Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\copyrightinfo{1997}{Marek Rychlik}

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\begin{document}

\title{The Equichordal Point Problem}

\author{Marek Rychlik}
\address{Department of Mathematics,
University of Arizona, 
Tucson, AZ 85721}


\email{rychlik@math.arizona.edu}
\thanks{This research has been supported in part by the National
Science Foundation under grant no. DMS 9404419.}



\subjclass{Primary 52A10, 39A; Secondary 39B, 58F23, 30D05}

\commby{Krystyna Kuperberg}



\date{September 15, 1996}



\keywords{Equichordal, heteroclinic, convex, multi-valued}
\begin{abstract}
If $C$ is a Jordan curve on the plane and $P, Q\in C$, then the
segment $\overline{PQ}$ is called a {\em chord} of the curve
$C$. A point inside the curve is called {\em equichordal} if
every two chords through this point have the same length.
Fujiwara in 1916 and independently Blaschke, Rothe and
Weitzenb\"ock in 1917 asked whether there exists a curve with two
distinct equichordal points $O_1$ and $O_2$. This problem has
been fully solved in the negative by the author of this
announcement just recently. The proof (published elsewhere)
reduces the question to that of existence of heteroclinic
connections for multi-valued, algebraic mappings.  In the current
paper we outline the methods used in the course of the proof,
discuss their further applications and formulate new problems.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{The origins of the problem}
The Equichordal Point Problem (\EPP) was originally posed by
Fujiwara in 1916 \cite{fujiwara} and probably independently by
Blaschke, Rothe and Weitzenb\"ock in 1917 \cite{blaschke}.
\begin{problem}
Let $C$ be a Jordan curve on the plane and let $O$ be a point
inside the curve. We will call $O$ an {\em equichordal point} if
every chord of the curve $C$ passing through $O$ has the same
length (cf. Figure~\ref{basic-figure}). The \EPP\ asks whether
there is a curve with two distinct equichordal points.
\end{problem}
\begin{figure}[htb]
\begin{picture}(0,0)\special{psfile=era15el-fig-1.eps}\end{picture}\setlength{\unitlength}{0.006250in}\begin{picture}(379,407)(100,390)
\put(120,705){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P$}}}
\put(330,520){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$Q$}}}
\put(355,675){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$R$}}}
\put(205,390){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$S$}}}
\put(290,425){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\|P-Q\|=\|R-S\|$}}}
\put(285,560){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$O$}}}
\put(247,781){\makebox(0,0)[lb]{\smash{\SetFigFont{9}{10.8}{rm}$r=.5+.2\sin\theta+.2\cos3\theta$}}}
\end{picture}
\caption{\label{basic-figure}An equichordal point $O$ of a curve given in polar coordinates.}
\end{figure}
Any Jordan curve $C$ with two equichordal points $O_1$ and $O_2$
will be called an {\em equichordal curve}.  By scaling, we may
assume that all chords of $C$ passing through $O_1$ have length
$1$. Since there is a chord passing through both $O_1$ and $O_2$,
all chords passing through $O_2$ will also have length 1.

Once the above normalization has been made, the quantity
$a=\|O_1-O_2\|$ becomes a parameter of the \EPP\ known as the
{\em eccentricity} of $C$. Thus, the \EPP\ can be considered for
any particular value $a\in(0,1)$.

Due in part to its elementary formulation the \EPP\ has been
studied in a number of papers, either using elementary methods or
advanced analytical tools. Fujiwara~\cite{fujiwara} proved by
elementary methods that there cannot be three equichordal
points. Clearly, there are many curves with one equichordal
point. Roughly speaking, 
they can be
obtained by choosing a
 ``half'' of the curve almost arbitrarily and then ``reflecting''
 in the point $O$. The reader will easily fill in the detailed
 conditions needed in this construction. For instance, the origin
 is an equichordal point of a curve given in polar coordinates by
 the equation $r=f(\theta)$, where $f:\reals\to\reals $ is
 continuous, periodic with period $2\pi$,
 $f(\theta+\pi)+f(\theta)=1$ and $00$, but in the Dynamical
Systems tradition, we will ignore this dependence in our
notation. 

Every point $A$ of the line $O_1O_2$ with the exception of $O_1$
and $O_2$ also possesses a local invariant manifold
$W^{s,u}(A)$. The character of stability depends on where the
point is located. The invariant manifolds computed for $A$ near
$A_1$ and $A_2$ will foliate the neighborhoods of these points.

From the existence of a foliation it follows that if $C$ is an
equichordal curve, then $C\supset W^s_{loc}(A_1)\cup
W^u_{loc}(A_2)$.

