% 50th Spring Topology and Dynamical Systems Conference
% Dynamical Systems, Organized by Lori Alvin, Krystyna Kuperberg, and Roland Roeder
% Friday, March 11, 2016
% slides for talk "Aperodicity at the boundary of chaos"
 
 
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\date[March 11, 2016]


  
\title[Kuperberg pseudogroup]{Aperodicity at the boundary of chaos}


\author{Steve Hurder\\joint work with Ana Rechtman}

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



\begin{document}

\frame{\titlepage} % # 1


 
  \section{Introduction}
 
 

\frame % 1
{
  \frametitle{Genericity of Dynamics  }
  
  
      
   {\bf Weak Palis Conjecture:} $M$ compact manifold, then the space of ${\rm Diff}^r(M)$ of $C^r$-diffeomorphisms ($r \geq 1$) contains a dense open set which decomposes as the union $\cM\cS \cup \cI$ of two disjoint open sets:  
   \begin{itemize}
\item $\cM\cS$ is the set of Morse-Smale diffeomorphisms, 
\item $\cI$ is the set of diffeomorphisms having transverse homoclinic intersection.
\end{itemize}

A diffeomorphism is Morse-Smale if its  non-wandering set consists of finitely many hyperbolic periodic orbits.

\medskip

$\bullet$ ~ {J.~Palis},
 {\it A global view of dynamics and a conjecture on the denseness of  finitude of attractors},
  {\bf Ast\'erisque},Vol. 261, 2000.

      
$\bullet$ ~ {J.~Palis},
 {\it On Open questions leading to a global perspective in dynamics},
  {\bf Nonlinearity},21:, 2008.

        
\vfill   

}

 \frame % 1
{
  \frametitle{}
 
    
  
\begin{figure}[!htbp]
\centering
The Weak Palis Conjecture

 \includegraphics[width=80mm]{pix/PalisConjecture.png}
 
 (Illustration   from Palis, \emph{Nonlinearity} 2008)
\end{figure}

  

   \vfill   

}


\frame % 1
{
  \frametitle{ }
We are interested in the ``structural stability''   of the following class of examples, called ``Kuperberg flows'':
 
 \bigskip
 
{\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 no periodic orbits.}
 

\bigskip
 

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



$\bullet$ ~ {{\'E}~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.  
 
 
$\bullet$ ~ {S.~Hurder \& A.~Rechtman},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque}, Vol. 377 (216), 250 pages.

      
}

 
\vfill   

}


\frame % 1
{
  \frametitle{ }
  
  What   we know of the dynamics of the Kuperberg flows:
 
 \medskip
 
  {\bf Theorem (A. Katok, 1980)} {\it Let $M$ be a closed, orientable 3-manifold. Then an aperiodic flow $\phi_t$  on $M$ has  entropy zero. }

\medskip

{\bf Theorem (Ghys, Matsumoto, 1995)} The Kuperberg flow has a unique minimal set $\fM \subset M$.
 
 \medskip


  {\bf Theorem (Hurder \& Rechtman, 2015)} {\it Let $\Phi_t$ be a generic Kuperberg flow on a plug $\mK$. 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 $\Phi_t$.

 Moreover, the   flow restricted to $\fM$ has non-zero ``slow entropy'', for exponent $\alpha = 1/2$.} 
 
 So, a generic Kuperberg flow \emph{almost} has positive entropy.
  
   
   
 \vfill   

}

   
\frame % 1
{
  \frametitle{}
 
    
  
  {\bf Question:} Where do the Kuperberg flows sit in the scheme of the Weak Palis Conjecture?
  
\medskip
 
{\bf Theorem 1:} {\it Let $\Phi_t$ be  a Kuperberg flow on a plug $\mK$. Then 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 $\Phi_t^{\e}$ is ``partially Morse-Smale'' and so has entropy $0$.}
 
 
 \medskip

{\bf Theorem 2:} {\it Let $\Phi_t$ be  a Kuperberg flow on a plug $\mK$. Then 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 a ``horseshoe'',  and so has positive entropy.}

  \medskip

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

  \medskip
  
$\bullet$ ~ {S.~Hurder \& A.~Rechtman},
 {\it Aperiodic flows at the boundary of chaos},
 {\it in preparation}, available March 2016.

 


\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{}
 
    
  Modified Wilson Plug $\mW$ (sort of Morse-Smale)
\begin{figure}[!htbp]
\centering
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=76mm]{figures_png/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=100mm]{figures_png/Figure07.png}
\end{figure}
\vspace{-14pt}
Embed so that the Reeb cylinder $\{r=2\}$ is tangent to itself. 
  
   \vfill   

}

   
     
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
   The insertion map as it appears in the face $E_1$
   \medskip
   
 \includegraphics[width=50mm]{pix/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{}
 
      {\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/modifiedradius-1.png}
\includegraphics[width=50mm]{pix/modifiedradius-3.png}
\vspace{-6pt}
\end{figure}

   
   \vfill   

}

\section{Theorem 1}

  
 \frame % 1
{
  \frametitle{}
 
{\bf Proposition:}
{\it Let $\Phi_t^\e$ be a Kuperberg 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.  

\bigskip

Varying the radius of the insertion for $\e < 0$, we obtain Theorem 1.
   \vfill   

}



\section{Theorem 2}

  
 \frame % 1
{
  \frametitle{}
 
  The idea of the proof of Theorem 2 is to study the dynamics of Kuperberg flows $\Phi_t^\e$ for $\e \geq 2$.
  
  \medskip
  
  Our approach in the Asterisque paper introduced the technique of comparing 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]{figures_png/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/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/dynamicsr=2e.png}
  
     
 \vspace{-14pt}

\end{figure}

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

   
 
   \vfill   

}

   
  
\frame % 1
{
  \frametitle{ }
 
 Instead, we 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=60mm]{pix/Figure18.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=60mm]{pix/Figure20.png}
  
     
 
\end{figure}


\vfill   

 }
  
      
  
\frame % 1
{
  \frametitle{ }
 We then show that for $\e > 0$ well-chosen with respect to the choice of $k$ above, the restriction of the map $\vp$ to $U_0$ is defined by the return map of $\Phi_t^e$ and hence $\Phi_t^e$  has positive entropy. 
 
 \bigskip
 
 Conclude with two remarks: 
 
 \medskip
 
 $\bullet$ ~ For $\e < 0$, the dynamics of the map $\Phi_t^\e$ 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 the map $\Phi_t^\e$ is chaotic, but making calculations of entropy for example, is only possible for well-chosen embeddings. We have no intuition, for example, of how to describe the nonwandering set for the flows $\Phi_t^\e$. 
 
 
 
 
\vfill   

 }
  

  
\frame % 1
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  \frametitle{ }
\begin{center}
\Large{ Thank you for your attention!}
\end{center} 

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 }
  
  
\end{document}

