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% Beginning of amspaper.tex
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% Robert W. Ghrist
% PACM, Princeton Univ.
% rwghrist@math.princeton.edu
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% Last modified: 19 Sept 1995
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%\issueinfo{00}% volume number
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%  {}%         % month
%  {1995}%     % year

%\copyrightinfo{1995}%            % copyright year
%  {American Mathematical Society}% copyright holder

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

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\begin{document}

\title{Flows on $S^3$ Supporting all Links as Orbits}

\author{Robert W. Ghrist}
\address{Center for Applied Mathematics, Cornell University, Ithaca NY, 14853}
\curraddr{Program in Applied and Computational Mathematics, Princeton
	University, Princeton NJ, 08544-1000; Institute for Advanced Study,
	Princeton NJ, 08540}
\email{rwghrist@math.princeton.edu; robg@math.ias.edu}
\thanks{The author was supported in part by an NSF Graduate Research 
	Fellowship.}
\thanks{The author wishes to thank Philip Holmes for his encouragement
	and support.}


\subjclass{Primary 57M25, 58F22; Secondary 58F25, 34C35}

\def\currentvolume{1}
\def\currentissue{2}

\date{June 16, 1995}

\communicated_by{Krystyna Kuperberg}

\setcounter{page}{91}


%\date{June 14, 1995.}

\keywords{Knots, links, branched 2-manifolds, flows.}

\begin{abstract}
We construct counterexamples to some conjectures of 
J. Birman and R. F. Williams concerning
the knotting and linking of closed orbits of flows on 3-manifolds.
By establishing the existence of
``universal templates,'' we produce examples of flows on
$S^3$ containing closed orbits of all knot and link types simultaneously.
In particular, the set of closed orbits of
any flow transverse to a 
fibration of the complement of the figure-eight knot in $S^3$
over $S^1$ contains representatives of every (tame) knot and
link isotopy class. Our methods involve semiflows on branched
2-manifolds, or {\em templates}. 
\end{abstract}

\maketitle


\section*{} In this announcement, we answer some questions raised
by Birman and Williams in their original examination of the link
of closed orbits in the flow on $S^3$ induced by the fibration of the
complement of a fibred knot or link \cite{BW83b}.
In their work, they proposed the following conjecture:
\begin{conjecture}[Birman and Williams, 1983] \label{conj_1}
        \hskip-1.5mm The figure-eight knot ($K_8$) does not appear as a closed orbit
        of the flow induced by the fibration of the complement
        of the figure-eight knot.
\end{conjecture}
By ``the'' flow is meant the flow obtained by integrating the
gradient of $p$, where $p:S^3\setminus K_8\rightarrow S^1$ is the unique
fibration over $S^1$ whose
monodromy is the pseudo-Anosov representative of its isotopy class, with
respect to the Nielsen-Thurston classification \cite{Thu88,FLP79}.

We have resolved this conjecture in the negative and, in so doing, have 
discovered interesting examples of flows on 3-manifolds and semiflows 
on branched 2-manifolds. More specifically, in Theorem \ref{thm_fig8},
we show that any fibration of the complement of the figure-eight knot in 
$S^3$ over $S^1$ induces a flow on $S^3$ containing every tame
knot and link as closed orbits.
We include only those definitions and results which are
relevant for this announcement, leaving 
details to a separate work \cite{Ghr94b}.

%*****************************************************************************
\section{Template Theory}
%*****************************************************************************

{}\hskip-2pt Periodic orbits of a flow are embedded circles. When the flow is
three-dimensional, periodic orbits are knots and the collection
of periodic orbits forms a link which often is nontrivial.
A valuable tool for examining knotted periodic orbits in three
dimensional flows is the template construction of Birman and Williams
\cite{BW83a,BW83b}.
% ###############################
\newpage
\begin{definition}\label{def_template}
A {\em template} (also known as a {\em knotholder})
is a compact branched
two-manifold with boundary and smooth expanding semiflow
built from a finite number of {\em branch line charts},
as given in Figure \ref{fig_strips}(a).
\end{definition}
% ###############################

For a more detailed definition, along with examples, see
\cite{BW83a,BW83b}.
Each {\em branch line} of a template appears (locally)
as in Figure \ref{fig_strips}(a): there will be a certain 
number ($\geq 2$) of
incoming strips which completely cover the branch line (expanding the
incoming semiflow), and a certain number ($\geq 2$)
of outgoing strips. The incoming and outgoing strips of all the
branch lines are then connected bijectively. 
In Figure \ref{fig_strips}(b), we display the simplest example
of a template: the Lorenz template \cite{BW83a}, consisting
of one branch line and two strips.

