% FOLIATIONS 2016 � Pawel Walczak's 50 years in mathematics
% July 11-17, 2016.
% Wednesday, July 13, 15:00 - 15:30

% slides for talk "On the dynamics of derived from Kuperberg flows"
 
 
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\date[March 11, 2016]


  
\title[]{Dynamics of Derived from Kuperberg flows}


\author{Steve Hurder\\joint work with Ana Rechtman\\ B\c{e}dlewo, Poland -- 13 July  2016}

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

 
\begin{document}

\frame{\titlepage} % # 1


 
  \section{Introduction}
 
 
 

\frame % 1
{
  \frametitle{ }
 At the \emph{International Symposium/Workshop on Geometric Study of Foliations} in November 1993, there was an evening seminar on a recent preprint by Krystyna Kuperberg, proving:

  
\medskip
 
{\bf Theorem (K. Kuperberg, 1994)} {\it Let $M$ be a closed, orientable 3-manifold. Then $M$ admits a $C^{\infty}$  non-vanishing vector field whose flow $\phi_t$ has \underline{no periodic orbits}.}
 
 
\bigskip
 % \pause

 { \footnotesize 
$\bullet$ ~ {Krystyna~Kuperberg},
 {\it A smooth counterexample to the {S}eifert conjecture},
 {\bf Ann. of Math. (2)}, 140:723--732, 1994.  



$\bullet$ ~ {\'Etienne~Ghys},
 {\it 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},
 {\bf Ast\'erisque}, 227: 283--307, 1995.  
 
$\bullet$ ~ {Shigenori~Matsumoto},
 {\it K.{M}. {K}uperberg's {$C^\infty$} counterexample to the {S}eifert conjecture},
{\bf S\=ugaku}, Mathematical Society of Japan, Vol. 47:38--45, 1995.
{Translation:} {\bf Sugaku Expositions}, A.M.S., Vol. 11:39--49, 1998. 
 
 $\bullet$ ~ {Greg \& Krystyna~Kuperberg},
 {\it Generalized counterexamples to the {S}eifert conjecture},
 {\bf Ann. of Math. (2)}, 144:239--268,, 1996.  

 
}

 
\vfill   

}


\frame % 1
{
  \frametitle{ }
  
These papers also studied the dynamics of Kuperberg   flows:
 
  
\bigskip

{\bf Theorem (Ghys, Matsumoto, 1995)} {\it The Kuperberg flow has a \underline{unique minimal set} $\cZ \subset M$.}
 
 \bigskip

{\bf Theorem (Matsumoto, 1995)} {\it The Kuperberg flow has an \underline{open set} $\fW$ \underline{of wandering points} whose forward orbits limit to the unique minimal set.} 

There cannot be an invariant measure equivalent to Lebesgue.
 
 \bigskip
 
  {\bf Theorem (A. Katok, 1980)} {\it Let $M$ be a closed, orientable 3-manifold. Then an aperiodic flow $\phi_t$  on $M$ has  \underline{zero entropy}. }


 

   
 \vfill   

}


\frame % 1
{
  \frametitle{ }
 
 The other ``clue'' to the dynamics of Kuperberg flows, was given by the computer model of Bruno  S\'evennec
 of the intersection of the unique minimal set with a cross section. Very mysterious!
 
 \includegraphics[width=110mm]{pix_bedlewo/minimal-set}
 

\vfill   

}


  \section{Results}


\frame % 1
{
  \frametitle{ }
  
 {\bf Problem:} [Kuperberg]  Describe the topological shape of the unique minimal set $\cZ$. Does $\cZ$ have \emph{stable shape}?
   
   \medskip
   
 This was the motivation for the study  by   Ana Rechtman and  myself of the dynamical properties of Kuperberg flows, starting in 2010: 
  
 \bigskip
 % \pause
 
  { \footnotesize 
  
$\bullet$ ~ {H. \& R.},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque}, Vol. 377.

      
 $\bullet$ ~ {H. \& R.},
 {\it Aperiodic flows at the boundary of chaos},
 {\it  arXiv:1603.07877}.

 $\bullet$ ~ {H. \& R.},
 {\it Perspectives on Kuperberg flows},
 {\it  arXiv:1607.00731}.

 $\bullet$ ~ {H. \& R.},
 {\it The dynamics of Derived from Kuperberg flows},
 {\it  in preparation}.

}

   
 \vfill   

}

   
\frame % 1
{
  \frametitle{}
 
 The first observation of our work was that there are many variations on the Kuperberg construction. 
  A flow is \emph{generic} if it is actually    like its illustrations (to be made precise).
  
  Let $\Phi_t$ be a  {generic} Kuperberg flow on a plug $\mK$.
  
