% 31st Summer Conference on Topology and its Applications 


% August 1-5, 2016.
% Thursday, August 4, 2016

% slides for talk "Perspectives on Kuperberg flows"
 
 
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\date[March 11, 2016]


  
\title[]{Perspectives on Kuperberg flows}


\author{Steve Hurder\\{\small joint work with Ana Rechtman\\ Leicester, UK -- 4 August 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} {\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}.}

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




\bigskip
 
The result seemed to appear miraculously out of the aether.

% \pause

Ana Rechtman and I   investigate these flows in our manuscript: 
  
$\bullet$ ~ {H. \& R.},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque}, Vol. 377, 2016; 250 pages.


   
\vfill   

}


\frame % 2
{
  \frametitle{Wilson Plug}
 
  Kuperberg's proof  introduced a very clever construction of an aperiodic ``plug'' for    flows on 3-manifolds,  to cut open periodic orbits.
The construction of   plugs has a history.

% \pause

 Wilson's fundamental idea in 1966 was the construction of a plug which trapped content, and all trapped orbits have limit set a periodic orbit contained in the plug.    The   result of repeatedly inserting this plug is a flow with at most two periodic orbits.
 
 \begin{center}
\includegraphics[height=40mm]{pix_leicester/Wilson_flow.png}\qquad
\includegraphics[height=40mm]{pix_leicester/Wilson_merged_orbits.png}
\end{center}


 
  \vfill   

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

  
    

  Paul Schweitzer in 1974 had two deep insights. In the Wilson Plug: 

$\bullet$ Each periodic circle  can be replaced by  a Denjoy minimal set for a flow    on a punctured $2$-torus;

$\bullet$ the new minimal set does not have to be in a planar flow, but may be contained in a surface flow which embeds in $\mR^3$.
  
   
  
   
 \begin{center}
\includegraphics[width=0.5\textwidth]{pix_leicester/Schweitzer_plug.png}
\end{center}
  
   

  
 
   \vfill   

}


\frame % 3
{
  \frametitle{}

   
  {\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. 
  
\bigskip

% \pause

An alternate embedding  of the Denjoy minimal set in $\mR^3$ was later used by Jenny Harrison in a very difficult proof to show the same result for $C^2$-flows.

 $\bullet$ ~ {J.C.~Harrison},
 {\it {$C^2$} counterexamples to the {S}eifert conjecture},
 {\bf Topology}, 27:249--278, 1988.  
 
 
 \bigskip
 
 So, the reaction to Krystyna's short paper, proving ever so much more, was pure astonishment. 
 
  
  \vfill   

}

 
 
\frame % 7
{
  \frametitle{Kuperberg's ``Big Idea''}
  
    
Shigenori Matsumoto's summary: 
 \begin{center}
\includegraphics[width=\textwidth]{pix_leicester/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 % 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=54mm]{pix_leicester/Figure03b.png}
 
The two periodic orbits are now unstable.

\end{figure}

   
 
   \vfill   

}


\frame % 6
{
 \frametitle{}
  
   \begin{center}
   Deform the modified Wilson Plug to have a pair of ``horns''
   
   \medskip
   
\includegraphics[width= .6\textwidth]{pix_leicester/Kuperberg_ingredients2.png}
\end{center}
  
  \vfill   

}


\frame % 6
{
 \frametitle{ }
  
     
 

   
   \begin{center}
     Insert the horns with a twist and a bend, matching  the flow lines on the boundaries, to obtain Kuperberg Plug $\mK$


\medskip


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

}

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

 

 \includegraphics[width=66mm]{pix_leicester/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_leicester/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{ }

Three subsequent papers   explored the properties of these flows:
 
 \bigskip
 
   { \footnotesize 
 
$\bullet$ ~ {\'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.  

\medskip
 
$\bullet$ ~ {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. 
 
 \medskip
 
 $\bullet$ ~ {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!
 
     
\begin{center}
\includegraphics[width=.5\textwidth]{pix_leicester/MinimalSet_model.png}
\end{center}

 

\vfill   

}

  

  \section{Questions}

 

\frame % 1
{
  \frametitle{ }
  
     
{\bf Problem:}   Describe the geometric properties of the unique minimal set $\cZ$.  For example, when is the minimal set $1$-dimensional? Or, is it always $2$-dimensional, and then what is the growth type of the path connected components of the minimal set?
   
   \medskip
   
 {\bf Problem:}   Describe the topological shape of the unique minimal set $\cZ$. Does $\cZ$ have \emph{stable shape}? Is the shape of $\cZ$ \emph{movable}, a notion introduced by Karol Borsuk.
    
