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\documentclass{era-l} \pagespan{108}{123} \PII{S 1079-6762(96)00015-7} \usepackage{epsfig} \copyrightinfo{1997}{Marek Rychlik} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \theoremstyle{remark} \newtheorem{problem}[theorem]{Problem} \numberwithin{equation}{section} \newcommand\ints{{\mathbb Z}} \newcommand\nats{{\mathbb N}} \newcommand\reals{{\mathbb R}} \newcommand\complex{{\mathbb C}} \newcommand\disk{{\mathbb D}} \newcommand\torus{{\mathbb T}} \newcommand\proj{{\mathbb P}} \newcommand\bksl{{\backslash}} \newcommand\norm{\|} \newcommand\rank{\mathop{\rm rank}\nolimits} \newcommand\dist{\mathop{\rm dist}\nolimits} \newcommand\graph{\mathop{\rm graph}\nolimits} \newcommand\const{{\rm const}} \newcommand\sgn{\mathop{\rm sgn}\nolimits} \newcommand\coh[1]{\mathop{\sim}\limits_{#1}} \newcommand\ecoh[1]{\mathop{\sim}\limits_{#1}^{\epsilon}} \newcommand\closure[1]{\overline{#1}} \newcommand\mapright[1]{\smash{ \mathop{\longrightarrow}\limits^{#1}}} \newcommand\mapdown[1]{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \newcommand\rectangle[8] {\begin{matrix} #1 &\mapright{#2} \cr \mapdown{#4} & &\mapdown{#5} \cr #6 &\mapright{#7} \cr \end{matrix} } \newcommand\cA{{\mathcal A}} \newcommand\cB{{\mathcal B}} \newcommand\cC{{\mathcal C}} \newcommand\cD{{\mathcal D}} \newcommand\cE{{\mathcal E}} \newcommand\cF{{\mathcal F}} \newcommand\cG{{\mathcal G}} \newcommand\cH{{\mathcal H}} \newcommand\cJ{{\mathcal J}} \newcommand\cK{{\mathcal K}} \newcommand\cL{{\mathcal L}} \newcommand\cN{{\mathcal N}} \newcommand\cO{{\mathcal O}} \newcommand\cP{{\mathcal P}} \newcommand\cQ{{\mathcal Q}} \newcommand\cR{{\mathcal R}} \newcommand\cS{{\mathcal S}} \newcommand\cT{{\mathcal T}} \newcommand\cU{{\mathcal U}} \newcommand\cV{{\mathcal V}} \newcommand\cW{{\mathcal W}} \newcommand\cM{{\mathcal M}} \newcommand\cX{{\mathcal X}} \newcommand\cY{{\mathcal Y}} \newcommand\cZ{{\mathcal Z}} \newcommand\newcommandeq{{\buildrel \mathrm{def} \over =}} \newcommand\Tr{\,\hbox{Tr}} \newcommand\ad{\,\hbox{ad}} \newcommand\Ad{\,\hbox{Ad}} \newcommand\myspace{\torus\times\caplie{G}} \newcommand\E{\mathop{\hbox{$\mathbb E$}}\nolimits} \newcommand\D{\mathop{\hbox{$\mathbb D$}}\nolimits} \newcommand\diam{\mathop{\rm diam}\nolimits} \newcommand\Lip{\mathop{\rm Lip}\nolimits} \newcommand\Card{\mathop{\rm Card}\nolimits} \newcommand\Res{\mathop{\rm Res}\nolimits} \newcommand\supess{\mathop{\rm sup\,ess}} \newcommand\supinf{\mathop{\rm inf\,ess}} \def\fiverm{\usefont {\encodingdefault}{\seriesdefault}{\familydefault}{\shapedefault}} \newcommand\EPP{${\mathcal{EPP}}$} \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] \begin{picture}(0,0)\special{psfile=era15el-fig-2.eps}\end{picture}\setlength{\unitlength}{0.006250in}\begin{picture}(637,429)(40,360) \put(355,365){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$1$}}} \put(350,420){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$a$}}} \put(105,430){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$A_1$}}} \put(220,470){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$O_1$}}} \put(370,470){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$O$}}} \put(465,470){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$O_2$}}} \put(590,430){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$A_2$}}} \put(370,765){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P_0$}}} \put(215,665){