% slides completed July 23, 2011
% for NYC Meeting seminar




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\date[July 28, 2011]


  
\title[Kuperberg minimal sets]{Renormalization and dimension \\for Kuperberg minimal sets}


\author{Steven Hurder \& Ana Rechtman}

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



\begin{document}

\frame{\titlepage} % # 1



 \section{Construction}
 
  
 
\frame % 1
{
 \frametitle{Kuperberg's Theorem}

{\bf Theorem:} [Kuperberg, 1994] Let $M$ be a closed  $3$-manifold $M$. Then there exists a smooth, non-vanishing  vector $\cX$ on $M$ with no periodic orbits.  

 
   
  
 
\setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(40,3.7)
\put(1.5,0.1){\includegraphics[width=80mm]{pix/Kuperberg_ghys.png}}
 \end{picture}
 \vspace{-20pt}
\end{figure}

  \vfill   

}

  

\frame % 2
{
  \frametitle{Wilson Plug}
 
 Wilson's fundamental idea was the construction of a plug which trapped content, and all trapped orbits have limit set a periodic orbit contained in the plug.    The two periodic orbits are attractors:
 
 \begin{center}
\includegraphics[width=0.2\textwidth]{pix/Wilson_flow.png}\qquad
\includegraphics[width=0.5\textwidth]{pix/Wilson_merged_orbits.png}
\end{center}


 
  \vfill   

}
 
\frame % 3
{
  \frametitle{Schweitzer's Theorem}

  
  
  {\bf Theorem:} [Schweitzer, 1974] Every homotopy class of non-singular vector fields on a closed $3$-manifold $M$ contains a $C^1$-vector field with no closed orbits. 
  
\medskip
 

In early 1970's, Paul Schweitzer had two deep insights: 

In the Wilson Plug: 

$\bullet$ the periodic circles can be replaced by a minimal set for a flow  without periodic orbits, such as the Denjoy minimal set;

$\bullet$ the new minimal set does not have to be in a planar flow, but may be contained in a surface flow which embeds in $\mR^3$.
  
  
   
  
 
   \vfill   

}


\frame % 3
{
  \frametitle{Schweitzer Plug}

   
  The circular orbits of the Wilson Plug are replaced by Denjoy minimal sets, embedded as follows:
  
   
 \begin{center}
\includegraphics[width=0.5\textwidth]{pix/Schweitzer_plug.png}
\end{center}


  
  \vfill   

}

 
 
\frame % 7
{
  \frametitle{Kuperberg's ``Big Idea''}
  
    
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   

}

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

 The  vector field $\cW
 = f(r,z) \partial /\partial z + g(r,z) \partial /\partial \theta$ on the plug $(r,\theta, z) \in [1,3] \times \mS^1 \times [-2,2] = W$ 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 % 6
{
 \frametitle{Form and twist the horns}
  
   \begin{center}
\includegraphics[width= .6\textwidth]{pix/Kuperberg_ingredients2.png}
\end{center}
  
  \vfill   

}


\frame % 6
{
 \frametitle{Insert the twisted horns}
  
 The vector field $\cW
$ induces a field $\cK$ on the surgered manifold.  
  
   \begin{center}
\includegraphics[width=.7\textwidth]{pix/Kuperberg_plug22.png}
\end{center}
  
  \vfill   

}

 
 
 
  
 \section{Dynamics of the   Plug}


\frame % 7
{
  \frametitle{Minimal set for the Kuperberg Plug}


If $x \in P$ has forward orbit trapped, then the $\omega$-limit set $\omega_x$ is a closed invariant subset of the interior of $P$, hence contains a minimal set $K$ for the flow. 

 {\bf Theorem:} 
   The  Kuperberg flow has a unique minimal set $\fK$, given by the closure of the orbit of the image of a \emph{special point} from either of the periodic orbits in the Wilson Plug.

\medskip
 

{\bf Problem:} Describe the dynamics of $\cX$ restricted to $\fK$.  

{\bf Problem:}  What is the topological shape of $\fK$?

 
  
 
   \vfill   

}
 


 

\frame % 7
{
  \frametitle{Computer model of $K$}
  
     
\begin{center}
\includegraphics[width=.5\textwidth]{pix/MinimalSet_model.png}
\end{center}

 \vfill   

}
 

\frame % 7
{
  \frametitle{Zippered laminations}
  
   
    {\bf Theorem:}[H \& R]
   The   Kuperberg flow preserves a ``zippered lamination'' $\fM$, where:
   
   $\bullet$ $\fM$ is defined by the closure of the orbit of the   Reeb Cylinder. 
   