     
   \medskip
   
    
$\bullet$ ~   {\it On movable compacta}, {\bf  Fund. Math.}, 66:137--146, 1969.

 
\vfill   

}


\frame % 1
{
  \frametitle{Karol Borsuk}
  
  
{\footnotesize 
  Karol Borsuk (May 8, 1905 -- January 24, 1982) was a Polish mathematician. His main interest was topology.\\
  
\smallskip

Borsuk introduced the theory of absolute retracts (AR's) and absolute neighborhood retracts (ANR's), and the cohomotopy groups, later called Borsuk-Spanier cohomotopy groups. He also founded the so-called Shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century.\\

\smallskip
Borsuk received his master's degree and doctorate from Warsaw University in 1927 and 1930, respectively; his Ph.D. thesis advisor was Stefan Mazurkiewicz. He was a member of the Polish Academy of Sciences from 1952. Borsuk's students included Samuel Eilenberg, Jan Jaworowski, \emph{Krystyna Kuperberg}, W{\l}odzimierz Kuperberg, and Andrzej Trybulec.
 
}
    
    $\mathtt{https://en.wikipedia.org/wiki/Karol~Borsuk}  $ 

    
   \medskip
   
 
 
\vfill   

}

 
\frame % 1
{
  \frametitle{ }
  
     
{\bf Problem:}   Describe the geometric properties of the unique minimal set $\cZ$.  For example, when is the minimal set $1$-dimensional? Or, is it always $2$-dimensional, and then what is the growth type of the path connected components of the minimal set?
   
   \medskip
   
 {\bf Problem:}   Describe the topological shape of the unique minimal set $\cZ$. Does $\cZ$ have \emph{stable shape}? Is the shape of $\cZ$ \emph{movable}?   
   \medskip

{\bf Problem:}  Why does a Kuperberg flow have zero entropy? For example, can one calculate the growth rates of $\e$-separated sets for the flow?  
   
   \medskip
   
{\bf Problem:}   What are the dynamical properties of the flows which are smooth perturbations of a Kuperberg flow? These are called ``Derived from Kuperberg'' flows, or DK flows.
   
   \medskip
   
 
 
\vfill   

}
 
 

\frame % 1
{
  \frametitle{ }

The question of the shape properties of the unique minimal set $\cZ$ was suggested by Alex Clark, and was the  motivation for the study  by   Ana Rechtman and  myself, starting in 2010,  of the dynamical properties of Kuperberg flows.
What we discovered about the shape properties of this minimal set, it  would have made   Borsuk proud of Krystyna's mathematical legacy!
  
 \bigskip
 
 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. That is to say, all choices are assumed to be  not too pathological.  

 
   
 \vfill   

}

  
 \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.
  
    \vfill   

}
 
    
\frame % 1
{
  \frametitle{}
 
    
  Let $\Phi_t$ be a  {generic} Kuperberg flow on a plug $\mK$.
  
  \medskip
     
  {\bf Theorem (H \& R, 2015)} {\it   The \emph{unique} minimal set $\cZ$ for the flow is a $2$-{dimensional lamination}  ``with dense boundary''   and $\cZ$  equals  the non-wandering set of $\Phi_t$.}

      
\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.

     
 \medskip

  {\bf Theorem (H \& R, 2015)} {\it The       minimal set $\cZ$   has {unstable shape}; but may be moveable.}
  
  This implies that it is wildly embedded in $\mR^3$.
 

    \medskip
  
  % \pause
  
  {\bf Problem:} {\it How do the dynamical properties of the non-generic Kuperberg flows differ from those of  the generic flows?}
  


\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.}

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



\vfill   

}
   
  

\frame % 1
{
  \frametitle{}
 
  
  \medskip
  
 There is still much about the dynamics of \emph{Derived from Kuperberg} flows  that we still don't understand. 

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

\medskip

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


\bigskip

For the rest of this talk, will try to explain why the Kuperberg flows have unstable minimal sets, and what this means for the dynamical properties of these flows.