% =========================================
\begin{figure}[htb]
\begin{center}
%\begin{psfrags}
%        \psfrag{a}[][]{(a)}
%        \psfrag{b}[][]{(b)}
        \epsfxsize=3.5in\leavevmode\epsfbox{fig_1.eps}
%\end{psfrags}
\end{center}
\caption{(a) Strips meet at a branch line; (b) the (embedded) 
Lorenz template.} \label{fig_strips}
\end{figure}
% =========================================

Upon embedding a template in $S^3$, the periodic orbits of the semiflow
form links. The relationship between embedded templates and
links of periodic orbits
in three dimensional flows is expressed in the Template Theorem
of Birman and Williams \cite{BW83b,BW83a}:

% ###############################
\begin{theorem}[Birman and Williams, 1983] \label{thm_temp}
Given a flow on a 3-manifold $M\!$ having a hyperbolic chain-recurrent
set (i.e., Axiom A plus no-cycle), the link of periodic orbits is in
bijective correspondence with the link of periodic orbits on a
particular embedded template ${\mathcal T}\subset M$ (with at most two 
exceptions). On any finite sublink, this correspondence is via ambient 
isotopy.
\end{theorem}
% ###############################

Aside from their relevance to the dynamics of flows,
templates are in their own right a fascinating class of objects. 
Birman and Williams \cite{BW83a,BW83b},
Holmes and Williams \cite{HW85}, M. Sullivan \cite{Sul94a}, and
others have discovered a number of remarkable properties of templates
and template knots. One outstanding conjecture about
templates was posed in \cite{BW83b} and revisited in \cite{Sul94a}:
\begin{conjecture}[Birman and Williams, 1983] \label{conj_2}
        There does not exist an embedded template which supports all
        (tame) knots as periodic orbits of the semiflow:
        i.e., a {\em universal template}.
\end{conjecture}

We will discuss the resolution of Conjecture \ref{conj_2} in 
\S\ref{sec_uni} and its relation to Conjecture \ref{conj_1} in
\S\ref{sec_fibr}.

Our primary focus in the investigation of templates is on
{\em subtemplates}:
% ------------------------------------------------------------------------
\begin{definition} \label{def_sub}
A {\em subtemplate} ${\mathcal S}$ of a template ${\mathcal T}$ is 
a subset of ${\mathcal T}$ which, under the induced semiflow, is 
itself a template: we write ${\mathcal S}\subset{\mathcal T}$.
\end{definition}
% ------------------------------------------------------------------------