The {\em global} invariant manifolds of $A_1$ and $A_2$ are
defined via the following formulas:
\begin{eqnarray*}
W^s(A_1)&=&\left\{P\in \reals^2\,|\,\lim_{n\to\infty} U^n(P)=A_1\right\},\\
W^u(A_2)&=&\left\{P\in \reals^2\,|\,\lim_{n\to-\infty}U^n(P)=A_2\right\}.
\end{eqnarray*}
It is clear that
\begin{eqnarray}
\label{invariant-manifold-formulas}
W^s(A_1)&=&\bigcup_{n=0}^\infty U^{-n}\left(W^s_{loc}(A_1)\right),\nonumber\\
W^u(A_2)&=&\bigcup_{n=0}^\infty U^{n}\left(W^u_{loc}(A_2)\right).
\end{eqnarray}
It is easy to see that 
\begin{eqnarray*}
W^s(A_1)&=&C\backslash\{A_2\},\\
W^u(A_2)&=&C\backslash\{A_1\}.
\end{eqnarray*}
For instance, $W^s(A_1)$ is an open arc of a Jordan curve,
invariant under an orienta\-tion-preserving homeomorphism
$U|C$. Thus, the endpoints of this arc are fixed points of $U|C$. But
there is only one fixed point of $U$ different from $A_1$, namely
$A_2$.  Hence, the complement of $W^s(A_1)$ in $C$ is $\{A_2\}$.

In conclusion, an equichordal curve admits the formula
\begin{equation}
C=W^s(A_1)\cup W^u(A_2).
\end{equation}
This proves that for any value of the eccentricity $a$ there can
be at most one equichordal curve. This formula and the symmetries
of $U$ imply that $C$ is symmetric with respect to reflections in
both axes.

More importantly, if $C$ existed, then $W^s(A_1)$ and $W^u(A_2)$
would form a {\em heteroclinic connection}, i.e. there would be
an arc in $W^s(A_1)\cup W^u(A_2)$ connecting $A_1$ to $A_2$.

We note that if $C$ does not exist, the sets $W^s(A_1)$ and
$W^u(A_2)$ may not even be well defined, as iterations of some
points will leave the domain of $U$.
\subsection{A summary of the elementary results}
These results which can be obtained from the rather elementary
Dynamical Systems considerations can be summarized in the
following:
\begin{theorem}
For any given value of the parameter $a$ there exists at most one
equichordal curve, up to rotations and dilations.  This curve is
a union of the invariant curves of the equichordal map $T$:
\[C=W^s(A_1)\cup W^u(A_2),\]
where $A_1=(-1/2,0)$ and $A_2=(1/2,0)$. If $C$ exists, then it is real-analytic
and symmetric with respect to reflections about both axes.

The necessary and sufficient condition of the existence of an
equichordal curve for a fixed $a$ is that the sets
(each consisting of two open arcs) $W^s(A_1)\backslash\{A_1\}$ and
$W^u(A_2)\backslash\{A_2\}$, coincide, i.e. that there is a
heteroclinic connection between $A_1$ and $A_2$.
\end{theorem}
\subsection{Oscillatory behavior and numerics}
It is known that if the set $W^s(A_1)\cap W^u(A_2)$ is discrete,
then $W^s(A_1)\cup W^u(A_2)$ forms a topologically complex
structure, schematically shown in Figure~\ref{homoclinic-figure}.
This structure indicates that the invariant manifolds $\Gamma(A_1)=W^s(A_1)$
and $\Gamma(A_2)=W^u(A_2)$ oscillate near $A_2$ and $A_1$, respectively.
\begin{figure}[htb]
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\put(580,540){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P_{-2}$}}}
\end{picture}

\caption{\label{homoclinic-figure}Intersecting invariant curves.}
\end{figure}
The invariant curves in the real domain can be easily
approximated by numerical methods for values of $a$ not close to
0. For instance, Figure~\ref{numerical-figure} and
Figure~\ref{magnified-figure} created with $a=0.6$ illustrate the
fact that the unstable invariant curve can be continued in the
real domain indefinitely and that it possesses oscillatory
behavior near the point $A_1$. The existence of an equichordal
curve would mean that for some parameter value the oscillation
would cease, so that the curve can be closed with an analytic
piece passing through $A_1$.
\begin{figure}[htb]
\centerline{\psfig{figure=era15el-fig-3.eps,height=3.5in}}\caption{\label{numerical-figure}The curve $\Gamma(A_2)$ for eccentricity $a=0.6$.}
\end{figure}
\begin{figure}[htb]
\centerline{\psfig{figure=era15el-fig-4.eps,height=3.5in}}\caption{\label{magnified-figure}The curve $\Gamma(A_2)$ for eccentricity $a=0.6$ 
magnified near $A_1$.}
\end{figure}
\section{Nonexistence of heteroclinic and homoclinic connections}
In this section we will focus on a range of techniques that can
be used to prove the nonexistence of heteroclinic or homoclinic
connections in ``regular'' dynamical systems.
\subsection{The complexification and multi-valuedness} 
As we have mentioned, the map $U$ admits an extension to the
complex domain as a 2-valued map. More precisely, except for the
points $(x,y)\in\complex^2$ for which $(x-b)^2+y^2=0$,
formula~\eqref{u-cartesian-formula} yields two {\em analytic}
branches of $U$.