   $\bullet$ The boundary   $\cZ = \partial \fM$ is the flow of a Cantor set (the \emph{zipper}),   
   defined by   $2$-dimensional  ping-pong dynamics. 
   
   $\bullet$ The dynamics of flow $\cX$ restricted to $\fM$ is ``Denjoy type'' - either polynomial (dense) or exponential class (Cantor)
   
    \medskip
     
    
   {\bf Theorem:}[H \& R] The minimal set $\fK$ has \emph{unstable shape}.
       
    \vfill   

}

 
 


\frame % 7
{
  \frametitle{The minimal set}
  
   
   {\bf Theorem:}[H \& R]  \quad  $K =    \overline{\cL}$ for  a ``generic'' Kuperberg Plug.
  
  By generic, we mean that the singularities for the vanishing of the vertical vector field $\cW$ used to define $\cK$ are quadratic type, and the insertion yields a quadratic radius function.
  
   The two papers below contain particular constructions of Kuperberg Plugs   which discuss the existence of open disks in $K$.
   
 $\bullet$  {\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   

}


\section{Strategy}

\frame % 7
{
  \frametitle{Families of orbits}
  
 
 Strategy is to consider the orbits of transversal segments to the Kuperberg flow, obtaining global dynamical information as well. 
  
   
Radius inequality:     $\overline{r}(x) \geq r(x)$, where $\overline{r}(x)$ is the radius of the image of $x$ in   the inserted region, with equality only at the periodic orbit entry point. Thus, passing through the face of an insertion   increases the radius.
         
\begin{center}
\includegraphics[width=.4\textwidth]{pix/radius_kuperberg.png}
\end{center}

    \vfill   

}
 
\frame % 7
{
  \frametitle{Tongues in Wilson Plug}

    \begin{center}
\includegraphics[width=.45\textwidth]{pix/Bands3.png}
\end{center}

   \vfill   

}


\frame % 7
{
  \frametitle{Propellors in Kuperberg  Plug}
  When a tongue as above is combined with the recursive Wilson dynamics of the Kuperberg flow,  we obtain a ``propellor'':
   \begin{center}
\includegraphics[width=.5\textwidth]{pix/propellor.png}
\end{center}

   \vfill   

}

\frame % 7
{
  \frametitle{Propellors produce leaves of the lamination}
The orbit of the Reeb cylinder $\cR = \{-1 \leq z \leq 1\}$ produces arcs:
  
   \begin{center}
\includegraphics[width=.3\textwidth]{pix/Arcs1.png}
\end{center}

   \vfill   

}

\frame % 7
{
  \frametitle{Ellipses capture full dynamics}
The orbit of the full cylinder $\cC = \{-2 \leq z \leq 2\}$ produces ellipses.
  
  Going to tip of propellor $\equiv$ moving toward left most  ellipse.
  
   \begin{center}
\includegraphics[width=.3\textwidth]{pix/Ellipses5.png}
\end{center}

   \vfill   

}

\frame % 7
{
  \frametitle{Re-insertion of the propellors}
Dynamics are \emph{renormalizable}, so nesting within nesting  \ldots
  
    Left arcs are lamination $\fM$. ~ Right arcs are recurrence sets. 
 
   \begin{center}
\includegraphics[width=.5\textwidth]{pix/transform2.png}
\end{center}

   \vfill   

}
  
 
 
\section{Renormalization}

\frame % 7
{
  \frametitle{Coding}
  
 The pseudogroup dynamics of the flow admits a coding. 
 
 $E_i$ are entry faces of insertions, $S_i$ are exit faces of insertions.
 
 Orbits in $\fK$ have quasi-tiling property: ``tiles'' are the codes for Wilson orbit segments.
  
\setlength{\unitlength}{1cm}
\begin{figure}[H]
 \begin{picture}(40,3.4)
\put(3,0.1){\includegraphics[width=50mm]{pix/Coding.png}}
 \end{picture}
 \vspace{-20pt}
\end{figure}

   \vfill   

}

 
\section{Renormalization}

\frame % 7
{
  \frametitle{Wilson sub-dynamics}
  
  Let $\Phi_t$ denote flow of $\cK$.
   Let $p_1$ be special point - image of Wilson periodic orbit in transversal $E_1$.
 
