% The Mathematical Congress of the Americas 2017


% July 24-26, 2017, 2016.
% talk on July 27, 2017

% slides for talk "Smooth flows with fractional entropy dimension"
 
 
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\date[March 11, 2016]


  
\title[Slow entropy]{Smooth flows with fractional entropy dimension}


\author{Steve Hurder\\ {\small MCA -- Montreal, July 27, 2017}}

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

 
\begin{document}

\frame{\titlepage} % # 1


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

 {\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
 

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

\pause

 There  are many choices   in the construction of   Kuperberg plugs:    
\begin{quote}
Ghys: \emph{Par ailleurs, on peut construire beaucoup de pi\`eges de Kuperberg et il n'est pas clair qu'ils aient le m\^eme dynamique.}
\end{quote}

 {\footnotesize 
$\bullet$ ~ {{\'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.}
 
    
 
\vfill   

}


\frame % 1
{
  \frametitle{ }

{\bf Problem:}  Investigate the  invariants of ``Kuperberg flows'':

$\bullet$ ~ dynamical invariants of the smooth   flow in plug $\mW$

$\bullet$ ~   topological invariants of unique minimal set $\Sigma$

$\bullet$ ~ relations with their smooth deformations.

\medskip
\pause
 
{\bf Theorem (Katok, 1980)} {\it Let $M$ be a closed, orientable 3-manifold. A smooth aperiodic flow $\phi_t$  on $M$ has  entropy zero. }


 \medskip
 
{\bf Theorem (H \& Rechtman, 2016)} {\it  The  minimal set $\Sigma$ of a generic Kuperberg flow is a 2-dimensional ``zippered lamination'', which has unstable shape.}

\medskip
 
{\bf Theorem (Ingebretson, 2017)} {\it  The Hausdorff dimension of the  minimal set $\Sigma$ for a generic Kuperberg flow has $2 < HD(\Sigma) < 3$.}

 

\vfill   

}

  \section{Entropy}
  
\frame % 1
{
  \frametitle{ }
  
 Let $\vp_t \colon M \to M$ be a smooth non-vanishing flow on a compact Riemannian manifold. 
  For   $\e, T > 0$, two points $p,q\in M$ are said to be 
\emph{$(\vp_t , T, \e)$-separated} if 
$$
d_M(\vp_t(p),\vp_t(q))>\e \quad \mbox{for some} \quad -T\leq t\leq T ~ .
$$
A set $E \subset M$ is \emph{$(\vp_t , T, \e)$-separated} if  all pairs of distinct 
points in $E$ are $(\vp_t , T, \e)$-separated. Let $s(\vp_t , T, \e)$ be the maximal
cardinality of a $(\vp_t , T, \e)$-separated set in $X$.  
   
   \medskip
     
     The topological entropy of the flow $\vp_t$ is then defined by
$$
h_{top}(\vp_t)= \frac{1}{2} \cdot \lim_{\e\to  0} \left\{ \limsup_{T\to\infty}\frac{1}{T}\log(s(\vp_t , T, \e)) \right\} \ ,
$$
which is independent of the choice of metric $d_M$.
 

\vfill   

}


 

\frame % 1
{
  \frametitle{ }
  
 For a flow with zero entropy, de Carvalho, and independently Katok and Thouvenot, introduced  the notion of \emph{slow entropy} as a measure of the complexity of the flow.  
 The slow entropy measures the subexponential growth of the $\e$-separated points.

{\bf Definition.} 
For   $0 < \alpha < 1$, the $\alpha$-slow entropy of $\vp_t$ is given by 
$$
h^{\alpha}_{top}(\vp_t)= \frac{1}{2} \cdot \lim_{\e\to  0} \left\{ \limsup_{T\to\infty}\frac{1}{T^{\alpha}}\log \{s(\vp_t , T, \e) \} \right\} \ .
$$

\pause 
Kyewon Park and her coauthors   introduced the notion of the \emph{entropy dimension} of a flow $\vp_t$:
$${\rm{Dim}}_{h}(\vp_t)=  \inf_{\alpha > 0} \left\{  h^{\alpha}_{top}(\vp_t) \right\}  = 0\ .
$$
 
$\bullet$~ Are there non-trivial entropy-like invariants for Kuperberg flows?
 
\vfill   

}

  \section{Constructions}

\frame % 2
{
  \frametitle{}


 $\mW = [-2,2] \times [1,3] \times \mS^1$ 
   with  non-vanishing vector field 
 $$\vec{W} =  g \frac{f}{dz} + f \frac{f}{d\theta}$$
 $\bullet$ ~ $f$ is asymmetric in the vertical coordinate $z$ about $z=0$ \\
$\bullet$ ~  $g \geq 0$ is constant in the $\mS^1$ factor, and vanishes only along the circles $\cO_{i} = \{(-1)^i\} \times \{2\} \times \mS^1$
 
  \begin{figure}[!htbp]
  \includegraphics[width=50mm]{pix_montreal/Figure03b.png}
\end{figure}



 
    \vfill   

}
 
 
  
\frame % 6
{
 \frametitle{ }
  
 By symmetry on $g \geq 0$, it must vanish to an even order  along  $\cO_{i}$.
  