The two branches of $U$ cannot be separated in a natural way for
the same reason that the two branches of $\sqrt{\cdot}$ cannot be
separated, i.e. each branch is an analytic continuation of the
other, and thus only choosing branch cuts can produce
single-valued branches. However, the introduction of the branch
cuts is a drastic operation that precludes any application of
global methods.  Thus, both branches of $U$ must be given equal
priority.

We note that the restriction of the complexification of $U$ back
to the real domain is a 2-valued real map. It is given by the
formula
\begin{equation}
\label{equichordal-walk-formula}
U(P)=-P\pm\frac{P-O_2}{\|P-O_2\|}.
\end{equation}
The computation of $U(P)$ may be described as a two-step process:
\begin{enumerate}
\item We select one of the two points of the line $PO_2$ which
are distant by 1 from $P$.
\item We reflect the point selected in the first step in the
origin $O$.
\end{enumerate}
The process of computing the $n$th iteration $U^n(P)$ involves
$2^n$ choices of the sign in
formula~\eqref{equichordal-walk-formula} and it somewhat resembles
a random walk.
\subsection{The single-valued case} 
The multi-valued character of $U$ is a source of numerous
technical difficulties in our solution of the \EPP. Therefore, it
will be beneficial to examine a very simple argument, first
published by Ushiki in \cite{ushiki-standard-map}, which is
applicable if instead of $U$ we consider a single-valued map
$F:\complex^2\to\complex^2$ such that $F^{-1}$ exists and is
single-valued as well. It is not difficult to see that in this
situation $F$ is a biholomorphic map.
\begin{theorem}
\label{ushiki-theorem}
Let $F:\complex^2\to\complex^2$ be a biholomorphic map. Let us
assume that $F$ has two hyperbolic fixed points $A_1$ and $A_2$
(not necessarily distinct).  The intersection $W^s(A_1)\cap
W^u(A_2)$ is at most a countable set.
\end{theorem}
In this theorem the sets $W^s(A_1)$ and $W^u(A_2)$ are the stable
manifold of $A_1$ and unstable manifold of $A_2$, respectively.
As in the real case, they are defined as follows:
\begin{eqnarray*}
W^s(A_1)&=&\left\{P\in \complex^2\,|\,\lim_{n\to\infty}F^n(P)=A_1\right\},\\
W^u(A_2)&=&\left\{P\in \complex^2\,|\,\lim_{n\to-\infty}F^n(P)=A_2\right\}.
\end{eqnarray*}
The fundamental theorem of Hadamard-Perron implies that these
sets are embedded copies of $\complex$.

We will say that there is a heteroclinic connection between $A_1$
and $A_2$ if there is a homeomorphism $\gamma:[0,1]\to\complex^2$
such that $\gamma(0)=A_1$, $\gamma(1)=A_2$ and
$\gamma(]0,1[)\subseteq W^s(A_1)\cap W^u(A_2)$ (if $A_1=A_2$, then
the term ``homoclinic connection'' is used).  Thus, the result of
Ushiki implies nonexistence of heteroclinic and homoclinic
connections for biholomorphic maps of $\complex^2$.