Of particular interest to us are subtemplates which are diffeomorphic 
to their ``parents,'' for they induce a map from the template to the
subtemplate:
% -------------------------------------------------------------------
\begin{definition} \label{def_inflat}
A {\em template inflation} of a template ${\mathcal T}$
is a map ${\mbox{{\bf R}}}:{\mathcal T}\hookrightarrow{\mathcal T}$ 
taking orbits to orbits which is a diffeomorphism onto its image.
\end{definition}
% -------------------------------------------------------------------
% ::::::::::::::::::::::::::::::::::::::::::::::
\begin{remark} 
The image of a template inflation 
${\mbox{{\bf R}}}:{\mathcal T}\hookrightarrow{\mathcal T}$ is 
a subtemplate.
\end{remark}
% ::::::::::::::::::::::::::::::::::::::::::::::
For ${\mathcal T}$ an embedded template, a template inflation
${\mbox{{\bf R}}}:{\mathcal T}\hookrightarrow{\mathcal T}$ induces a 
topological action on 
periodic orbits of ${\mathcal T}$. Inflations
which preserve the isotopy class of the link of periodic orbits
are of particular importance.
% -------------------------------------------------------------------
\begin{definition} \label{def_iso}
Let ${\mbox{{\bf R}}}:{\mathcal T}\hookrightarrow{\mathcal T}$ be an 
inflation of a template ${\mathcal T}\subset S^3$, and
let $i_{{\mathcal T}}$ denote inclusion of ${\mathcal T}$ into $S^3$.
If $i_{{\mathcal T}}$ and $i_{{\mathcal T}}\circ{\mbox{{\bf R}}}$ are 
isotopic embeddings of ${\mathcal T}$ in $S^3$, then ${\mbox{{\bf R}}}$ 
is an {\em isotopic inflation}.
\end{definition}
% -------------------------------------------------------------------
% ::::::::::::::::::::::::::::::::::::::::::::::
\begin{remark}
The image of an isotopic inflation ${\mbox{{\bf R}}}$ is a subtemplate 
${\mbox{{\bf R}}}({\mathcal T})$ isotopic in $S^3$ to ${\mathcal T}$. 
\end{remark}
% ::::::::::::::::::::::::::::::::::::::::::::::
% ::::::::::::::::::::::::::::::::::::::::::::::
\begin{example} \label{ex_Mike}
The first example of template containing an isotopic subtemplate was 
given by M. Sullivan \cite{Sul94a}, for a template ${\mathcal V}$, 
pictured in Figure \ref{fig_v_sub_v}(a). In Figures 
\ref{fig_v_sub_v}(b) and \ref{fig_v_sub_v}(c)
we show a subtemplate within ${\mathcal V}$ and removed from 
${\mathcal V}$ respectively. The reader may verify that the template 
of Figure \ref{fig_v_sub_v}(c) is isotopic to ${\mathcal V}$.
\end{example}
% ::::::::::::::::::::::::::::::::::::::::::::::

% =========================================
\begin{figure}[htb]
\begin{center}
%\begin{psfrags}
%        \psfrag{a}[][]{(a)}
%        \psfrag{b}[][]{(b)}
%        \psfrag{c}[][]{(c)}
        \epsfxsize=3.75in\leavevmode\epsfbox{fig_2.eps}
%\end{psfrags}
\end{center}
\caption{(a) The template ${\mathcal V}$; (b) a subtemplate of 
${\mathcal V}$; (c) removed from ${\mathcal V}$.} \label{fig_v_sub_v}
\end{figure}
% =========================================


Our primary tool in the analysis of templates is the application
of symbolic dynamics \cite{BW83b,HW85}. By crushing out the
transverse direction of the semiflow, a template ${\mathcal T}$ 
becomes a directed graph; hence, orbits on the template 
${\mathcal T}$ may be placed in a bijective correspondence with 
the space of symbolic itineraries $\Sigma_{\mathcal T}$ in a 
subshift of finite type \cite{Bow78}.

With this symbolic (Markov) structure, template inflations can be
represented as a symbolic action on the associated
Markov partition: each symbol in the sequence is inflated
to a particular word. This allows for the efficient computation
of very ``deep'' subtemplates through composing the inflations
and their associated symbolic actions.
% ::::::::::::::::::::::::::::::::::::::::::::::
\begin{example}
In Example \ref{ex_Mike}, the subtemplate of ${\mathcal V}$ 
induces an isotopic template inflation 
${\mbox{{\bf D}}}:{\mathcal V}\hookrightarrow{\mathcal V}$ which
in turn induces an action on $\Sigma_{\mathcal V}$ given by,
\begin{equation}
{{\mbox{{\bf D}}}}:
\Sigma_{\mathcal V}\hookrightarrow\Sigma_{\mathcal V}
\;\;\;\;\;\;\;\;\left\{
        \begin{array}{l}        x_1\mapsto x_1 \\
                                x_2\mapsto x_1x_2 \\
                                x_3\mapsto x_3 \\
                                x_4\mapsto x_3x_4 \end{array}\right. ,
\end{equation}
where $\{x_i: i=1\ldots 4\}$ is the Markov partition for ${\mathcal V}$ 
represented in Figure \ref{fig_v_sub_v}(a).
\end{example}
% ::::::::::::::::::::::::::::::::::::::::::::::

Our strategy involves using combinations of
isotopic inflations along with the induced actions
on symbol sequences. This allows us to keep track of the
topology of orbits on complicated subtemplates,
while also permitting symbolic ``coordinates'' for 
tracking deeply embedded orbits.

