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This article is cited in 2 scientific papers (total in 2 papers) Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology V. Dragovićab, S. Gasiorekc, M. Radnovićcb a Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, USA b Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrad, Serbia c School of Mathematics and Statistics, University of Sydney, Sydney, Australia
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 9, DOI: https://doi.org/10.4213/sm9662 
Bibliographic databases:
    § 1. Symmetric $2$–$2$-relations, elliptic functions and elliptical billiardsThis work is dedicated to Rodney Baxter on the occasion of his 80th anniversary. The last section of Baxter’s celebrated book ([9], p. 471) starts with: “In the Ising, eight-vertex and hard hexagon models weencounter symmetric biquadratic relations, of the form (1.1)”. Here
$$
\begin{equation}
E\colon au^2v^2+b(u^2v+uv^2)+c(u^2+v^2)+2duv+e(u+v)+f=0.
\end{equation}
\tag{1.1}
$$
In the sequel of the last section of [9] Baxter derives an elliptic parametrization of a symmetric biquadratic relation, providing an effective proof of Euler’s classical theorem, which denoted the beginning of the study of elliptic functions and related addition theorems (see [30]). Theorem 1.1 (Euler’s theorem, 1766). For a general symmetric $2$–$2$-correspondence (1.1) there exists an even elliptic function $\phi$ of the second degree and a constant shift $c$ such that Elliptic functions and their addition formulae play a prominent role in the entire Baxter opus [5]–[8] and in the theory of integrable systems in general. In his book, in particular, they already appear in the second paragraph of the preface. For further references we list well-known identities for Jacobi elliptic functions (see [4], for instance):
$$
\begin{equation}
\begin{gathered} \, \kappa^2\operatorname{sn}^2z+\operatorname{dn}^2z=1, \\ \operatorname{sn}(z+w)=\frac{\operatorname{sn} z\operatorname{cn}w\operatorname{dn}w + \operatorname{sn} w\operatorname{cn}z\operatorname{dn}z}{1-\kappa^2\operatorname{sn}^2 z \operatorname{sn} ^2 w}, \qquad \operatorname{sn} (K-z)=\frac{\operatorname{cn}z}{\operatorname{dn}z}, \\ K=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-\kappa^2t^2)}}. \end{gathered}
\end{equation}
\tag{1.2}
$$
Here $\kappa$ is a constant different from $0$ and $1$. By using the above argumentation, Baxter managed to get his celebrated $R$-matrix, which is also known as the $XYZ$ $R$-matrix and the Eight Vertex Model $R$-matrix because of its fundamental role in both of these very important models of quantum and statistical mechanics, respectively. Symmetric biquadratic relations (1.1) also play an important role in Poncelet’s theorem and related questions of integrable billiards within conics. Let us start with the situation of Poncelet’s theorem. Suppose conics $\Gamma$ and $\mathcal{K}$ are given. Consider the $2$–$2$-correspondence on $\Gamma$ induced by $\mathcal{K}$ in the following way. To a point $M\in\Gamma$ one can assign points $M_1$ and $M_1'$ on $\Gamma$ such that the lines $L_{MM_1}$ and $L_{MM_1'}$ are tangent to the conic $\mathcal{K}$. In this way a symmetric $2$–$2$-correspondence is defined. Moreover, every symmetric $2$–$2$-correspondence on a conic is defined in this way. The Italian mathematician Trudi studied Poncelet’s theorem around 1853 in terms of compositions of such symmetric $2$–$2$-relations and provided another proof in [43] and [44]. Recall that the addition formulae for Jacobi elliptic functions were used by Jacobi himself in his proof of Poncelet’s theorem for circles. More about symmetric $2$–$2$-relations and their role in integrable systems can be found in [47], [48], [29] and [22]. In the present paper we study a new instance of symmetric $2$–$2$-relations, which appears in integrable billiard dynamics in the Minkowski