Two well-known examples of biholomorphic maps of $\complex^2$ are
\begin{enumerate}
\item $U(x,y)=(1-ax^2+y,bx)$, where $b\ne 0$ (the H\'enon map);
\item $U(x,y)=(2x-y+k\sin x, x)$, where $k\ne 0$ (the so-called
standard map).
\end{enumerate}
The original interest in these examples concerned only the real
domain.  The result of Ushiki by using complex-analytic methods
simplified the proof of chaotic behavior of these mappings for
all parameter values.
\begin{remark}
It is not known whether Theorem~\ref{ushiki-theorem} holds for a
biholomorphic map $F:\complex^n\to\complex^n$, $n\ge 3$.
\end{remark}
\subsection{A proof of Theorem~\ref{ushiki-theorem}}
Let us suppose that $W^s(A_1)\cap W^u(A_2)$ is uncountable. Let
us consider the set $X=W^s(A_1)\cup W^u(A_2)$. This set is a
connected Riemann surface. Moreover, $F(X)=X$, i.e. $G=F|X$ is an
automorphism of $X$. Let $\tilde X$ be the universal cover of $X$
and let $\tilde G$ be the lifting of $G$ to $\tilde X$. The map
$\tilde G$ is clearly an automorphism of $\tilde X$. Moreover,
$\tilde G$ has at least two fixed points in $X$, one attracting
and one repelling. The Uniformization Theorem tells us that
$\tilde X$ is isomorphic either to $\complex$, $\disk$ or 
$\hat\complex$ (Riemann sphere). The first two surfaces do not
admit automorphisms with two nonelliptic fixed points, and thus
$\tilde X$ is isomorphic to the Riemann sphere. Moreover, $\tilde
G$ is conjugate to a multiplication by a number. In addition
$X=\tilde X$. But $X\subseteq\complex^2$. We obtain a
contradiction by applying Liouville's theorem, as $\hat\complex$
cannot be embedded into $\complex^2$.
\subsection{Remarks on Theorem~\ref{ushiki-theorem} and \EPP}
The proof we have just presented is based on the study of the map
$F|X$ where the set $X=W^s(A_1)\cup W^u(A_2)$.  When there is a
heteroclinic or homoclinic connection, $X$ becomes a connected
Riemann surface, while $F|X$ is its automorphism. This
observation outlines our strategy for handling heteroclinic and
homoclinic connections for multi-valued mappings. The main
complication is that $F|X$ is no longer an automorphism but 
a multi-valued mapping itself. Therefore, a straightforward
application of the Uniformization Theorem needs to be
replaced by a more involved argument.
\subsection{The invariant manifolds of multi-valued maps}
The local invariant manifold theory  can be applied to the branches
of a multi-valued map. For instance, $U$ has the principal branch
$U_+$ defined by the principal branch of $\sqrt{\cdot}$. This
branch, when considered in the complex domain, is nonsingular at
$A_1$ and $A_2$. Therefore, there exist local invariant manifolds
$W^s_{loc}(A_1)$ and $W^u_{loc}(A_2)$, this time considered as 
subsets of $\complex^2$. These manifolds are analytically
embedded disks.

The maps $U_+|W^s(A_1)$ and $U_+|W^u(A_2)$ may be analytically
linearized, according to a result dating back to
Poincar\'e. Thus, there exist functions
$\psi_1:W^s(A_1)\to\complex$ and $\psi_2:W^u(A_2)\to\complex$,
biholomorphic in a neighborhood of $A_1$ and $A_2$ respectively,
such that for $i=1,2$ we have
\begin{equation}
\psi_i\circ U_+ = \lambda_i\cdot\psi_i
\end{equation}
where $\lambda_i=(1\mp a)/(1\pm a)$ are the eigenvalues of
$DU_+(A_i)$ corresponding to the vertical direction.
The functions $\psi_i$, $i=1,2$ will be called the {\em local
linearizing parameters}.

The global invariant manifolds $W^s(A_1)$ and $W^u(A_2)$ cannot
be simply constructed via
formulas~\eqref{invariant-manifold-formulas}. The formulas
themselves could be made sense of by allowing {\em all} images or
preimages under $U$. However, the resulting sets would not have a
nice local structure.

It is possible to define two Riemann surfaces, which we will denote
by $W^s(A_1)$ and $W^u(A_2)$, in a way that will make them much
more useful to answer the question about the existence of
heteroclinic or homoclinic connections. The construction will be
performed in two stages. The first stage consists in constructing
the {\em unbranched} invariant manifolds ${_0W}^s(A_1)$ and
${_0W}^u(A_2)$, which will possibly have
countable numbers of punctures. The Riemann surfaces $W^s(A_1)$
and $W^u(A_2)$ are obtained by simply filling in the punctures.

Let us first define the unbranched stable manifold ${_0W}(A_1)$.
Informally, it consists of all germs of orbits
$(y_n)_{n=0}^\infty$ of the multi-valued map $U$ such that for
sufficiently large $n$ the point $y_n\in W^s_{loc}(A_1)$ and
$y_{n+1}=U_+(y_n)$. We are about to give a more detailed
definition.
\begin{figure}[thb]
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\put(196,621){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\phi_{N-1}$}}}
\put( 78,527){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\phi_N$}}}
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\end{picture}

\caption{\label{stable-manifold-figure}The unbranched stable
manifold of $A=A_1$ constructed from $W=W^s_{loc}(A_1)$.}
\end{figure}
The reader should consult Figure~\ref{stable-manifold-figure} for
a graphic illustration.
\begin{definition}
The unbranched stable manifold of the point $A=A_1$ is a Riemann
surface ${_0W}^s(A)$  which as a set consists of sequences of germs
$(m_n)_{n=0}^\infty$, where each $m_n$ is a germ of a curve
$V_n$ at a point $y_n$. We will require the following additional
properties:
\begin{enumerate}
\item for every $n\ge 0$ there is a unique regular local branch
$\phi_n$ of the relation $U$ such that $\phi_n(V_n)=V_{n+1}$ and
$\phi_n(y_n)=y_{n+1}$;
\item for sufficiently large $n$ we have $V_n\subseteq
W_{loc}^s(A)$ and $\phi_n=F$.
\end{enumerate}
\end{definition}
The branch points, resulting in punctures, appear when one of the
curves $V_n$ intersects the branch manifold of $U$, i.e. the set
of those $(x,y)$ for which $(x-b)^2+y^2=0$, or $x-b=\pm iy$. One
also has to consider possible branch points at infinity. No
singularity more complicated than a branch point can be
generated, due to the fact that $U$ is algebraic.