      In the generic case, $g$ vanishes to second order.
      
      Consider the case where  $g$ vanishes to   order $2n$ for $n > 1$. As $n$ increases, the speed of approach to the orbits $\cO_i$ slows down.
 
    \begin{figure}[!htbp]
  \includegraphics[width=40mm]{pix_montreal/Figure03a.png}
\end{figure}
 
  \vfill   

}


  
\frame % 2
{
  \frametitle{}
 
 
 
   
   \begin{center}
     Self-insert the Wilson plug with a twist and a bend, matching  the flow lines on the boundaries, to obtain Kuperberg Plug $\mK$


\medskip


 \includegraphics[width=90mm]{pix_montreal/Figure15.png}
 
 Embed so that the Reeb cylinder $\{r=2\}$ is tangent to itself. 
 
 The degree of tangency influences the dynamics.
  
 \end{center}
   
 
   \vfill   

}

\section{Lamination}
 
\frame % 6
{
 \frametitle{ }
  
  
     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*}

 
   
  \vfill   

}

 
 \frame % 1
{
  \frametitle{}
 
      
Consider  the flow of the image   $\tau(\cR') \subset \mK$  
$$ \fM_0 ~ \equiv ~  \{\Phi_t(\tau(\cR')) \mid -\infty < t < \infty \}  $$
   The surface $\fM_0$ is  union of embedded ``tongues'' in $\mK$, where each tongue wraps around the Reeb cylinder $\tau(\cR')$.

    \begin{figure}[!htbp]
\centering
 
 \includegraphics[width=56mm]{pix_montreal/Figure18a.png}
 \end{figure}


 
  
     \vfill   

}
  
     
  
 
  
    
  

 \frame % 1
{
  \frametitle{}
 
  $\fM_0$ is an infinite union of tongues at increasing levels, corresponding to the level filtration
  $$\fM_0^0 \subset \fM_0^1 \subset \fM_0^2 \subset \cdots $$

  
  
The closure $ \fM ~ \equiv ~ \overline{\fM_0}     \subset \mK$ is a lamination with boundary.
 

The topology of $\fM$ is highly complex.       

\medskip
    
  $\fM_0$ is a ``fat tree''  whose leaves at higher levels are  recurrent on themselves, corresponding to  the branching of the tree below.



     \vfill   

}
  
     
  
  

 \frame % 1
{
  \frametitle{}




      \begin{figure}[!htbp]
\centering
  
 \includegraphics[width=90mm]{pix_montreal/Figure27.png}
 \end{figure}


 

     \vfill   

}
 
\section{Growth}

\frame % 2
{
  \frametitle{}
 
 Choose a Riemannian metric on the plug $\mK$.
 
  Then $\fM_0 \subset \mK$   inherits a Riemannian metric. 
   
   Let  $d_{\fM}$ denote the associated path-distance function on $\fM_0$.
   
 Fix  the basepoint $\omega_0 = (2, \pi, 0) \in \tau(\cR')$ and  let 
 $$ B_{\omega_0}(s)  = \{ x \in \fM_0 \mid d_{\fM}(\omega_0 , x) \leq s\}$$  
be the closed ball of radius $s$ about the basepoint $\omega_0$. 

Let $A(X)$ denote the Riemannian area of a Borel subset $X \subset \fM_0$. 

Then $\mathrm{Gr}(\fM_0, s) = A(B_{\omega_0}(s))$ is   the \emph{growth function} of $\fM_0$.

Given functions $f_1, f_2 \colon [0,\infty) \to [0, \infty)$, we say that $f_1 \lesssim f_2$ if there exists constants $A, B, C > 0$ such that for all $s \geq 0$, we have that $  f_2(s) ~ \leq ~ A \cdot f_1(B \cdot s) + C$.

 Say that    $f_1 \sim f_2$ if both $f_1 \lesssim f_2$ and $f_2 \lesssim f_1$ hold.     

  

     \vfill   

}
 

\frame % 2
{
  \frametitle{}

 $f_1 \sim f_2$ defines  an equivalence relation on functions, which is used to define their   \emph{growth type}.
 
 {\bf Theorem (H \& Ingebretson, 2017)} There exists Kuperberg flows such that the growth type $\mathrm{Gr}(\fM_0, s)$      satisfies
$$  \mathrm{Gr}(\fM_0, s) \sim \exp(s^{\alpha})$$
for $\alpha > 0$ arbitrarily small.  
  