space, on a hyperboloid of one sheet. Such billiards were recently introduced in [37], where Poncelet-type theorems and the corresponding analytic conditions for periodicity were derived. In this work we study periodicity conditions for the dynamics in accordance with the general ideology from [27]: we relate them to extremal polynomials on unions of two intervals. As is known from classics (see [51] and [4]), such polynomials are parametrized by elliptic functions. Identities and addition formulae for elliptic functions, like (1.2), will play a significant role in parametrizing periodic trajectories of this dynamical system. This paper is organized as follows. In § 2 we provide a necessary review of confocal families and billiards on a one-sheeted hyperboloid in the Minkowski space. We conclude § 2 with new results on the elliptic periodicity of such billiards, which we collect in § 2.4. In § 3 we describe the topological properties of integrable billiards from § 2 in terms of Fomenko graphs. In § 4 we show through examples that the analytic conditions for closed billiard trajectories lead to discriminantly factorizable polynomials, a generalized form of discriminantly separable polynomials. In § 5 we conclude the paper by establishing a connection of those conditions with generalized extremal polynomials on two intervals, so-called Zolotarev polynomials, and deduce some properties of the corresponding rotation numbers. § 2. Confocal conics and billiards on a hyperboloid of one sheetIn this section we present the main notions and results regarding confocal families of conics and corresponding billiards on a one-sheeted hyperboloid in the Minkowski space. The three-dimensional Minkowski space $\mathbf{M}^{3}$ is the real three-dimensional vector space $\mathbb{R}^{3}$ with the symmetric nondegenerate bilinear form On the hyperboloid of one sheet in $\mathbf{M}^{3}$, the metric $ds^2 = -dx^2 + dy^2 + dz^2$ is a Lorentz metric of constant curvature. Geodesics of this metric are the intersections of $\mathcal{H}$ and planes containing the origin. We call these geodesics space-, time- or light-like if so are their tangent vectors $v$, that is, if $\langle v, v \rangle$ is, respectively, positive, negative, or zero. Note that the light-like geodesics on $\mathcal{H}$ are exactly its generatrices.2.1. Conics on the hyperboloidA conic on $\mathcal{H}$ is defined as the intersection of $\mathcal{H}$ with the following cone: We assume that the cone is not symmetric, that is, $b\neq c$. Moreover, then without loss of generality we can assume that $b<c$. Its intersection with $\mathcal{H}$ bounds a compact domain on $\mathcal{H}$ if and only if all the generatrices of the cone are space-like: see [37]. This happens exactly in one of the following two cases.
The conic is called a collared $\mathcal{H}$-ellipse and consists of two components, which are symmetric to each other with respect to the coordinate $yz$-plane (see Figure 1, a).
The conic is called a transverse $\mathcal{H}$-ellipse and consists of two components, which are symmetric to each other with respect to the coordinate $xy$-plane (see Figure 1, b). 2.2. Confocal familiesThe family of conics which are confocal to the conic given by (2.2) on $\mathcal{H}$ is given by
$$
\begin{equation}
\mathcal{C}_{\lambda} \colon -\frac{x^2}{a-\lambda}+\frac{y^2}{b-\lambda}+\frac{z^2}{c-\lambda}=0.
\end{equation}
\tag{2.3}
$$
Note that the cone (2.2) is denoted by $\mathcal{C}_0$. We describe the confocal family in more detail and provide illustrations in Figure 2. If $\mathcal{C}_0$ is a collared $\mathcal{H}$-ellipse, that is, $0<a<b<c$, then the confocal family (2.3) consists of two types of conic:
The family also contains the following degenerate conics:
If $\mathcal{C}_0$ is a transverse $\mathcal{H}$-ellipse, that is, $b<0<a<c$, then the confocal family (2.3) consists of four subfamilies of conics.