The branch points appear a finite number at a time, i.e. if we
put an upper bound of $n$ in the above construction, then only a
finite number of branch points are involved. Due to this property,
we may recursively fill in the punctures and obtain a Riemann
surface $W^s(A_1)$.

The construction of the unstable manifold is carried out in an
analogous way, by considering the germs of trajectories
$(y_n)_{n=-\infty}^0$ such that for sufficiently large negative
$n$ we have $y_n\in W^u_{loc}(A_2)$ and $y_{n+1}=U_+(y_n)$.
\subsection{The connection surface}
\begin{figure}[thb]
\begin{picture}(0,0)\special{psfile=era15el-fig-6.eps}\end{picture}\setlength{\unitlength}{0.006250in}\begin{picture}(639,370)(20,449)
\put(122,504){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$W_1$}}}
\put(184,662){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_1$}}}
\put(193,737){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$R$}}}
\put(511,666){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_2$}}}
\put(454,500){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$W_2$}}}
\put(285,779){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$R$}}}
\put(395,750){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$R$}}}
\put(152,571){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$A_1$}}}
\put(475,563){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$A_2$}}}
\end{picture}

\caption{\label{heteroclinic-figure}The connection surface.}
\end{figure}
When there exists a heteroclinic connection, a third Riemann
surface, called the {\em connection surface} and denoted by
$\cH$, can be constructed by considering the germs of the
double-sided trajectories $(y_n)_{n=-\infty}^\infty$ of $U$ such
that for sufficiently large $n$ we have $y_n\in W^s_{loc}(A_1)$,
$y_{n+1}=U_+(y_n)$ and for sufficently large negative $n$ we have
$y_n\in W^u_{loc}(A_2)$, $y_{n+1}=U_+(y_n)$
(cf. Figure~\ref{heteroclinic-figure}). Again, with a small
effort we may fill in the punctures.

The two mappings which truncate the double-sided sequences
$(y_n)_{n=-\infty}^\infty$ to the one-sided sequences
$(y_n)_{n=0}^\infty$ and  $(y_n)_{n=-\infty}^0$ induce two
holomorphic mappings $p_1:\cH\to W^s(A_1)$ and $p_2:\cH\to
W^u(A_2)$, which we will call {\em the projections}.
We note that in the single-valued case these mappings are
injective and allow us to write $\cH=W^s(A_1)\cap W^u(A_2)$.
In the multi-valued situation, we only have the following diagram
of holomorphic mappings:
\begin{equation}
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\thicklines
\put(250,545){\vector(-3,-4){ 45}}
\put(275,545){\vector( 3,-4){ 45}}
\put(315,505){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$p_2$}}}
\put(185,465){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$W^s(A_1)$}}}
\put(230,505){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$p_1$}}}
\put(255,550){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\cH$}}}
\put(310,465){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$W^u(A_2)$}}}
\end{picture}
\end{equation}
\subsection{The shift maps, projections and linearizing parameters}
The shift (to the left) on the sequences $(y_n)_{n=0}^\infty$
induces a (single-valued) map $\sigma_1:W^s(A_1)\to
W^s(A_1)$. Without difficulty we verify that $\sigma_1$ is
analytic. In a similar way, the shift to the right induces an analytic map
$\sigma_2:W^u(A_2)\to W^u(A_2)$.

When there is a heteroclinic or homoclinic connection, the shift
to the left on double-sided sequences induces a biholomorphic map
$\sigma:\cH\to\cH$.

