% AMS Fall Southeastern Sectional Meeting
% Special Session on The Analysis, Geometry, and Topology of Groupoids
% Sunday, October 18, 2015
% slides for talk "Entropy of the Kuperberg pseudogroup"
% slides completed October 16, 2015

% Abstract: The purpose of this talk is to describe some novel aspects of the dynamics of the Kuperberg pseudogroup. Krystyna Kuperberg showed that the smooth Seifert Conjecture is false, by constructing smooth aperiodic plugs in 3-dimensions, which can be used to construct smooth aperiodic flows on any compact 3-manifold. The flows in these aperiodic plugs have a unique minimal set. In the paper ÓThe dynamics of generic Kuperberg flowsÓ by Hurder and Rechtman, we analyze the dynamics and topology of these minimal sets. A key tool is the introduction of the Kuperberg pseudogroup, defined by the return maps of the flow to a section, where the section is allowed to have tangencies to the flow, and thus the pseudogroup generators have singularities. While the usual entropy of an smooth aperiodic flow must always vanish, the Óslow entropyÓ need not, and we discuss why the slow entropy does not vanish for the Kuperberg pseudogroup, and also is not zero for the flow.


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\date[October 18, 2015]


  
\title[Kuperberg pseudogroup]{Entropy of the Kuperberg pseudogroup}


\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{ }
 Counter-example to the Seifert Conjecture:
 
{\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.}
 

\medskip
\pause

 {\footnotesize 
$\bullet$ ~ {K.~Kuperberg},
 {\it A smooth counterexample to the {S}eifert conjecture},
 {\bf Ann. of Math. (2)}, 140:723--732, 1994. (9 pages)



$\bullet$ ~ {{\'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. (24 pages)
 
 
$\bullet$ ~ {S.~Hurder \& A.~Rechtman},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque}, to appear, 170 pages in TeX.

}

\medskip
\pause
{\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. }

\vfill   

}

\section{Entropy variation}

\frame % 1
{
  \frametitle{}
 

{\bf Theorem (Hurder \& Rechtman, 2015)} {\it Let $M$ be a closed, orientable 3-manifold. Then $M$ admits a $C^{\infty}$-family of non-vanishing vector fields 
$\vec{X}_{t}$ for $-\epsilon < t < \epsilon$ whose flows:}

$\bullet$ have entropy 0 for $t < 0$

$\bullet$ have no periodic orbits for $t=0$

$\bullet$ have positive entropy   for $t > 0$.

\medskip
\pause

Solution of Palis Conjecture for surface diffeomorphisms:

 {\footnotesize 


$\bullet$ ~ {E.~Pujals and M.~Sambarino},
 {\it On Homoclinic tangencies and hyperbolicity for surface diffeomorphisms},
  {\bf Ann. of Math. (2)},151:961--1023, 2000.

}
\pause

Explicit family of constructions:

 {\footnotesize 

$\bullet$ ~  {S.~Hurder and A.~Rechtman},
 {\it Zippered laminations at the boundary of hyperbolicity},
 {preprint},   2015.


}


\vfill   

}

\section{Sections}

 \frame % 1
{
  \frametitle{}
 
  $\phi_t \colon M \to M$ a smooth flow. A \emph{complete section} is a closed surface $T \subset M$ which is everywhere transverse to the flow.  
 
     Return map of flow induces diffeomorphism $f_{\phi} \colon T \to T$
     
     Dynamics of $\phi_t$ $\leftrightarrow$ dynamics of $f_{\phi}$

\bigskip
\pause

$\phi_t \colon M \to M$ a smooth flow. A section is a   surface $T \subset M$ which is generically  transverse to the flow.  

Return map of flow induces a smooth pseudogroup $\cG_{\phi}$ on $T$

 Dynamics of $\phi_t$ $\leftrightarrow$ dynamics of $\cG_{\phi}$ ??

   \vfill   

}

\section{Minimal sets}

 \frame % 1
{
  \frametitle{}
 
  $\Sigma \subset M$ is minimal set for $\phi_t$ if:
  
  $\bullet$~ $\phi_t(\Sigma) = \Sigma$ for all $t$
  
  $\bullet$~ $\Sigma$ is closed
  
  $\bullet$~ $\Sigma$ is minimal with respect to these two properties.

\medskip
\pause

Suppose that section $T \cap \Sigma$ is contained in the interior of  $\Sigma$

$\Longrightarrow$  Dynamics of $\phi_t$ $\leftrightarrow$ dynamics of $\cG_{\phi}$

 

   \vfill   

}


 \frame % 1
{
  \frametitle{}
 
{\bf Theorem (Ghys, Matsumoto, 1995)} The Kuperberg flow has a unique minimal set $\Sigma \subset M$, with all other points wandering. 
 