We note that the conics in this confocal family have joint tangent lines, which are generatrices of $\mathcal{H}$ touching $\mathcal{C}_0$. There are eight such generatrices, and their points of intersection are the foci of the confocal family:
These eight generatrices divide $\mathcal{H}$ into 20 domains. Twelve of these domains contain conics from the confocal family, while the eight remaining domains contain no conics. The family also contains degenerate conics $\mathcal{C}_a$, $\mathcal{C}_b$ and $\mathcal{C}_c$, which are contained in the corresponding coordinate planes, and $\mathcal{C}_{\infty}$, which is placed on the plane at infinity and can be regarded as the intersection of $\mathcal{H}$ with the cone $-x^2+y^2+z^2=0$. For each point $(x,y,z) \in \mathcal{H}$ equation (2.3) has two solutions with respect to $\lambda$, which we call the generalized Jacobi coordinates or elliptic coordinates of this point. If $\mathcal{C}_0$ is a collared $\mathcal{H}$-ellipse, these solutions are real and distinct, one belonging to $(-\infty,a]$, and the other to $[b,c]$, which means that each point on $\mathcal{H}$ is the point of intersection of one collared $\mathcal{H}$-ellipse and a confocal conic of hyperbolic type. The generalized Jacobi coordinates of any point within $\mathcal{C}_0$ satisfy $0 < \lambda_1 \leqslant a$ and $b \leqslant \lambda_2 \leqslant c$. If $\mathcal{C}_0$ is a transverse $\mathcal{H}$-ellipse, the equation has two distinct real solutions within the $12$ domains bounded by joint light-like tangent lines, one double real solution on these lines, and no real solutions within the remaining eight domains. The generalized Jacobi coordinates of any point within $\mathcal{C}_0$ satisfy $b \leqslant \lambda_1 < 0 <\lambda_2 \leqslant a$. 2.3. Billiards and periodic trajectoriesOn $\mathcal{H}$ we define the billiard motion as the geodesic flow until the trajectory meets the boundary ellipse, then satisfying the billiard reflection law on it, using the bilinear form (2.1) and the normal vector to the boundary at the point of reflection. Note that the normal vector is not defined at those points of the boundary where the tangent line is light-like, so generally speaking, the reflection cannot be defined there. On the other hand, as shown in [38], when such a situation occurs on a conic, one can continuously extend the billiard flow to such points, defining the reflection there as returning along the same segment in the opposite direction. For a detailed discussion of the billiard reflection law in the pseudo-Euclidean setting, see [38], [23] and [24]. In [46] and [40] a method was proposed to determine the integrability of a discrete dynamical system by reducing the problem to the factorization of matrix polynomials. Specific applications include billiards in an ellipsoid in Euclidean and Minkowski spaces. As shown in [37], this technique extends to $\mathcal{H}$. The geometric manifestation of integrability can be seen through the existence of caustics. Namely, all segments of a given billiard trajectory within an $\mathcal{H}$-ellipse are tangent to the same conic, which is confocal with the boundary. Since both the geodesic flow and the reflection preserve the type of a vector, we have that each billiard trajectory is space-like, time-like or light-like. Moreover, this type is the same for all trajectories sharing the same caustic (see Remarks 3.2 and 3.4 in this paper). In general billiard problems, the study of periodic orbits and their geometric properties is of considerable interest. For integrable cases the works [36], [18], [19], [22], [23], [27] and [1] among many others characterize periodic trajectories in terms of an underlying elliptic curve and prove versions of a Poncelet-type theorem. Such a theorem will also hold in the setting of this paper: given a periodic billiard trajectory in an $\mathcal{H}$-ellipse, any billiard trajectory which shares the same caustic is also periodic with the same period. The work of Cayley (see, for instance, [13] and [14]) in the 19th century on Poncelet’s theorem provided analytic conditions relating the period of a billiard trajectory to its caustics. (A modern account of these results can be found in the paper by Griffiths and Harris [36] from the 1970s.) Such conditions for a generalized Poncelet theorem within an ellipsoid in $d$-dimensional Euclidean spaces, the