%*****************************************************************************
\section{Universal Templates} \label{sec_uni}
%*****************************************************************************

The following result resolves Conjecture \ref{conj_2} in the negative:
% ----------------------------------------------------
\begin{theorem} \label{thm_all}
The template ${\mathcal V}$ of Example \ref{ex_Mike}
contains an isotopic copy of every 
(tamely embedded) knot and link as periodic orbits of the semiflow.
\end{theorem}
% ----------------------------------------------------

\subsection*{Idea of proof:} 

Consider the templates ${\mathcal W}_q$, $q>0$, pictured in Figure 
\ref{fig_W_q}. These templates are embedded $q$-fold covers of 
${\mathcal V}$ which have an alternating sequence of $2q$ ``ears''.

% =========================================
\begin{figure}[htb]
\begin{center}
%\begin{psfrags}
%        \psfrag{q}[b][t]{$2q$}
        \epsfysize=0.8in\leavevmode\epsfbox{fig_3.eps}
%\end{psfrags}
\end{center}
\caption{The template ${\mathcal W}_q$ is a $q$-fold cover of 
		${\mathcal V}$.}
\label{fig_W_q}
\end{figure}
% =========================================


% ----------------------------------------------------
\begin{lemma} \label{lem_braids}
Let $b\in B_n$ be a braid on $n$-strands. Then $\bar{b}$,
the closure of $b$, appears as a (set of) periodic orbit(s)
on ${\mathcal W}_q$ for sufficiently large $q<\infty$.
\end{lemma}
% ----------------------------------------------------

\begin{proof} 
The concatenation of alternating positive and negative
ears on ${\mathcal W}_q$ mimics the group operation of $B_n$. It 
is simple to find a generating set for $B_n$, each element of
which ``fits'' on a finite concatenation of alternating ears,
as occurs in ${\mathcal W}_q$. 
\end{proof}

Given ${\mathcal T}$ a subtemplate of ${\mathcal V}$, 
we can consider the space of orbit symbol sequences 
$\Sigma_{\mathcal T}\subset\Sigma_{\mathcal V}$. By
specifying a form of ``symbolic surgery'' on $\Sigma_{\mathcal T}$, 
we modify ${\mathcal T}$ to create a new subtemplate ${\mathcal T}^+$ 
which has an additional ``ear'': the topological action
of this surgery is depicted in Figure \ref{fig_append}.

% =========================================
\begin{figure}[htb]
\begin{center}
%\begin{psfrags}
%        \psfrag{T}[][]{${\mathcal T}$}
%        \psfrag{P}[][]{${\mathcal T}^+$}
        \epsfxsize=3.5in\leavevmode\epsfbox{fig_4.eps}
%\end{psfrags}
\end{center}
\caption{(a) A closeup of the top ear of ${\mathcal V}$ and a subtemplate
	${\mathcal T}$; (b) appending an ear to ${\mathcal T}$
	yields ${\mathcal T}^+$.} \label{fig_append}
\end{figure}
% =========================================

% ----------------------------------------------------
\begin{proposition} \label{prop_W}
The template ${\mathcal W}_q$ appears as a subtemplate of 
${\mathcal V}$ for all $q>0$.
\end{proposition}
% ----------------------------------------------------
\begin{proof}
The technical part of the proof is to show that we can append 
ears to a copy of ${\mathcal V}$ in alternating fashion. To do so, we
first must map ${\mathcal V}$ into itself in such a way as to
avoid certain edges of ${\mathcal V}$, as suggested in Figure 
\ref{fig_append}. Then, after appending the appropriate ears, we 
send this latter ${\mathcal V}$ back into itself in such a way that 
we may iterate the procedure. The procedure is diagrammatically 
represented in Equation (\ref{eq_seq}). This step depends greatly 
on our ability to work with subtemplates symbolically.

