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| jstor.org/stable/2974690
Idiosyncrasy. Use "$a\coloneqq b $" to mean "$a$ is defined to be $b$". We use the usual symbol, "$B^{c}$", for the complement of a set $B$; let "$A \stackrel{c}{—} B$ " denote the set-difference $A \cap B^{c}$. Symbol $\sqcup_{1}^{\infty} B_{k}$ indicates that the sets $\left\{B_{k}\right\}_{k}$ in the union happen to be disjoint.
If $\lambda$ is a measure such as "length" on a space $Y$, then a "$\lambda$-nullset" $B \subset Y$ has zero length, $\lambda(B)=0$. An example is a set $B$ consisting of finitely many points. The pair $(Y, \lambda)$ is called a measure space. A map $\varphi: X \rightarrow Y$ between two measure spaces $(X, \mu)$ and $(Y, \lambda)$ is measure-preserving if
$$
\mu\left(\varphi^{-1} B\right)=\lambda(B), \text { for all subsets } B \text { of } Y .
$$
A measure-preserving map $R: X \rightarrow X$ from a space to itself is called a transformation, which we write in full as $(R: X, \mu)$. Finally, for a point $z \in X$ let $R^{n}(z)$ denote the $n$-fold composition $R\left(R\left(.\mathclap{^n}.. R(z) \ldots\right)\right)$. The sequence of points $z, R(z), R^{2}(z), \ldots$ is the orbit of $z$ under $R$.
§1 PONCELET'S THEOREM. From any point outside of $\mathbf{E}$, a "righthand tangent to $\mathbf{E}$" can be consistently chosen. In particular, $\mathbf{E}$ gives rise to a "righthand homeomorphism," $R$, mapping $\mathbf{C}$ to itself.
![37306D38042[1].png](data/attachment/forum/month_2204/2204252202f66b6a0460ebed99.png "37306D38042[1].png")
Figure 1.1. Two ellipses, with $\mathbf{E}$ properly inside of $\mathbf{C}$. An observer standing at $z$ peering inward at $\mathbf{E}$ sees a "righthand" and a "lefthand" tangent to E. Call the righthand tangent map $R: C \rightarrow C$. Notation: From a point $z$ outside $\mathbf{E}$, let $|z, \mathbf{E}|_{L}$ and $|z, \mathbf{E}|_{R}$ denote the distance from $z$ to the lefthand and righthand tangent points on $\mathbf{E}$. In the case $\mathbf{E}$ is a circle, ie. $|z, \mathbf{E}|_{L}=|z, \mathbf{E}|_{R}$, agree to write $|z, \mathbf{E}|$ for the common value.
Viewed this way, Poncelet's theorem becomes a statement about the dynamics of the map $R$ and explains why it is called a Closure Theorem; if the orbit of one point $z$ "closes up" in $n$ steps, that $R^{n}(z)=z$, then the orbit of any point closes up and also in $n$ steps.
In closing up, if the orbit of $z$ "winds around $\mathbf{E}$" $p$ times ($p=1$ and $n=4$ in Figure P.1(a); in P.1 (b), $p=2$ and $n=5$) then every point's orbit would have to wind $p$ times before closing up, since $R$ preserves order along $\mathbf{C}$. This suggests that — after a suitable change of coordinates — $R$ is simply a rigid rotation of a circle by rational rotation number $p / n$. Let $\mathbb{K}:=[0,1)$ be the half-open interval topologized as a circle, and let $\oplus$ and $\ominus$ denote addition and subtraction modulo 1. For a rotation number $\alpha \in \mathbb{R}$, let
$$
\rho_{\alpha}: \mathbb{K} \rightarrow \mathbb{K}: x \mapsto x \oplus \alpha
$$
be the corresponding rigid rotation. A "change of coordinates" would then be a homeomorphism $\varphi: \mathbf{C} \rightarrow \mathbb{K}$ with a commutative diagram
$$\xymatrix{
\bf C \ar[d]^{φ} \ar[r]^R &\bf C\ar[d]^{φ}\\
\mathbb K \ar[r]^{ρ_α} &\mathbb K}$$
Such a $\varphi$ satisfying $\varphi \circ R=\rho_{\alpha} \circ \varphi$ is called a topological conjugacy from $R$ to $\rho_{\alpha}$. Poncelet's theorem would thus follow from
Lemma 1.3. For ellipses $\mathbf{C}, \mathbf{E}$ and mapping $R$ as in Figure 1.1, there is a rotation number $\alpha \in[0,1)$ and topological conjugacy $\varphi$ carrying $R$ to $\rho_{\alpha}$.