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\Gamma(A_1)$}}} \put(535,665){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\Gamma(A_2)$}}} \put(155,535){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P_2$}}} \put(570,570){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P_{-1}$}}} \put(160,565){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$P_1$}}} \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] \begin{picture}(0,0)\special{psfile=era15el-fig-5.eps}\end{picture}\setlength{\unitlength}{0.006250in}\begin{picture}(576,391)(25,344) \put(157,456){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$A$}}} \put(127,389){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$W$}}} \put(189,547){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_N$}}} \put(179,507){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_{N+1}$}}} \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}$\phi_N$}}} \put(198,707){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_{N-1}$}}} \put(381,711){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\phi_1$}}} \put(601,551){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_0$}}} \put(551,631){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\phi_0$}}} \put(281,666){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$\phi_{N-2}$}}} \put(450,630){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}$V_1$}}} \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} \setlength{\unitlength}{0.012500in}\begin{picture}(135,96)(185,465) \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} \setlength{\unitlength}{0.010000in}\begin{picture}(345,171)(85,410) \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} \bibitem{blaschke} W.~Blaschke, W.~Rothe, and R.~Weitzenb{\"o}ck, \emph{Aufgabe 552}, Arch. Math. Phys. \textbf{27} (1917), 82. \bibitem{ehrhart} R.~Ehrhart, \emph{Un ovale {\`a} deux points isocordes?}, Enseignement Math. \textbf{13} (1967), 119--124. \MR{37:823} \bibitem{fujiwara} M.~Fujiwara, \emph{{\"U}ber die {M}ittelkurve zweier geschlossenen konvexen {K}urven in {B}ezug auf einen {P}unkt}, T{\^o}hoku Math J. \textbf{10} (1916), 99--103. \bibitem{gelfreich-lazutkin-svanidze} V. G. Gelfreich, V. F. Lazutkin, and N. V. Svanidze, \emph{Refined formula to separatrix splitting for the standard map}, Preprint, November 1992. \MR{95c:58152} \bibitem{michelacci-volcic} G.~Michelacci and A.~Volci{\v c}, \emph{A better bound for the excentricities not admitting the equichordal body}, Arch. Math. \textbf{55} (1990), 599--609. \MR{91m:52002} \bibitem{rychlik-equichordal} Marek Rychlik, \emph{A complete solution to the equichordal point problem of {F}ujiwara, {B}laschke, {R}othe and {W}eitzenb{\"o}ck}, Inventiones Math. (1996), accepted for publication. \bibitem{shaemke-volkmer} R.~Sch{\"a}fke and H.~Volkmer, \emph{Asymptotic analysis of the equichordal problem}, J. Reine Angew. Math. \textbf{425} (1992), 9--60. \MR{93d:52003} \bibitem{suss} W.~S{\"u}ss, \emph{Eilbereiche mit ausgezeichneten Punkten}, T{\^o}hoku Math J. (II) \textbf{24} (1925), 86--98. \bibitem{ushiki-standard-map} Shigehiro Ushiki, \emph{Sur les liaisons-cols des systemes dynamiques analytiques}, C.\ R.\ Acad.\ Sci.\ Paris \textbf{291} (1980), no.~7, 447--449. \MR{82b:58045} \bibitem{wirsing} E.~Wirsing, \emph{Zur {A}nalytizit{\"a}t von {D}oppelspeichenkurven}, Arch. Math. \textbf{9} (1958), 300--307. \MR{21:2205} \end{thebibliography} \end{document}