The linearizing parameters allow extensions to $W^s(A_1)$ and
$W^u(A_2)$. Indeed, we may naturally consider
$W^s_{loc}(A_1)\subset W^s(A_1)$ and $W^u_{loc}(A_2)\subset
W^u(A_2)$.  For instance, any point $y_0\in W^s_{loc}(A_1)$ is
identified with the unique sequence $(y_n)_{n=0}^\infty$ such
that for all $n\ge 1$ we have $y_{n+1}=U_+(y_n)$. Subsequently,
we define
\begin{eqnarray*}
\psi_1(m)&=&\lim_{n\to\infty}\lambda_1^{-n}\psi_1(\sigma_1^n(m)),\\
\psi_2(m)&=&\lim_{n\to\infty}\lambda_2^{n}\psi_2(\sigma_2^n(m)),
\end{eqnarray*}
where $m\in W^s(A_1)$ and $m\in W^u(A_2)$ respectively. The
functions $\psi_1:W^s(A_1)\to\complex$ and
$\psi_2:W^u(A_2)\to\complex$ defined in this way are analytic,
surjective and satisfy the equations
\begin{eqnarray*}
\psi_1\circ \sigma_1 &=& \lambda_1\cdot\psi_1,\\
\psi_2\circ \sigma_2 &=& \lambda_2^{-1}\cdot\psi_2.
\end{eqnarray*}
The following diagram summarizes the relationships between
various objects that we have constructed. In particular, the two
parallelograms contained in it commute:
\begin{equation}
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\thicklines
\put(150,490){\vector(-3,-4){ 45}}
\put(175,490){\vector( 3,-4){ 45}}
\put(360,565){\vector(-3,-4){ 45}}
\put(385,565){\vector( 3,-4){ 45}}
\put(110,430){\vector( 3, 1){180}}
\put(410,490){\vector(-3,-1){180}}
\put(185,510){\vector( 3, 1){165}}
\put(345,570){\vector(-3,-1){165}}
\put(215,450){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$p_2$}}}
\put( 85,410){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$W^s(A_1)$}}}
\put(130,450){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$p_1$}}}
\put(155,495){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\cH$}}}
\put(210,410){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$W^u(A_2)$}}}
\put(425,525){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$p_2$}}}
\put(295,485){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$W^s(A_1)$}}}
\put(340,525){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$p_1$}}}
\put(365,570){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\cH$}}}
\put(420,485){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$W^u(A_2)$}}}
\put(240,520){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\sigma$}}}
\put(250,465){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\sigma_1$}}}
\put(315,440){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\sigma_2$}}}
\put(230,540){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\sigma^{-1}$}}}
\end{picture}
\end{equation}
\subsection{The shadow maps}
There are three maps $Sh_1:W^s(A_1)\to\hat\complex^2$,
$Sh_2:W^u(A_2)\to\hat\complex^2$ and $Sh:\cH\to\hat\complex^2$
which will be called the {\em shadow maps}.  Each of them is
induced by mapping a sequence $(y_n)$ to $y_0$.  The images lie
in $\hat\complex^2$ as certain branch points map to $\infty$. The
images of the shadow maps are complicated subsets of
$\hat\complex^2$ which are roughly the same as the result of
application of the formulas~\eqref{invariant-manifold-formulas}.
Roughly speaking, the Riemann surfaces $W^s(A_1)$, $W^u(A_2)$ and
$\cH$ are desingularizations of the subsets $Sh_1(W^s(A_1))$,
$Sh_2(W^u(A_2))$ and $Sh(\cH)$ of $\hat\complex^2$. We leave it
to the reader to define the last three sets more directly by
using trajectories.

\subsection{The classification of components}


Our main effort will be to show that there is an algebraic curve
$V\subseteq \hat\complex^2$ such that 
\begin{equation}
\label{compact-curve-condition}
Sh_1(W^s(A_1))\cup Sh_2(W^u(A_2))\subseteq V.
\end{equation}
In view of Chow's Theorem, it is sufficient to find a compact
analytic variety $V$ with the above property, as then this is
automatically an algebraic variety. The implication for the \EPP\
would be that the equichordal curve, if it existed, would be
algebraic. Showing nonexistence of an algebraic equichordal curve
proves to be a relatively easy task, and thus our strategy leads to
the solution of the \EPP. For a detailed argument the reader
should consult \cite{rychlik-equichordal}.

The connected components of the connection surface $\cH$ are 
permuted by the automorphism $\sigma$. Thus, $\cH$ splits into a
union of cyclically permuted components, where infinite cycles
are allowed. A connected component $M$ of $\cH$ will be called
{\em elliptic}, {\em parabolic} or {\em hyperbolic}, depending on
whether the universal covering space of $M$ is isomorphic to
$\hat\complex$ (Riemann sphere), $\complex$ or $\disk$
(Poincar\'e disk). Each type of component is analyzed
by a separate argument. 
\subsection{Elliptic and parabolic components}
The following result admits an easy proof:
\begin{lemma}
There are no elliptic connected components of $\cH$.
\end{lemma}
A bit longer argument leads to the following classification of
parabolic components:
\begin{theorem}
If $M$ is a parabolic connected component  of $\cH$, then
$M$ is a part of a cycle of length $1$,
i.e. $\sigma(M)=M$. Furthermore, $W^s(A_1)$ and $W^u(A_2)$ are
isomorphic to $\complex$, and $\cH$ is isomorphic to $\complex_*$
(cylinder). There exists a unique algebraic curve $V$ of genus
$0$ such that
\begin{equation}
W^s_{loc}(A_1)\cup W^u_{loc}(A_2)\subseteq V.
\end{equation}
Moreover, $Sh_1(W^s(A_1))\cup Sh_2(W^u(A_2))\subseteq V$.
\end{theorem}
Thus $V$ is a rational variety isomorphic to $\hat \complex=\proj_1$
(the projective space of dimension 1). This situation corresponds
to the result stated in Theorem~\ref{ushiki-theorem}. It is easy
to check that $U|V$ has a single-valued branch conjugate to
multiplication by a number, just as in Theorem~\ref{ushiki-theorem}.
\subsection{Hyperbolic components}
The most subtle point of our solution of the \EPP\ is an analysis
of the hyperbolic components of the connection surface $\cH$. We
refer the reader to \cite{rychlik-equichordal} for details. We
note that the analysis uses theorems of Fatou and Riesz
concerning the boundary behavior of complex functions defined on
the unit disk.
\subsection{The invariant parameter and compactness}
The function \[\psi=(\psi_1\circ p_1)\cdot(\psi_2\circ p_2)\] is
well defined and analytic on $\cH$ and it has the property $\psi\circ
\sigma=\psi$, due to the {\em resonance condition}
$\lambda_1\lambda_2=1$. Thus, it is natural to call $\psi$ the
{\em invariant parameter}.