% =========================================
\begin{equation} \label{eq_seq}
\begin{array}{cccccccc}
{\mathcal W}_1 & \stackrel{\mbox{\tiny append }}{\longrightarrow} &
{\mathcal W}_2 & \stackrel{\mbox{\tiny append }}{\longrightarrow} &
{\mathcal W}_3 & \stackrel{\mbox{\tiny append }}{\longrightarrow} &
{\mathcal W}_4 & \stackrel{\mbox{\tiny append }}{\longrightarrow} \\
\downarrow{\mbox{\tiny$=$}} & &
        \downarrow{\mbox{\tiny$\subset$}} & &
        \downarrow{\mbox{\tiny$\subset$}} & &
        \downarrow{\mbox{\tiny$\subset$}} & \\
{\mathcal V} & \hookrightarrow &
{\mathcal V} & \hookrightarrow &
{\mathcal V} & \hookrightarrow &
{\mathcal V} & \hookrightarrow 
\end{array} \; \cdots
\end{equation}
% =========================================
\end{proof}
% ----------------------------------------------------
\begin{proof}[Proof of Theorem \ref{thm_all}]
By Lemma \ref{lem_braids}, we can find any closed braid
as a periodic orbit set on some ${\mathcal W}_q$.
By Proposition \ref{prop_W}, every 
${\mathcal W}_q\subset{\mathcal V}$; hence, ${\mathcal V}$
contains all closed braids. By a theorem of Alexander
(see \cite{Ale23,Bir74}), ${\mathcal V}$ contains all knots 
and links.
\end{proof}

\begin{remark}\label{rem_8}
The proof is constructive, but does not necessarily yield
the ``simplest'' version of a closed braid in ${\mathcal V}$.
Consider Conjecture \ref{conj_1} concerning the existence of a
figure-eight knot ($K_8$) in a particular flow. A careful attempt 
to draw $K_8$ on ${\mathcal V}$ will frustrate the reader.
By computing symbol sequences, we calculate a
representation of the figure-eight knot in ${\mathcal V}$ (the 
simplest known example) which crosses the branch lines
$11,358,338$ times (i.e., this is the minimal period of the 
itinerary). The symbolic methods used in
these proofs extract very deep information.
\end{remark}

By employing a version of Alexander's Theorem due to Franks
and Williams \cite{FW85} along with our techniques, we can
extend Theorem \ref{thm_all}:

% ----------------------------------------------------
\begin{theorem}\label{thm_all_templates}
The template ${\mathcal V}$ contains all orientable templates as 
subtemplates. These may be chosen to be disjoint and completely 
unlinked.
\end{theorem}
% ----------------------------------------------------

This hints at a classification of (orientable) templates: a template
${\mathcal T}$ is {\em universal} $\Leftrightarrow$ ${\mathcal T}$ 
contains ${\mathcal V}$ $\Leftrightarrow$ ${\mathcal T}$ 
contains all templates. 

%*****************************************************************************
\section{Fibrations of Knot Complements} \label{sec_fibr}
%*****************************************************************************

Besides applications to template theory, Theorem \ref{thm_all}
has implications for flows induced by fibrations of knot and
link complements whose monodromies are of pseudo-Anosov type 
(in the Thurston
classification \cite{Thu88,FLP79}).
%A knot or link $K$ in $S^3$ is {\em fibred} if $S^3\setminus K$
%fibres over $S^1$ with fibre a Seifert spanning surface $M^2$
%\cite{Rol77}. In filling up the complement, $M^2$ induces a flow
%on $S^3\setminus K$ which induces a diffeomorphism $\Phi$ on the fibre
%called the {\em monodromy} of the fibration. By using the Thurston
%classification of surface diffeomorphisms \cite{Thu88,FLP79}, we
%can speak of ``the'' fibration for $K$. For example, t
The complement of the figure-eight knot in $S^3$ is fibred 
with fibre a punctured torus and monodromy isotopic to
the pseudo-Anosov map \cite{Thu82}
\[
        \Phi = \left[\begin{array}{cc}
		{2}&{1}\\{1}&{1}\end{array}\right]
		:{\mathbb R}^2\setminus{\mathbb Z}^2\rightarrow
			{\mathbb R}^2\setminus{\mathbb Z}^2 .
\]
The closed orbits
of the induced flow (the embedded suspension of $\Phi$) are the 
subject of \cite{BW83b} and of Conjecture \ref{conj_1}, which we 
now resolve.