\vfill   

}
   
  
 
 \section{Lamination}
 
 
  
 \frame % 1
{
  \frametitle{The hidden dynamics of Kuperberg flows}
 
    Define the closed subsets of $\ds \mW=[1,3] \times \mS^1\times[-2,2] \cong {\mathbf R} \times \mS^1$

\medskip
 $\cD_i = \sigma_i(D_i)$ for $i =1,2$   are   solid $3$-disks embedded in $\mW$.
$$
\mW' ~   \equiv   ~ \mW - \left \{ \cD_1  \cup \cD_2 \right\} \quad , \quad 
\wmW~   \equiv   ~ \overline{\mW - \left \{ \cD_1  \cup \cD_2 \right\}} ~ .
$$
\begin{eqnarray*}
\cC ~ & \equiv & ~  \{r=2\}   \quad \text{[\emph{Full ~ Cylinder}]}\\
\cR ~ & \equiv & ~  \{(2,\theta,z) \mid  -1 \leq z \leq 1\} \quad \text{[\emph{Reeb ~ Cylinder}]}\\
\cR' ~ & \equiv & ~  \cR \cap \wmW \quad \text{[\emph{Notched ~ Reeb ~ Cylinder}]}\\
\cO_i ~ & \equiv & ~  \{(2,\theta,(-1)^i)  \},   i=1,2  \quad \text{[\emph{Periodic Orbits}]}
\end{eqnarray*}
  $\cO_1$ is the lower   boundary circle  of the Reeb cylinder $\cR$,  \\ $\cO_2$ is the upper boundary circle. 

 
 
   \vfill   

}

 
 \frame % 1
{
  \frametitle{}
 
   \begin{figure}[!htbp]
\centering
 
  
 
 \includegraphics[width=70mm]{pix_leicester/Figure17.png}
 \end{figure}

 
     
 
 
   \vfill   

}
  

 \frame % 1
{
  \frametitle{}
 
   \begin{figure}[!htbp]
\centering
 
  The image $\tau(\cR') \subset \mK$
 
 \medskip
 
 \includegraphics[width=100mm]{pix_leicester/Figure23.png}
 \end{figure}

 
     
 
 
   \vfill   

}
  

 \frame % 1
{
  \frametitle{}
 
  
     
Consider  the $\cK$-orbit of the image   $\tau(\cR') \subset \mK$ and its closure
$$
\fM_0 ~ \equiv ~  \{\Phi_t(\tau(\cR')) \mid -\infty < t < \infty \} \quad  , \quad  \fM ~ \equiv ~ \overline{\fM_0}     \subset \mK ~.
$$

Then $\cO_i \cap \wmW \subset \cR'$ hence the minimal set $\cZ \subset \fM$. 
 
The non-wandering set of $\Phi_t$ in $\mK$ is denoted $\Omega$.

\medskip

{\bf Theorem:} 
{\it Let $\Phi_t$ be a generic Kuperberg flow,  then   
$\cZ =   \Omega = \fM$.}
 
 \medskip
 
{\bf Theorem:} 
{\it For a generic Kuperberg flow,  the  space $\fM$ has the structure of a   \underline{zippered  lamination}  with $2$-dimensional  leaves. }  
 
{\it Proof:} The closures of the boundary orbits of $\fM_0$ are dense in $\fM$. Hence, the boundary of each leaf of $\fM$ is dense in itself. 

    \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}

\medskip

   
   % \pause
   
  
{\bf Proposition:} {\it The shape of $\cZ$ is equal to the shape of $\fM_0$.}

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 Strategy:} Use ``almost closed paths'' in $\fM_0$ to calculate $\widehat{\pi}_1(\cZ , x_*)$.
 
\vfill   

 }
  
  

 \frame % 1
{
  \frametitle{}
 
  {\bf Problem:} Give an explicit description of the embedded space $\fM_0$.
  
  This is done in the Ast\'erisque volume. The key observation is that $\fM_0$ is an infinite union of propellers at increasing levels. We give  a sequence of illustrations to suggest how this works. 
  
     \begin{figure}[!htbp]
\centering

Finite propeller = flow of entry arc $\gamma$ in region $r > 2$ 
   
 \medskip
 
 \includegraphics[width=56mm]{pix_leicester/Figure18a.png}
 \end{figure}

 

     \vfill   

}
  
  \frame % 1
{
  \frametitle{}
 
      \begin{figure}[!htbp]
\centering

 Infinite propeller with infinite sequence of notches to which finite propellers are attached. 
   
 \medskip
 
 \includegraphics[width=100mm]{pix_leicester/Figure24.png}
 \end{figure}


   
   \vfill   

}



 \frame % 1
{
  \frametitle{}
 
  
       \begin{figure}[!htbp]
\centering

 The surface $\fM_0$ is  embedded in $\mK$, where each propeller wraps around the Reeb cylinder $\tau(\cR')$.
   