  \medskip
    % \pause
    
  {\bf Theorem (H \& R, 2015)} {\it   There the \emph{unique} minimal set $\fM$ for the flow is a $2$-{dimensional lamination}  ``with boundary''   which is equal to the non-wandering set of $\Phi_t$.}

\medskip

  {\bf Theorem (H \& R, 2015)} {\it The       minimal set $\fM$   has {unstable shape}.}
  
  This implies that it is wildly embedded in $\mR^3$.
 
     
\medskip

  {\bf Theorem (H \& R, 2015)} {\it  The flow $\Phi_t$ has {positive} ``{slow entropy}'', for exponent $\alpha = 1/2$.}   
  
  Thus, a generic Kuperberg flow \emph{almost} has positive entropy.

     
 


\vfill   

}

  
\frame % 1
{
  \frametitle{}
 
  We also considered   ``Derived from Kuperberg'' constructions, which deform a generic Kuperberg flow $\Phi_t = \Phi_t^0$ on a plug $\mK$. 
 
 \medskip
 
{\bf Theorem  (H \& R, 2016)} {\it There is a $C^{\infty}$-family of flows $\Phi_t^{\e}$ on $\mK$, for $-1 < \e \leq 0$, with $\Phi_t^0 = \Phi_t$, such that each flow $\Phi_t^{\e}$ has \underline{two periodic orbits},  and  all   orbits are properly embedded.}
 
 
 \medskip

{\bf Theorem  (H \& R, 2016)} {\it There is a $C^{\infty}$-family of flows $\Phi_t^{\e}$ on $\mK$, for $0 \leq  \e < a$, with $\Phi_t^0 = \Phi_t$, such that each flow $\Phi_t^{\e}$ admits \underline{countably many families of  ``horseshoes''} with dense periodic orbits, and    so has positive entropy.}

  \medskip
% \pause
{\bf Conclusion 1:} {\it The generic Kuperberg flows lie at the boundary of chaos (entropy $> 0$) and the boundary of   tame dynamics.}

  \medskip

{\bf Conclusion 2:} {\it There is still much about the dynamics of \emph{Derived from Kuperberg} flows  that we still don't understand.}

  
 


\vfill   

}

\section{Plugs}


 \frame % 1
{
  \frametitle{}
 
  {\bf Definition:} A plug is a 3-manifold with boundary of the form $P = D \times [-1,1]$ with $D$ a compact surface with boundary. $P$ is endowed with a non-vanishing vector field $\vec{X}$, such that:
  
  
  $\bullet$~ $\vec{X}$ is vertical in a neighborhood of $\partial P$, that is $\vec{X} = \frac{d}{dz}$. Thus $\vec{X}$ is inward transverse along $D \times \{-1\}$ and outward transverse along $D \times \{1\}$, and parallel to the rest of $\partial P$.
  
  $\bullet$~ There is at least one point $p \in D \times \{-1\}$ whose positive orbit is trapped in $P$.
  
  $\bullet$~ If the orbit of $q \in D \times \{-1\}$ is not trapped then its orbit intersects $D \times \{1\}$ in the facing point.
  
    $\bullet$~ There is an embedding of $P$ into $\mR^3$ preserving the vertical direction.




   \vfill   

}

  
     
 \frame % 1
{
  \frametitle{}
 
    
%  \underline{Modified Wilson Plug} $\mW$  
\begin{figure}[!htbp]
\centering
  \underline{Modified Wilson Plug} $\mW$  
  
Consider the rectangle $R \times \mS^1$ with the vector field $\vec{W} = \vec{W}_1 + f \frac{f}{d\theta}$\\
  $f$ is asymmetric in $z$ and  $\vec{W}_1 = g \frac{f}{dz}$ is vertical.

 \includegraphics[width=70mm]{pix_bedlewo/Figure03b.png}
 


\end{figure}

   
 
   \vfill   

}

 
    
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 Grow horns and embed them to obtain Kuperberg Plug $\mK$, \\matching the flow lines on the boundaries.

 

 \includegraphics[width=90mm]{pix_bedlewo/Figure07.png}
\end{figure}
\vspace{-14pt}
Embed so that the Reeb cylinder $\{r=2\}$ is tangent to itself. 
  
  That's it. The resulting flow is aperiodic!
   \vfill   

}

   
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 Close up view of the lower embedding $\sigma_1$

 

 \includegraphics[width=66mm]{pix_bedlewo/Figure06.png}
\end{figure}
\vspace{-14pt}
   
   \vfill   

}

   
     
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
   The insertion map as it appears in the face $E_1$
   \medskip
   
 \includegraphics[width=50mm]{pix_bedlewo/modifiedradius-2.png}
  
 
\end{figure}

 
   {\it  Radius Inequality}: 
   
   For  all $x' = (r', \theta',  -2) \in  L_i$, let $x = (r, \theta,z) =
  \sigma_i^\e(r', \theta',  -2) \in \cL_i$,  then $r < r'$ unless  $x' = (2,\theta_i, -2)$ and then $r=2$.