We give two  applications of this construction.

 

     \vfill   

}
 

\frame % 2
{
  \frametitle{}
 


The Kuperberg pseudogroup $\cGF$ is generated by the holonomy of the lamination $\fM$ for the section $\bRt$.

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

The  complexity of $\cGF$ is   the \emph{ growth type} of 
$ \ell \mapsto  h(\cGF, d, \e, \ell)  $

 {\bf Theorem (H \& Ingebretson, 2017)} There exists Kuperberg flows such that for $\e > 0$ sufficiently small, the growth type     satisfies
$\ds   h(\cGF, d, \e, \ell)    \sim \exp(\ell^{\alpha})$, 
for $\alpha > 0$ arbitrarily small.  

\bigskip

These examples have non-trivial \emph{lamination} slow entropy. 

The relation between the lamination slow entropy and the flow slow entropy is complicated.
     \vfill   

}
 
\section{Shape}


\frame % 2
{
  \frametitle{}
 
The open sets
$\ds U_{\ell} = \{ x \in \mK \mid  d_{\mK}(x, \Sigma) < \e_{\ell}\}$  where we have  $0 <  \e_{\ell +1} < \e_{\ell}$ for all $\ell \geq 1$, and  $\ds \lim_{\ell \to \infty}  \, \e_{\ell}   =   0$, give a shape approximation to $\Sigma$. 

For $\alpha > 0$, an $\alpha$-pseudo-orbit for the Kuperberg flow $\vp_t$ determines a path   in $U_{\ell}$ if $\alpha < \e_{\ell}$. 


\medskip

{\bf Theorem (Misiurewicz, 1984)} $h_{top}(\vp_t) = h_{\psi}(\vp_t)$   where $h_{\psi}(\vp_t)$ denotes the entropy of $\vp_t$ calculated using pseudo-orbits. 

\medskip 
{\bf Theorem (Barge \& Swanson, 1990)} $h_{top}(\vp_t) =  H_{\psi}(\vp_t)$ where $H_{\psi}(\vp_t)$ denotes the growth rate  of separated  periodic pseudo-orbits for $\vp_t$. 

\medskip 

 {\bf Conjecture:}     The   expansion growth function for $\vp_t$ defined using pseudo-orbits has the same growth type as for $\fM_0$.  


   \vfill   

}
 
 
  
 
 \frame % 1
{
  \frametitle{}

 We also state   an additional shape property for the minimal set of a generic Kuperberg flow. 
 
{\bf Theorem (H \& Rechtman, 2016)} 
Let $\Sigma$ be the minimal set for a generic Kuperberg flow. Then the Mittag-Leffler condition   for homology groups is satisfied. That is, given a shape approximation $\fU = \{U_{\ell}\}$ for $\Sigma$,
then  for any $\ell\geq 1$ there exists $p>\ell$ such that for any $q\geq p$ 
$$
Image\{H_1(U_p;\mZ)\to H_1(U_\ell;\mZ)\} = Image \{H_1(U_q;\mZ)\to H_1(U_\ell;\mZ)\}.
$$

 A shape $1$-cycle is an ``almost closed'' path  with endpoints sufficiently close (see picture below.)
  \medskip

   {\bf Problem:} How are the shape $1$-cycles related to the periodic pseudo-orbits for $\vp_t$?
  
  
     \vfill   

}
  
  
 
\frame % 2
{
  \frametitle{}
   
    
      \begin{figure} 
\centering

  
 \includegraphics[width=77mm]{pix_montreal/Figure56a.png}
 \end{figure}

\pause

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



        \vfill   

}
 

  
    
\frame % 1
{
  \frametitle{References}
 {\footnotesize     
 

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


 
{S.~Hurder and A.~Rechtman},
 {\it The dynamics of  generic Kuperberg flows},
 {\bf Ast\'erisque},   Vol. 377, 2016.


{S.~Hurder and A.~Rechtman},
 {\it Aperiodicity at the boundary of chaos },
{\bf  Ergodic Theory Dynamical Systems}, to appear, 2017.

 
{S.~Hurder and A.~Rechtman},
 {\it Perspectives on Kuperberg flows},
{\bf  Topology Proceedings}, to appear, 2017.

{D.~Ingebretson},
 {\it Hausdorff dimension of generic Kuperberg minimal sets},
{\bf Thesis}, University of Illinois at Chicago,  August 2017.


{S.~Hurder and D.~Ingebretson},
 {\it Smooth flows with fractional entropy dimension},
{in preparation}, 2017.

{S.~Hurder and A.~Rechtman},
 {\it The dynamics of Derived from Kuperberg flows},
{in preparation},  2017.

  
    

}
}
\vfill   

 

 
  


\end{document}