The following class of connected components of $\cH$ will play a
critical role in our solution of the \EPP:
\begin{definition}
A connected component $M$ of $\cH$ is called {\em regular} iff the
invariant parameter $\psi|M$ is constant.
\end{definition}
By $\cH_{reg}$ we denote the union of all regular components.
Our next major goal is to prove that $\cH_{reg}\neq\emptyset$.
It will be accomplished by variational methods.
\subsection{The extreme property of $A_1$ and $A_2$}
At the heart of our method there is a variational method. It is
based on the observation that $A_1$ and $A_2$ have a special
extreme property. We proceed to describe this property in the
case of $A_1$ in detail. Let $\lambda_1=(1-a)/(1+a)$ be the
eigenvalue of the linearization of $U$ at $A_1$ along the
vertical direction. Let us study the sequences of points 
$(P_n)_{n=0}^\infty$ with the property that $U(P_n)=P_{n+1}$ for
all $n\ge 0$ and $\|P_n-A_1\|\le K\lambda_1^n$, where $K$ is a
constant. It can be shown that for sufficiently large $n$ we have
$U=U_+$ where $U_+$ is the principal branch of $U$. In other
words, the only way to approach $A_1$ with the rate $\lambda_1$ is to
follow the principal branch of $U$. Other sequences $P_n$,
obtained by a different choice of the branches of $U$, may
still have the property $\lim_{n\to\infty}P_n=A_1$. However, they
will approach $A_1$ at a rate slower than $\lambda_1$.

The extreme property eventually produces compactness of the
regular components of $\cH$. This fact is crucial in our proof.
\appendix
\section{The result of Sh\"afke and Volkmer}
The result of Sh\"afke and Volkmer \cite{shaemke-volkmer}
addresses the problem of quantifying the oscillatory behavior of
the trajectories of $U$. We will formulate this result in the
notation used in this paper.
\begin{theorem}
Let $P_n=T^n(P_0)$ for all $n\ge 0$, where $P_0=(-b,1/2)\in
C$. Let $P_n=(x_n,y_n)$.  Then $\lim_{n\to\infty}x_n=-(1+h(a))/2$,
where
\begin{equation}
h(a)=\omega e^{-\frac{\pi^2}{2a}}\left[1+\frac{\pi^2}{24}a+O(a^2)\right],
\end{equation}
and $1.359\le \omega\le 1.361$.
\end{theorem}
We note the fact that $P_0$ belongs to the equichordal curve
$C$, should one exist. This is a consequence of the reflectional
symmetries of $C$. Moreover, if $C$ exists, then
$h(a)=0$. Therefore, the asymptotics of $h(a)$ given in the above
theorem implies nonexistence of equichordal curves for
sufficiently small $a$.  This result and the analyticity of $h$
imply that there may be only a finite number of values
of $a$ for which there is an equichordal curve.
\section{A historical sketch}
\subsection{The first result}
Fujiwara~\cite{fujiwara} showed that there are no convex curves
with three equichordal points. This result should be considered
elementary.
\subsection{The case of large eccentricities}
The progress on the \EPP\ was marked by results which gradually
decrease the range of eccentricities for which this theorem
holds. For instance, it is not too difficult to see that there are
no equichordal curves for $a\in (1/2,1)$ \cite{ehrhart}. As we decrease the
lower limit, nonexistence becomes gradually more difficult to
prove. A typical result of this kind is formulated as
\begin{theorem}
There are no equichordal curves with eccentricities $a\in
(\epsilon,1]$.
\end{theorem}
The basis for the results in this direction is a result of
G.~A. Dirac (1952), according to which the equichordal curve,
if it exists, lies inside the set $B(O_1,1/2+b)\cup
B(O_2,1/2+b)$ and outside of the set $B(O_1,1/2-b)\cup
B(O_2,1/2-b)$. It has also been known since the 1920's \cite{suss} that the
equichordal curve would have to be symmetric with respect to the
reflection in the line $O_1O_2$ as well as in the bisector of the
segment $O_1O_2$. Let $Z$ be the point defined by the property
that $Z$ lies on the perpendicular to the line $O_1O_2$ at $O_2$
and that $\|Z-O_2\|=1/2$. The symmetries imply that if $C$ is an
equichordal curve, then $Z\in C$. The strategy,
exemplified by \cite{michelacci-volcic} is to consider the
sequence of points $T^n(Z)$, where $T=T_1\circ T_2$ ($T^n$
denotes the $n$-fold composition).  It proves that this sequence
simultaneously converges to the line $O_1O_2$ and oscillates. Thus, 
it  will fail to satisfy the Dirac bounds.