 The there  exists times $0 < s_1 < s_2 < \cdots$ with $s_{\ell} \to \infty$ very rapidly, \\
 $\xi_{\ell} = \Phi_{s_{\ell}}(p_1)$ with, for $(r,\theta, z)$ coordinates on $E_1$ 
 
  $\bullet$ ~ radius $r(\xi_{\ell}) = r(p_1)$ constant, 
 
  $\bullet$ ~ angle $\theta(\xi_{\ell}) = \theta(\xi_1)$ constant;
 
 $\bullet$ ~ height $z(\xi_{\ell})$ monotone increasing to $-1 = z(p_1)$. 
 
  $\bullet$ ~ $| z(\xi_{\ell +1}) - z(\xi_{\ell})|$ decreases polynomial speed as $\ell \to \infty$.
   \vfill   

}


\frame % 7
{
  \frametitle{Re-insertion dynamics}
  
  $K_1 \colon E_1 \to E_1$ is reinsertion map.
   $\psi_i  = \Phi_{s_{\ell_i}} \circ K_1 \circ  \Phi_{s_{\ell_0}}$ yields
   
     \begin{center}
\includegraphics[width=.5\textwidth]{pix/transform2.png}
\end{center}

 
 
    \vfill   

}

  
\frame % 7
{
  \frametitle{Ping Pong dynamics}
  
 Pseudogroup generated by $\langle \psi_i , \psi_j \rangle$ for $i \ne j$ large generates ``free'' semi-sub-pseuodogroup.
 
$\bullet$ ~ The problem is that the domains are planar, not linear.
  
 $\bullet$ ~ Orbits are chaotic in open neighborhood of special point $p_1$ - the renormalization point. 
 
 $\bullet$ ~ Chaos is seen in tips of propellors. 
 
    \vfill   

}
  
\frame % 7
{
  \frametitle{Almost homoclinic dynamics}
 
The dynamics generated by $\psi_i$ maps resemble that of homoclinic tangencies of partially hyperbolic systems:
  
     \begin{center}
\includegraphics[width=.4\textwidth]{pix/homoclinic.png}
\end{center}

 
    \vfill   

}
  
\frame % 7
{
  \frametitle{Reeb circle}
  
The circle $\cS = \{(r,\theta, z) \mid ~ r=2 ~ , ~ z=0\}$ is called Reeb circle.
 
$\bullet$ ~Wilson orbit for $x$ with $0 < |r(x) -2 | < \e$ traverses $\cS$ within $\e$.

$\bullet$ ~Wilson flow is rotation on   circles  $\{(r,\theta, z) \mid ~ r \ne 2   , ~ z=cst\}$.

$\bullet$ ~ Flow of $\cS$ under $\Phi_t$ defines lamination $\fM$ (up to closure).
     \begin{center}
\includegraphics[width=.4\textwidth]{pix/Wilson_kuperberg.png}
\end{center}

    \vfill   

}
  
\frame % 7
{
  \frametitle{Density}
  
{\bf  Proposition:} The $\Phi_t$-orbits of special point is dense in an open tubular neighbor hood of $\cS$.
 
 \emph{Proof} (by G. \& K. Kuperberg): Choose vector field $\cK$ precisely, then show explicitly.
 
 \emph{Proof} (by \'{E}. Ghys): Choose vector field $\cK$ carefully, then calculate and invoke Baire for subset of insertions. 
 
 \emph{Proof} (by H. \& R.): Choose vector field $\cK$ and insertions generically, then special orbits $\{\xi_{\ell}\}$ have polynomial density, and follow image of this set under renormalization to get density in arbitrary  open neighborhoods  of points in the Reeb circle $\cS$. 
 
 
 
    \vfill   

}

\section{Bibliography}

\frame % 7
{
  \frametitle{Bibliography}
  {\small
  
$\bullet$  {\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   K.~Kuperberg, ``A smooth counterexample to the {S}eifert conjecture'',   Annals of Math,   1994.}\\
$\bullet$  {\footnotesize   G.~Kuperberg \& 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.}\\
$\bullet$  {\footnotesize   S.~Hurder \&  A.~Rechtman, ``Dynamics and topology of the Kuperberg minimal set'', 2011.}
  }
  
 
   \vfill   

}

 


     
 
\end{document}


 