Indeed, this lemma asserts something even if the $R$-orbit of $z$ fails to close up — the case when $\alpha$ is irrational.
The appearance of a measure. Arclength measure $\lambda$ is invariant under the rotation $\rho_{\alpha}$, that is,
$$
\lambda\left(\rho_{\alpha}^{-1}(B)\right)=\lambda(B)
$$
for any set $B$ included in $\mathbb{K}$. We normalize $\lambda$ to a probability measure, $\lambda(\mathbb{K})=1$. A conjugacy $\varphi$ would lift $\lambda$ to an $R$-invariant measure $\mu$ on $\mathbf{C}$,
$$
\mu(A):=\lambda(\varphi(A))
$$
which we will call good: finite, non-atomic (any individual point has zero $\mu$-length) and giving positive length to open intervals.
It is the converse which will help us out:Any $R$-invariant good $\mu$ gives rise to a topological conjugacy. In order to define this conjugacy, for points $z, y \in \mathbf{C}$ let $[z, y)$ denote the half-open interval on $\mathbf{C}$ going counterclockwise from $z$ to $y$. Next, normalize $\mu$ so that $\mu(\mathbf{C})=\lambda(\mathbb{K})=1$; now, for any three points $z, y, x$ on $\mathbf{C}$
$$
\mu([z, y)) \oplus \mu([y, x))=\mu([z, x))
$$
Fix any particular point $z_{0} \in \mathbf{C}$. Define $\varphi$ and $\alpha$ by
$$
\varphi(y):=\mu\left(\left[z_{0}, y\right)\right) \text { and } \alpha:=\mu\left(\left[z_{0}, R z_{0}\right)\right) .
$$
Thus $\varphi$ sends $z_{0}$ to 0 in $\mathbb{K}$ and is well-defined because of the normalization. Non-atomicity implies continuity of $\varphi$ and the "positive length" condition insures that $\varphi$ is invertible. Invariance yields that for any $x$,
$$
\begin{aligned}
\mu([x, R x)) &=\mu\left(\left[R z_{0}, R x\right)\right) \ominus \mu\left(\left[R z_{0}, x\right)\right) \\
&=\mu\left(\left[z_{0}, x\right)\right) \ominus \mu\left(\left[R z_{0}, x\right)\right) \\
&=\mu\left(\left[z_{0}, R z_{0}\right)\right)=\alpha .
\end{aligned}
$$
In other words, the rotation number $\alpha$ did not truly depend on the arbitrary point $z_{0}$, but only on the $R$-invariant measure $\mu$. With this, it is easy to verify that $\varphi$ carries $R$ to $\rho_{\alpha}$, as in (1.2).
Our previous lemma can be restated now like this.
Lemma 1.3'. For ellipses $\mathbf{C}$ and $\mathbf{E}$, with $R$ as in Figure 1.1: There exists an $R$-invariant good measure $\mu$.
When $\mathbf{C}$ and $\mathbf{E}$ are concentric circles, arclength measure along $\mathbf{C}$ is $R$-invariant. However, the case where they are non-concentric circles seems not as apparent, and the general elliptical case is less evident still. Happily, we can at least assume that the outer ellipse $\mathbf{C}$ is a circle — since any linear map of the plane carries ellipses to ellipses and tangent chords to tangent chords and so carries Figure 1.1 to another just like it. So before even starting to construct $\mu$, one can linearly compress the figure along the major axis of $\mathbf{C}$ to arrange that now $\mathbf{C}$ is a circle.