 
   \vfill   

}

  
   
   
 \frame % 1
{
  \frametitle{}
 
{\bf Definition:}  A \emph{Derived from Kuperberg} (DK) flow is obtained by choosing the embeddings so that we have:
 
 \medskip
 
      {\it Parametrized Radius Inequality}: For all $x' = (r', \theta',  -2) \in  L_i$, let $x = (r, \theta,z) =
  \sigma_i^\e(r', \theta',  -2) \in \cL_i$,  then $r < r'+\e$ unless  $x' = (2,\theta_i, -2)$ and then $r=2+\e$.

\bigskip

  The modified radius inequality for  the cases $\e<0$ and  $\e>0$:
\begin{figure}[!htbp]
\centering

 \includegraphics[width=50mm]{pix_bedlewo/modifiedradius-1.png}
\includegraphics[width=50mm]{pix_bedlewo/modifiedradius-3.png}
\vspace{-6pt}
\end{figure}

   
   \vfill   

}

   
 \frame % 1
{
  \frametitle{}
 
{\bf Proposition:}
{\it Let $\Phi_t^\e$ be a DK flow for which the insertion map satisfies the  Parametrized Radius Inequality with $\e<0$. 
Then  the flow in the plug $\mK_\e$ has two periodic orbits that
  bound an invariant cylinder, and the flow has topological entropy
  zero.}
  
  \medskip
  
{\it Idea of the proof:}  This follows from the   techniques for the standard flow when $\e =0$, which imply that every flow orbit of a point $x$ with radius $r(x) \ne 2$ entering an insertion, exits at the same radius.  

    \vfill   

}



\section{Generic}

  
  
 \frame % 1
{
  \frametitle{}
 
There are two types of \emph{generic} conditions.

{\bf Generic 1:}
Consider the rectangle ${\bf R} =  [1,3] \times [-2,2]$,  with a   vertical vector field $\vec{W}_1 = g \frac{f}{dz}$ where $g(r,z)$ vanishes at $(2,-1)$ and $(2,+1)$. We require that $g$   \emph{vanish  to second order with positive definite Jacobian} at these two points.

Then the Wilson field on $\mW = {\bf R} \times \mS^1$ is  $\vec{W} = \vec{W}_1 + f \frac{f}{d\theta}$ where 
  $f$ is asymmetric in $z$, and vanishes near the boundary.

  
  \medskip
  
  {\bf Generic 2:} These are conditions on the insertion maps $\sigma_i \colon D_i \to \cD_i$. We require that the $r$-coordinate of the image \emph{depends quadratically} on the $\theta$-coordinate of the domain, for values of $r$ near $r=2$. Much stronger than the basic radius inequality.
  
    \medskip
  
  {\bf Problem:} {\it What are the dynamical properties of the non-generic Kuperberg flows? Shape of its minimal set?}
  
   \vfill   

}
 
 \section{Pseudogroups}
 
 
 \frame % 1
{
  \frametitle{}
 
 The basic idea for the study of the minimal set for a Kuperberg flow, and the dynamics of a Derived from Kuperberg flow $\Phi_t^\e$ where $\e > 0$, is to compare  the dynamics of the flow  $\Phi_t^0$ with that of an induced map on a (partial) section to the flow.     

\medskip

Return map of a flow    $\Phi_t^\e$ induces a smooth pseudogroup $\cG_{\Phi^\e}$ on $\bRt$

\medskip

{\bf Critical difficulty:}  {\it There is not always a direct relation between the continuous dynamics of the flow $\Phi_t^\e$ and the discrete dynamics of the action of the pseudogroup $\cG_{\Phi^\e}$.}

   \vfill   

}
 
  

  


 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 The section ${\bf R}_0 \subset \mK$ used to define pseudogroup $\cG_{\Phi^\e}$.
  
 
 \includegraphics[width=110mm]{pix_bedlewo/Figure15.png}
 \end{figure}

\vspace{-14pt}
 
 The flow of $\Phi_t^\e$ is tangent to ${\bf R}_0$ along the center plane $\{z=0\}$, so the action of the pseudogroup has singularities along this line. 