\subsection{The case of small eccentricities}
The asymptotic analysis of the \EPP\ was initiated by
E.~Wirsing~\cite{wirsing} in 1958 and it was continued by
R.~Sch\"afke and H.~Volkmer in \cite{shaemke-volkmer}. In this
approach, one studies the asymptotic problem as $a\to 0$. Wirsing
discovered that the problem belongs to the category of
perturbation theory ``beyond all orders'', the first problem of
this kind known to us and rigorously demonstrated not to be
solvable by a power series in powers of $a$. The result of
Sch\"afke and Volkmer proves the absence of equichordal
curves for sufficiently small $a$. They claim to have proven that
there are no equichordal curves for $a<.03$.  However, this
result requires a careful analysis of truncation and round-off
errors while juggling between a function, its Laplace transform
and the Taylor series.
\section{The measurable Equichordal Point Problem}
Yet another interpretation of \EPP\ is contained in the following
open problem:
\begin{problem} [The measurable \EPP]
Let $D$ be a measurable subset of the plane. A point $O$ of the
plane is called a {\em measurable equichordal point} if for every
straight line $\ell$ passing through $O$ the Lebesgue measure of
the intersection $\ell\cap D$ is nonzero, finite and it does not
depend on $\ell$. Is there a set $D$ with two measurable
equichordal points?
\end{problem}
This problem seems to require methods that are quite different
from the ones considered in the solution of the original \EPP.
\section{A one-dimensional classification problem}
A question arises concerning which Riemann surfaces can occur as a
connection surface $\cH$ for multi-valued algebraic maps similar
to the one that appears in the \EPP. This question will be reformulated
abstractly in this section.

Let $M$ be a noncompact Riemann surface and let $\sigma:M\to M$
be an automorphism such that the cyclic group
$\Gamma=\{\sigma^n:\;n\in\ints\}$ acts on $M$ freely and
discretely and co-compactly, i.e. the quotient $C/\Gamma$ is
compact.  Moreover, let $\psi:M\to\complex$ be a holomorphic
function on $M$ which satisfies the functional equation
\[\psi\circ\sigma=\mu\psi\] where $\mu\in\complex_*$ and
$|\mu|\ne 1$. 
\begin{problem}
Let $g$ be the genus of the compact Riemann surface
$M/\Gamma$. Is it possible that $g\ge 2$?
\end{problem}
This question is motivated by the question whether the connection
surface $\cH$ may contain a component $M$ such that $Sh(M)$ is
contained in an algebraic curve of genus $\ge 2$. 

Recently J.~Mather has given a positive answer to this question
(verbal communication). In this section we will explain briefly
how his example is constructed.

Let us assume that there exists a nonconstant map
$\phi:S\to\complex_*/\Lambda$, where
$\Lambda=\{\mu^j:\,j\in\ints\}$ is a multiplicative subgroup of
$\complex_*$ acting on $\complex_*$ by multiplication and $S$ is
a Riemann surface of genus $g$. The fiber product $M\subseteq
S\times\complex_*$ defined as the set of all pairs $(z,w)$ such
that $\phi(z)=\pi(w)$ where $\pi:\complex_*\to\complex_*/\Lambda$
is the natural projection, admits an automorphism $\sigma:M\to M$
mapping $(z,w)$ to $(z,\mu w)$. It is clear that the pair
$(M,\sigma)$ has the desired property. Indeed, $M$ has genus $\ge
g$. However, $M$ may or may not be connected. Nevertheless, we may
select a connected component of $M$ mapped into itself by some
power of $\sigma$, and obtain a connected example in this way.

We note that it is not difficult to construct the map $\phi$. We
need to consider a multi-valued function on $\complex/\Lambda'$,
for instance, $\sqrt{\wp(z)}$ where $\wp$ is the Weierstrass
function. The corresponding Riemann surface $N$ and the
branched covering $\phi:N\to\complex/\Lambda'$ satisfy the
requirements of our construction. The particular choice
$\sqrt{\wp(z)}$ yields $N$ of genus 2.


\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}




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R.~Ehrhart, \emph{Un ovale {\`a} deux points isocordes?}, Enseignement Math.
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\end{thebibliography}



\end{document}