% ----------------------------------------------------
\begin{theorem} \label{thm_fig8}
Any 
transverse flow to a 
fibration of the complement of the figure-eight knot in $S^3$
over $S^1$ 
%induces a flow on $S^3$ 
contains every tame
knot and link as closed orbits.
\end{theorem}
% ----------------------------------------------------

\begin{proof}
In \cite{BW83b}, Birman and Williams derived a template for the
fibration-induced flow on the complement of the figure-eight knot.
In \cite{Sul94a}, it was shown that this template contains 
${\mathcal V}$ as a subtemplate. By Theorem \ref{thm_all}, the 
figure-eight template is universal. This corresponds to the 
flow induced by the gradient of the 
particular
fibration which has pseudo-Anosov monodromy; however, pseudo-Anosov 
maps minimize dynamics, so any 
other flow transverse to a
fibration in the isotopy class 
has at least the periodic orbit link that the pseudo-Anosov case has
\cite{AF83,Thu88}. 
\end{proof}

\begin{remark}
As in Remark \ref{rem_8}, the simplest-known copy of the figure-eight 
knot which appears as a closed orbit of the flow induced by the
monodromy has period $11,358,338$ in the monodromy.
\end{remark}
Knowing which fibred knots (links) do {\em not} support every 
link as closed orbits of the fibration-induced flow 
would be useful, since,
for fibred knots (links), the [minimal] 
link of closed orbits of the fibration-induced flow 
forms an invariant for the knot (link)\cite{BW83b}. 
We have found, using branched 
covering techniques of Birman \cite{Bir79}, an infinite family of 
fibred links which support a universal template in the 
induced flow on the complement:

% ----------------------------------------------------
\begin{proposition} \label{prop_fibr}
The closure of any braid of the form 
$\left(\sigma_1\sigma_2^{-1}\right)^k$
for $\vert k\vert>1$ is a fibred link with fibration whose induced
flow on the complement supports all
knots and links as closed orbits. In particular, the Borromean rings
($k=3$) shares this property.
\end{proposition}
% ----------------------------------------------------
\begin{remark}
Since pseudo-Anosov maps have dense periodic point sets, the
flows induced on the complements of these closed braids can be
chosen so that the closed orbits fill up $S^3$ densely.
\end{remark}

%*****************************************************************************
\section{Third-Order ODEs} \label{sec_ODEs}
%*****************************************************************************

There are other implications which we do not describe in
this announcement, mostly in connection with third-order 
ODEs and global
bifurcations of periodic orbits in parametrized flows.
Notable among them is the following, explored in detail in 
\cite{GH95b}:
% ----------------------------------------------------
\begin{theorem}\label{thm_chua}
There exists an open set of parameters $\beta\in[6.5, 10.5]$
for which periodic solutions to the differential equation
\begin{eqnarray} \label{eq_chua} \nonumber
\dot{x} &=& 7[y-\phi(x)], \\ \nonumber
\dot{y} &=& x-y+z,\\
\dot{z} &=& -\beta y,  \\ \nonumber
\phi(x) &=& \frac{2}{7}x-\frac{3}{14}\left[\vert{x+1}\vert
		-\vert{x-1}\vert\right],
\end{eqnarray}
contain representatives from every knot and link equivalence class.
\end{theorem}
% ----------------------------------------------------
Equation (\ref{eq_chua}) is a PL-vector field modeling an electric
circuit \cite{CKM86}. This is but one example of a class of
ODEs supporting a particular type of homoclinic connection which
induces the existence of a universal template within the flow.






%%%%%\bibliography{../../refs/books,../../refs/haves,../../refs/havenots}

\bibliographystyle{amsplain}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\end{thebibliography}

\end{document}

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