
{
  \frametitle{Shape of the minimal set}
  
    For $\epsilon > 0$, let $N_{\epsilon}(\fM) = \{x \in \mK \mid d(x, \fM) < \epsilon\}$.  

\medskip
    
  {\bf Definition:}   The shape of $\fM$ is $\cS(\fM) = \varprojlim   ~ \{ N_{\e} \mid \e > 0\}$.
  
  \medskip
  
 {\bf Definition:}  A continua $K \subset M$ is said to have \emph{stable shape} if there exists a cofinal sequence $\e_{\ell} > 0$ for $\ell \geq 1$, such that $\e_{\ell} \to 0$, and  the inclusion  $N_{\e_{\ell}} \subset N_{\ell_1}$ is   a homotopy equivalence for all $\ell \geq \ell_1$.

\medskip

{\bf Example:} The Denjoy minimal set has stable shape  $\cong \mS^1  \vee \mS^1$.
      
\bigskip
    We then realize the original motivation for this work:
  
    
   {\bf Theorem:} [H \& R]   The shape $\cS(\fM)$ is not stable.    
 

   \vfill   

}



\frame % 7
{
  \frametitle{}
  
     
   
{\it Proof:} The shape homotopy group $\pi_1^{sh}(\fM, \omega_1)$ is generated by paths in $\fM_0$ going out to the ends of the embedded tree, so is not stable!

   
  \setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(30,5.8)
  \put(1.8,0.2){\includegraphics[width=70mm]{pix/figure28final.png}}
 \end{picture}
\label{fig:KR7}
 \vspace{-20pt}
\end{figure}



   \vfill   

}

 
\frame % 7
{
  \frametitle{Remarks and Problems}
  
   {\bf Theorem:}  [H \& R] In every $C^1$-neighborhood of $\cK$, there exists a smooth flow $\Phi_t'$ on $\mK$ with positive entropy, and the associated invariant lamination $\fM'$ is the suspension of a horseshoe. There also exists a smooth flow $\Phi_t''$ on $\mK$ with no trapped orbits.
   
   \medskip
   
   {\bf Problem:} Give a description of   the dynamical properties of all flows $C^1$-close to a generic Kuperberg flow.

\medskip
   
    {\bf Theorem:}  [Kuperbergs, 1996] There exists a PL flow on $\mK$ for which the minimal set $\Sigma$ is $1$-dimensional.
    
   \medskip
   
   {\bf Problem:} Characterize the smooth (non-generic!) flows on $\fM$ for which $\Sigma$ is $1$-dimensional.

 \medskip
   
   {\bf Problem:} Understand the topological orbit-equivalence class of the   action $\cGM$ on $\fC$.
      
   \vfill   

}

 

 
 
\frame % 7
{
  \frametitle{Bibliography}
  
$\bullet$  {\footnotesize   H.~Seifert,  ``Closed integral curves in {$3$}-space and isotopic two-dimensional deformations'' ,   Proc. A.M.S., 1, 1950.}\\
$\bullet$  {\footnotesize   F.W.~Wilson, ``On the minimal sets of non-singular vector fields'',   Annals of Math,  1966.}\\
$\bullet$  {\footnotesize   P.~Schweitzer, ``Counterexamples to the {S}eifert conjecture and opening closed leaves of foliations'',   Annals of Math, 1974.}\\
$\bullet$  {\footnotesize   M.~Handel, ``One-dimensional minimal sets and the {S}eifert conjecture'',  Annals of Math,  1980.}\\
$\bullet$  {\footnotesize   J.~Harrison, ``$C^2$ counterexamples to the {S}eifert conjecture'',  Topology,  1988.}\\
$\bullet$  {\footnotesize   K.~Kuperberg, ``A smooth counterexample to the {S}eifert conjecture'',   Annals of Math,   1994.}\\
% $\bullet$  {\footnotesize   G.~Kuperberg and K.~Kuperberg, ``Generalized counterexamples to the {S}eifert conjecture'' ,   Annals of Math,   1996.}\\
$\bullet$  {\footnotesize   {\'E.}~Ghys, ``Construction de champs de vecteurs sans orbite p\'eriodique (d'apr\`es {K}rystyna {K}uperberg)'',    S{\'e}minaire Bourbaki, Vol. 1993/94, Exp.\ No.\ 785, 1995.}\\
$\bullet$  {\footnotesize   S.~Matsumoto, ``{K}uperberg's {$C^\infty$} counterexample to the {S}eifert conjecture'',  S\=ugaku, Mathematical Society of Japan, 1995.}
  
  
 
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}

 


     
 
\end{document}


 