 
   \vfill   

}
  

 
  
  
  \frame % 1
{
  \frametitle{}
We consider two maps with domain in $\bRt$

\medskip
  $\bullet$~ $\psi$ which is the return map of the \emph{Wilson flow} $\Psi_t$

  $\bullet$~ $\phi_1^\e$ which is the return map of the \emph{Kuperberg flow} $\Phi_t^e$ for orbits that go through the entry region $E_1$

   \medskip
   
   Form the pseudogroup they generate $\widehat{\cG}_\e = \langle \psi, \phi_1^\e\rangle$.
   
{\bf Proposition:} The restriction of   $\widehat{\cG}_\e$ to the region $\{r> 2\} \cap \bRt$ is a sub-pseudogroup of $\cG_{\Phi^\e}$
   
      \vfill   

}


  
 

  
 \frame % 1
{
  \frametitle{}
 
Action of  $\widehat{\cG}_0 = \langle \psi, \phi_1^\e\rangle$   on the line $r=2$ for $\e = 0$.


  \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=60mm]{pix_bedlewo/dynamicsr=2.png}
 
 \vspace{-14pt}
 
\end{figure}

  This looks like a ping-pong game, except that the play action is too slow to generate entropy. 

 
   
 
   \vfill   

}

   

  
 \frame % 1
{
  \frametitle{}
 
 Action of  $\widehat{\cG}_\e = \langle \psi, \phi_1^\e\rangle$   on the line $r=2$ for $\e > 0$.

  \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=60mm]{pix_bedlewo/dynamicsr=2e.png}
  
     
 \vspace{-14pt}

\end{figure}

 
   The dynamics of this action is actually too complicated to draw precisely, or calculate with.
   

   
 
   \vfill   

}

   \section{Horseshoes}
  
\frame % 1
{
  \frametitle{ }
 
Define a compact region  $U_0 \subset \bRt$ which is mapped to itself by the map $\vp = \psi^k \circ \phi_1^\e$ for $k$ sufficiently large.
 
   \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=70mm]{pix_bedlewo/horses1.png}
  
     
 
\end{figure}

\vfill   

 }
  
  
  
\frame % 1
{
  \frametitle{ }
 
 The images of the powers  $\vp^\ell$ of the map the map  $\vp$ form a $\delta$-separated set for the action of the pseudogroup $\widehat{\cG}_\e$.
 
  \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=70mm]{pix_bedlewo/horses2.png}
  
     
 
\end{figure}


\vfill   

 }
  
     
 \frame % 1
{
  \frametitle{}

 
 
 This is a horseshoe dynamical system for the \emph{pseudogroup action}: 
 
 \medskip


 \begin{figure}[!htbp]
\centering
{


 \includegraphics[width=80mm]{pix_bedlewo/horseshoe1.png}
 }
 \end{figure}




\vfill   

}

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

{\bf Definition:}  Let $\fZ \subset X$ be a continuum embedded in a metric space $X$.
  A \emph{shape approximation}   of $\fZ$ is a 
 sequence  $\fU = \{U_{\ell} \mid \ell =1,2,\ldots\}$  satisfying the conditions:
\begin{enumerate}
\item each $U_{\ell}$ is an open neighborhood of $\fZ$ in $X$ which is homotopy equivalent to a compact polyhedron;
\item $U_{\ell +1} \subset U_{\ell}$ for $\ell \geq 1$, and their closures satisfy $\ds \bigcap_{\ell \geq 1} ~ \oU_{\ell} = \fZ$.
\end{enumerate}

Shape homotopy groups:  
$$\widehat{\pi}_1(\cZ , x_*) \equiv \varprojlim \ \left\{ \pi_1(U_{\ell+1} , x_*) \to \pi_1(U_{\ell} , x_*)\right\} $$
 

{\bf Meta-Principle:} For each class $[\gamma] \in \widehat{\pi}_1(\fM , \omega_0)$ there is a horseshoe subdynamics for the pseudogroup   $\cG_{\Phi^\e}$ acting on $\bRt$ with a periodic orbit defining $[\gamma]$.
 
\vfill   

 }
  
  
\frame % 1
{
  \frametitle{ }
  
 Conclude with two remarks: 
 
 \medskip
 
 $\bullet$ ~ For $\e < 0$, the dynamics of a DK flow is tame, and completely predictable, except that as $\e \to 0$ the dynamics approaches that of the Kuperberg flow.
 
 \medskip
 
  $\bullet$ ~ For $\e > 0$, the dynamics of a DK flow is chaotic, but making calculations of entropy for example, is only possible in special instances. Also, we have no intuition, for example, of how to describe the nonwandering sets for DK flow with $\e > 0$.
 
 % \pause
 
 \begin{center}
\Large{ Thank you for your attention!}
\end{center} 

 
 
\vfill   

 }
  
   
  
\end{document}