Rolling the chord. The righthand map, $R$, generally stretches or shrinks sub-intervals $I \subset \mathbf{C}$, thus making the standard arclength non-invariant. A direct approach to making an $R$-invariant "length" $\mu$, is to compensate for stretch/shrink by integrating against arclength an appropriately chosen "height function" $h$ whose height varies so as to cancel out the distortion introduced by $R$. Then the "$\mu$-length" of a set $A$ would be
$$
\mu(A)=\mu_{h}(A):=\int_{A} h(z) d z \quad \begin{aligned}
&\text { where “$dz$" denotes arc- } \\
&\text { length measure on C. }
\end{aligned}
$$
If $h$ is a continuous function from $\mathbf{C}$ to the positive reals, then automatically $\mu$ will be non-atomic and give positive length to open intervals. The issue becomes: What property does $h(\cdot)$ need to fulfill for the measure $\mu=\mu_{h}$ to be $R$-invariant near a point $z$?
In Figure 1.7, the ratio $((z$-arc$) / \Delta z) \rightarrow 1$ as the angle $\theta \searrow 0$, and similarly $((y$-arc)$/ \Delta y) \rightarrow 1$. Consequently, the "infinitesimal ratio" of the $y$-arc to the $z$-arc is
$$
\lim _{\theta>0} \frac{y\text{-arc}}{z\text{-arc}}=\lim _{\theta>0} \frac{\Delta y}{\Delta z}=\frac{l_{y}}{l_{z}}
$$
this last equality comes from the equality $\measuredangle P y z=\measuredangle P z y$ of the base angles of isosceles triangle $y P z$ at the right of Figure 1.7.
![37306D38042[1].png](data/attachment/forum/month_2204/22042522397adc32f981379c9a.png "37306D38042[1].png")
Figure 1.7. Circle $\mathbf{C}$ surrounds ellipse $\mathbf{E}$. Let $z$ and $y$ be points at opposite ends of a chord tangent to $\mathbf{E}$, and let $\ell_{z}$ and $\ell_{y}$ denote their distance to the common point of tangency. Roll the chord through a small angle $\theta$. Its "$z$ end" sweeps out an arc on $\mathbf{C}$ as does its "$y$ end"; on the corresponding tangent lines, distances $\Delta z$ and $\Delta y$ are swept out.
Infinitesimally, the $\mu$-length of the "$z$-arc" is simply its arclength times $h(z)$. So for $\mu$ to be $R$-invariant at $z$, the function $h$ must satisfy that — infinitesimally — the product $h(z)$ Length ($z$-arc) equals $h(y)$ Length ($y$-arc). In consequence, the invariance condition required of $h(\cdot)$ is
$$
h(z) \cdot|z, \mathbf{E}|_{R}=h(y) \cdot|y, \mathbf{E}|_{L}, \text { where } y=R z .
$$
This uses the previous displayed-equation as well as $\ell_{z}=|z, \mathbf{E}|_{R}$ and $\ell_{y}=|y, \mathbf{E}|_{L}$.
Proof of Lemma 1.3': If our inside ellipse $\mathbf{E}$ happens to be a circle, then the lefthand and righthand tangential-distances equal a common function $z \mapsto|z, \mathbf{E}|$. Then
$$
h(z):=1 /|z, \mathbf{E}|
$$
satisfies (1.8a), and the corresponding $\mu$ is the desired $R$-invariant measure.
To handle the non-circular case, choose some linear map $\mathscr{A}$ which transforms $\mathbf{E}$ into a circle, as shown in Figure 1.9. Since a linear map preserves the ratio of lengths of parallel line-segments, we have that
$$
\frac{|y, \mathbf{E}|_{L}}{|z, \mathbf{E}|_{R}}=\frac{|\mathscr{A} y, \mathscr{A} \mathbf{E}|_{L}}{|\mathscr{A} z, \mathscr{A} \mathbf{E}|_{R}}
$$
But since $\mathscr{A} \mathbf{E}$ is a circle, the subscripts on the second ratio are unnecessary and it can be written $|\mathscr{A} y, \mathscr{A} \mathbf{E}| /|\mathscr{A} z, \mathscr{A} \mathbf{E}|$. Just as before, then,
$$
h(z):=\frac{1}{|\mathscr{A} z, \mathscr{A} \mathbf{E}|}, \quad \text { for all } z \in \mathbf{C},
$$
satisfies (1.8a) and makes the measure $\mu_{h}$-which we will call Poncelet-measure — invariant under $R$.