 \bigskip
 \pause
 
 {\bf Theorem (Hurder \& Rechtman, 2013)} The unique minimal set for a generic Kuperberg flow is a \emph{zippered lamination}, a stratified space with two strata:
 
 $\bullet$~ A 2-dimensional strata that has a laminated structure $\F$;

$\bullet$~ A $1$-dimensional strata that is transversally Cantor-like. 

$\bullet$~ $\Sigma \cap T$ defines a smooth pseudogroup $\cGF$
\bigskip
 \pause


$\Longrightarrow$  Dynamics of $\phi_t$ $\leftrightarrow$ dynamics of $\cG_{\phi}$ $\leftrightarrow$ dynamics of $\cGF$ ?

  \bigskip
 \pause

 
{\bf Theorem (Hurder \& Rechtman, 2015)} Entropy of Kuperberg flow vanishes if and only if   geometric entropy of $\cGF$ vanishes. 


 
   \vfill   

}

 
 \frame % 1
{
  \frametitle{}
 
  
 {\bf Strategy:}
 
 \bigskip
 

$\bullet$~ Construct smooth variations of the Kuperberg flow

\bigskip

$\bullet$~ Evaluate the entropy of these flows using pseudogroup entropy

   \vfill   

}

\section{Construction}

 \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{}
 
    
  
\begin{figure}[!htbp]
\centering
Construct a modified Wilson vector field $\vec{W}$ in a rectangle $R$

 \includegraphics[width=40mm]{figures_png/Figure01.png}
\end{figure}

  

   \vfill   

}

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


 \includegraphics[width=76mm]{figures_png/Figure03b.png}
 


\end{figure}

   
 
   \vfill   

}

 
     
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 Embed the Modified Wilson Plug in $\mR^3$:

\medskip

 \includegraphics[width=90mm]{figures_png/Figure04.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}
   The subtlety: embed so that the Reeb cylinder $\{r=2\}$ is tangent to itself. 
   Or, vary this parameter by $-\epsilon < t < \epsilon$ to get $\vec{X}_{t}$ on $\mK$
 
   \vfill   

}

   
     
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
$\epsilon =0$:   The insertion map as it appears in the face $E_1$
   \medskip
   
 \includegraphics[width=60mm]{figures_png/ezero.png}
 \medskip
 
\end{figure}

   
 
   \vfill   

}

    
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
$\epsilon < 0$:   The insertion map as it appears in the face $E_1$
   \medskip
   
 \includegraphics[width=50mm]{figures_png/enegative.png}
 \medskip
 
\end{figure}

   
 
   \vfill   

}

   
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
$\epsilon > 0$:   The insertion map as it appears in the face $E_1$
   \medskip
   
 \includegraphics[width=60mm]{figures_png/epositive.png}
 \medskip
 
\end{figure}

   
 
   \vfill   

}



 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 The section ${\bf R}_0 \subset \mK$ used to define pseudogroups $\cG_{\phi}$ and $\cGF$.
 \medskip
 
 \includegraphics[width=110mm]{figures_png/Figure15.png}
 \medskip
 
 The flow $\vec{W}$ is tangent to ${\bf R}_0$ along the center plane $\{z=0\}$. 
\end{figure}

   
 
   \vfill   

}
  
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 The notched cylinder $\cR'$ embedded in $\mW$ 
 \medskip
 
 \includegraphics[width=60mm]{figures_png/Figure17.png}
 
 \medskip
 $\mK$ is obtained from $\mW$ by a quotient map, $\tau \colon \mW \to \mK$
 
  \end{figure}

   
 
   \vfill   

}
 
 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 The flow of the cylinder $\cR'$ through one insertion is a simple propeller:
 \medskip
 
 \includegraphics[width=70mm]{figures_png/Figure20.png}
 \medskip
 
  \end{figure}

   
 
   \vfill   

}


 \frame % 1
{
  \frametitle{}
 
    
 \begin{figure}[!htbp]
\centering
 The flow of the cylinder $\cR'$ through infinite time is an infinite branched tree $\fM_0$ - a leaf of a lamination:
 \medskip
 
 \includegraphics[width=80mm]{figures_png/Figure34.png}
 \medskip
 
  \end{figure}

   
 
   \vfill   

}


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

The Kuperberg pseudogroup $\cGF$ is generated by the holonomy of the lamination $\fM$ defined by the flow of the Reeb cylinder

$$
\fM_0 ~ \equiv ~  \{\phi_t(\tau(\cR')) \mid -\infty < t < \infty \} $$

$$ \fM ~ \equiv ~ \overline{\fM_0}     \subset \mK 
$$

$\fM$ is wildly wicked, though   $\fM_0$ admits a level filtration 

$$\fM_0^0 \subset \fM_0^1 \subset \fM_0^2 \subset \cdots $$
 
which matches the branching of the tree above.
   