 \medskip
 
 \includegraphics[width=90mm]{pix_leicester/Figure27.png}
 \end{figure}


  
     \vfill   

}
  

 \frame % 1
{
  \frametitle{}
 
      \begin{figure}[!htbp]
\centering

An approximate cycle is an ``almost closed'' path  with endpoints sufficiently close.
This  yields a class in  $\widehat{\pi}_1(\cZ , x_*)$ by ``translating the loop out to infinity in open neighborhoods of $\fM_0$''.
   
 \medskip
 
 \includegraphics[width=70mm]{pix_leicester/Figure56a.png}
 \end{figure}

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

Need a systematic method to label the classes that arise from almost closed paths in $\fM_0$.
  {\bf   Idea:}  Introduce a section $\bRt \subset \mK$ to the flow $\Phi_t$ and study the intersections of $\fM_0 \cap \bRt$.
   
 \begin{figure}[!htbp]
\centering
 
  
 
 \includegraphics[width=90mm]{pix_leicester/Figure15.png}
 \end{figure}

 
 
The return map to the section   defines a pseudogroup $\cG_{\Phi}$
    
 
 
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     The flow of $\Phi_t$ is tangent to ${\bf R}_0$ along the center plane $\{z=0\}$, so the action of the pseudogroup has singularities along this line. 


\medskip

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

\bigskip

None the less, the introduction of the Kuperberg pseudogroup $\cG_{\Phi}$ is a fundamental tool for the study of the dynamics of the flow $\Phi_t$.

\medskip


The semi-group formed by  the generators of $\cG_{\Phi}$ are used to give a complete description of the embedded space $\fM_0$.
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}
 
 
  
  
  
  
  \frame % 1
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  \frametitle{}
We consider two maps with domain in $\bRt$

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

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

   \medskip
   
 They generate a pseudogroup $\whcG = \langle \psi, \phi_1\rangle$ acting on $\bRt$.
   
{\bf Proposition:} {\it  The restriction of   $\widehat{\cG}$ to the region $\{r> 2\} \cap \bRt$ is a sub-pseudogroup of $\cG_{\Phi}$
The action of $\whcG$ on the line segment $\cC \cap \bRt$   yields   families of nested ellipses containing $\bRt \cap \fM_0$.}

\medskip

{\bf Corollary:} {\it The elements of the pseudogroup $\whcG$ labels the lower branches of the tree structure of $\fM_0$.}


      \vfill   

}


  
 

  
 \frame % 1
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  \frametitle{}
 
 

  \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=66mm]{pix_leicester/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. 

 
   
 
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     \section{Horseshoes}
 
    
   
 \frame % 1
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{\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_leicester/modifiedradius-1.png}
\includegraphics[width=50mm]{pix_leicester/modifiedradius-3.png}
\vspace{-6pt}
\end{figure}

   
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}

 

  
 \frame % 1
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  \frametitle{}
 
 

{\bf Meta-Principle:} For $\e > 0$ and ``most'' classes $[\gamma] \in \widehat{\pi}_1(\fM , \omega_0)$, there is a horseshoe subdynamics for the pseudogroup       $\whcG_\e = \langle \psi, \phi_1^\e \rangle$  acting on $\bRt$ with a periodic orbit defining $[\gamma]$.


 Action of   $\whcG_\e$  on the line $r=2$ for $\e > 0$.

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

\end{figure}

     

   
 
   \vfill   

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\frame % 1
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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. ``$k$ large'' corresponds to translating the almost closed path far out in the surface $\fM_0$.
 
   \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=66mm]{pix_leicester/horses1.png}
  
     
 
\end{figure}

\vfill   

 }
  
  
  
\frame % 1
{
  \frametitle{ }
 
 The images of the powers  $\vp^\ell$   form a $\delta$-separated set for the action of the pseudogroup $\widehat{\cG}_\e$.
 
  \begin{figure}[!htbp]
\centering
    \medskip
   
 \includegraphics[width=66mm]{pix_leicester/horses2.png}
  
     
 
\end{figure}


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 }
  
     
 \frame % 1
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 This is a horseshoe dynamical system for the \emph{pseudogroup action}!
 
 Compare to the usual illustration of a horseshoe dynamical system. 
 
 \medskip


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


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




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}

   
  
\frame % 1
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  \frametitle{ }
  
 Concluding 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 $\e > 0$.
 
   
 \medskip
 
  $\bullet$ ~   There are many more open questions and results to prove about Kuperberg flows and their variations!
   
   
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 }
  
   
  
  
 
    
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\bigskip

   
  
 \begin{center}
\Large{ Thank you Krystyna for your flows!!}
\end{center} 

  




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\end{document}