Exercise: Poncelet-measure for a special case. Suppose $\mathbf{C}$ and $\mathbf{E}$ are confocal ellipses, with foci at $(0, \pm F)$ and with semi-minor axis lengths of 1 and $r$ respectively, ie.
$\mathbf{C}: x^{2}+\frac{y^{2}}{F^{2}+1^{2}}=1^{2} \quad$ and $\quad \mathbf{E}: \frac{x^{2}}{r^{2}}+\frac{y^{2}}{F^{2}+r^{2}}=1^{2}$,
with $0< r<1$. Then Poncelet-measure $\mu_{h}$ will be the integral, against arclength measure along ellipse $\mathbf{C}$, of the function
$$
h(x, y)=\text { const } \cdot \frac{1}{\sqrt{\left(1+x^{2} F^{2}\right)\left(r^{2}+x^{2} F^{2}\right)}},
$$
where the constant is the square root of $\left(F^{2}+r^{2}\right) /\left(1-r^{2}\right)$.□
![37306D38042[1].png](data/attachment/forum/month_2204/22042522504e0df5c5649d30a0.png "37306D38042[1].png")
Figure 1.9. Ellipse $\mathbf{E}$ is carried by linear map $\mathscr{A}$ to a circle $\mathscr{A}(\mathbf{E})$. For artistic convenience, the linear map illustrated simply compresses the major axis (vertical, in the diagram) of $\mathbf{E}$ until equality with $\mathbf{E}$ 's minor axis. Each point of chord $y t z$ is carried vertically downward to line-segment $\mathscr{A}(y) \mathscr{A}(t) \mathscr{A}(z) ;$ here $t$ is the point where the $y z$-chord is tangent to $\mathbf{E}$.
Remark. What can be said when $\bf CE$ has no circuminscribed polygon? —that is, when $R$ is conjugate to an irrational rotation. One starts drawing tangent-chords from some point $z$, as in Figure 1.1, but the polygon-in-progress never closes up. What will be true is that the vertices of this unfulfilled polygon will densely fill out $\mathbf{C}$:Under an irrational rotation $\rho=\rho_{\alpha}$, the orbit of 0 is dense.(1.11) To establish this, partition the circle $\mathbb{K}$ into $M$ subintervals
$$
\left[0, \frac{1}{M}\right), \ldots,\left[\frac{M-1}{M}, 1\right) \text {. }
$$
Take $N \geqslant 1$ smallest such that $\rho^{N}(0)$ falls inside $\left[0,\frac1M\right)$. (This certainly will happen for some $N \leqslant M$ since, by the Pigeonhole Principle, some two of the $M+1$ points $\left\{0, \rho(0), \rho^{2}(0), \ldots, \rho^{M}(0)\right\}$ fall into the same subinterval.) Thus the $\rho^{N}$-orbit of 0 is $\left(\frac{1}{M}\right)$-dense. So its $\rho$-orbit is $\varepsilon$-dense, for all $\varepsilon$.□
By the way, it is natural to wonder whether the Poncelet rotation-number $\alpha$ is essentially unique; is it independent of linear map $\mathscr{S}$ ? That it is, follows from this challenge:If rotations $\rho_{\alpha}$ and $\rho_{\beta}$ are topologically conjugate, where $0 \leqslant \alpha, \beta \leqslant \frac{1}{2}$, then $\alpha=\beta$. Another natural question, which is discussed in the appendix, is to wonder
Is Poncelet-measure unique? How does
$\mu_{h}$ depend on our choice of linear map $\mathscr{A}$?(1.12)
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