   \vfill   

}


  

   
  
 \frame % 1
{
  \frametitle{}
 
  
   
 \begin{figure}[!htbp]
\centering
Generators $\{Id, \phi_1^+, \phi_1^-, \phi_2^+, \phi_2^-,  \psi\}$ of the pseudogroup $\cGF$
 \medskip
 
 \includegraphics[width=34mm]{figures_png/Figure16a.png}
 \includegraphics[width=34mm]{figures_png/Figure16b.png}
 \includegraphics[width=34mm]{figures_png/Figure16c.png}
 \medskip
 
  \end{figure}
 
 
   
 
   \vfill   

}

     
\section{Entropy}

 \frame % 1
{
  \frametitle{}

   
  
The  \emph{word length} $\| g \| \leq m$ if $g \in \cGF$ can be expressed as the composition of at most $m$ generators. 

 
     

    
   \emph{Entropy}   for a $C^1$-pseudogroup action  [Ghys, Langevin \& Walczak,  1988] measures      the ``exponentiality'' of the orbits.

  
 
  

   Let $\e > 0$ and $\ell > 0$, and $d$ the metric on ${\bf R}_0$. 
     A subset $\cE \subset {\bf R}_0$ is said to be $(d, \e, \ell)$-separated if   for all $w,w' \in \cE$   there exists $g \in \cGF$ with $w,w' \in \mathrm{Dom}(g)$, and  $\|g\|_w \leq \ell $  so that  $d(g(w), g(w')) \geq \e$. 

 
The  ``expansion growth function'' is:
  $$h(\cGF, d, \e, \ell) =     \max \{ \# \cE \mid \cE \subset {\bf R}_0 ~ \text{is} ~ (d, \e,\ell) \text{-separated} \}   $$
   
   \vfill   

}


 

  \frame % 1
{
  \frametitle{}

The  entropy of $\cGF$ is   the \emph{asymptotic exponential growth type} of the expansion growth function:
$$
 h(\cGF)    =   \lim_{\e \to 0}\left\{  \limsup_{\ell \to  \infty} ~  \ln \left\{ h(\cGF, d, \e, \ell) \right\}/ \ell \right\}$$
 
 \medskip
 \pause
 {\bf Theorem (Hurder \& Rechtman, 2015)} Let $\vec{X}_t$ be the modified Kuperberg flow on $\mK$. Then the    ``expansion growth function'':
 
  $\bullet$~  for $-\e < t < 0$ has polynomial growth, hence  $h(\cGF) =0$
 
 $\bullet$~  for $t=0$ has growth rate $\sim ~ \exp(\sqrt{n})$, hence $h(\cGF) =0$
 
 $\bullet$~  for $0 < t < \epsilon$ has exponential growth, hence $h(\cGF) >0$.
 
   
    \vfill   

}

   
  \frame % 1
{
  \frametitle{}

{\bf Idea of proof:} 
 \medskip
  
   
 $\bullet$~ for $-\e < t < 0$, the surface $\fM_0$ is finitely recursive
 
 $\bullet$~  for $t=0$, the surface $\fM_0$ is partially recursive
 
 $\bullet$~  for $0 < t < \epsilon$, the surface $\fM_0$ is fully recursive.
 
 \bigskip
 
 Then use:
 
 $\bullet$~  Resulting growth estimates for number of words in $\cGF$, 
 
  $\bullet$~  estimate non-expansiveness of maps defined by the words  in $\cGF$.



 
    \vfill   

}


  
\frame % 1
{
  \frametitle{References}
 {\footnotesize     
 
{\'E.~Ghys, R.~Langevin, and P.~Walczak},
 {\it Entropie g\'eom\'etrique des feuilletages},
{\bf Acta Math.}, 160:105--142, 1988. 
     



 {{\'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.



{S.~Hurder and A.~Rechtman},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque}, to appear, arXiv:1306.5025.


{S.~Hurder and A.~Rechtman},
 {\it Zippered laminations at the boundary of hyperbolicity},
 {preprint}, 2015.

 

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

{G.~Kuperberg and K.~Kuperberg},
 {\it Generalized counterexamples to the {S}eifert conjecture},
 {\bf Ann. of Math. (2)}, 144:239--268, 1996.
 
  

}
}
\vfill   

 
  
  
\end{document}