Lobachevsky space and pseudo-Euclidean spaces have been derived in the last few decades; see [18], [19], [17] and [23]. In this paper we use recently obtained Cayley-type conditions for elliptic billiards on $\mathcal{H}$, which were derived using a divisor shift on the elliptic curve where $\varepsilon = \operatorname{sign}(b\nu)$. Theorem 2.1 (see [37]). A space-like or time-like billiard trajectory on the $\mathcal{H}$-ellipse (2.2) with caustic $\mathcal{C}_{\nu}$ from the family (2.3) is $n$-periodic if and only if A light-like billiard trajectory in the $\mathcal{H}$-ellipse is $n$-periodic if and only if ${n=2m \geqslant 4}$ and where is the Taylor expansion around $X=0$ and $\delta = \operatorname{sign}(b)$.For more details we refer to [37]. We also note the recent paper [50] devoted to geodesics on a hyperboloid in Euclidean space. 2.4. Elliptic periodic trajectoriesPoints in $\mathbf{M}^{3}$ that are symmetric about coordinate planes have the same elliptic coordinates, hence there are eight points on $\mathcal{H}$ with elliptic coordinates $\lambda_1$, $\lambda_2$. Because of this symmetry, each billiard trajectory which is $n$-periodic in elliptic coordinates is also periodic in the Cartesian coordinate system, but its period can be $n$ or $2n$. We define this scenario as follows. Definition 2.2. A billiard trajectory is $n$-elliptic periodic if it is $n$-periodic in the elliptic coordinates corresponding to the confocal family (2.3). Trajectories that connect directly two points with the same Jacobi elliptic coordinates directly are 1-elliptic periodic. We thus consider $n$-elliptic periodicity for ${n \geqslant 2}$. Next we derive algebro-geometric conditions for elliptic periodic billiard trajectories on $\mathcal{H}$. Theorem 2.3. A billiard trajectory within a collared $\mathcal{H}$-ellipse with caustic curve $\mathcal{C}_\nu$ is $n$-elliptic periodic but not $n$-periodic if and only if one of the following conditions is satisfied on the elliptic curve (2.4), where $Q_{\pm}$ are the two points over $X=0$ and $P_\beta$ is a point over $X=\beta$: $\bullet$ $n=2m$ and (i) $\nu \in (-\infty,0) \cup (b,c) \cup (c,\infty)$ and $m(Q_- - Q_+) \sim 0$, (ii) $\nu \in (-\infty, 0) \cup (c,\infty)$ and $(m+1)Q_- - (m-1)Q_+ - P_a - P_\nu \sim 0$, (iii) $\nu \in (b, c)$ and $(m+1)Q_- - (m-1)Q_+ - P_a - P_c \sim 0$; $\bullet$ $n=2m+1$ and (iv) $\nu \in (-\infty,0) \cup (b,c) \cup (c,\infty)$ and $(m+1)Q_- - mQ_+ -P_a \sim 0$, (v) $\nu \in (-\infty, 0) \cup (c,\infty)$ and $(m+1)Q_- - mQ_+ - P_\nu \sim 0$, (vi) $\nu \in (b, c)$ and $(m+1)Q_- - mQ_+ - P_c \sim 0$. A billiard trajectory within a transverse $\mathcal{H}$-ellipse with caustic curve $\mathcal{C}_\nu$ is $n$-elliptic periodic, but not $n$-periodic if and only if one of the following conditions is satisfied on the elliptic curve (2.4): $\bullet$ $n=2m$ and (vii) $\nu \in \mathbb{R}\setminus\{b,0,a,c\}$ and $m(Q_- - Q_+) \sim 0$, (viii) $\nu \in (-\infty, b) \cup (a, c) \cup (c,\infty)$ and $(m+1)Q_+ - (m-1)Q_- - P_a - P_b \sim 0$, (ix) $\nu \in (b, 0)$ and $(m+1)Q_+ - (m-1)Q_- - P_a - P_\nu \sim 0$, (x) $\nu \in (0, a)$ and $(m+1)Q_+ - (m-1)Q_- - P_b - P_\nu \sim 0$; $\bullet$ $n=2m+1$ and (xi) $\nu \in (-\infty,b) \cup (b,0) \cup (a, c) \cup (c,\infty)$ and $(m+1)Q_+ - mQ_- -P_a \sim 0$, (xii) $\nu \in (-\infty,b) \cup (0,a) \cup (a, c) \cup (c,\infty)$ and $(m+1)Q_+ - mQ_- -P_b \sim 0$. Proof. Let $\mathcal{P}(x) = \varepsilon (x-a)(x-b)(x-c)(x-\nu)$, where $\varepsilon = \operatorname{sign}(b\nu)$, and consider the differential equation
$$
\begin{equation}
\frac{d\lambda_1}{\sqrt{\mathcal{P}(\lambda_1)}}+ \frac{d\lambda_2}{\sqrt{\mathcal{P}(\lambda_2)}}=0.
\end{equation}
\tag{2.5}
$$
along a given billiard trajectory. If $p_0$ is the initial point of an $n$-elliptic periodic trajectory and $p_1$ is the next point along this trajectory with the same elliptic coordinates as $p_0$, then integrating differential equation (2.5) from $p_0$ to $p_1$ results in one of the following:
$$
\begin{equation*}
\begin{aligned} \, m_0(P_a-Q_+)+m_1(P_c-P_b) &\sim 0, \\ m_2(P_\nu-Q_+)+m_3(P_c-P_b) &\sim 0, \\ m_4(P_a-Q_+)+m_5(P_\nu-P_b) &\sim 0 \end{aligned}
\end{equation*}
\notag
$$
in the case of a collared $\mathcal{H}$-ellipse. The period $n$ is $ m_0 = m_2=m_4$ and the $m_i$ can be even or odd. Cases (i) and (v) follow from the proof of Theorem 5.5 in [37], where all the $m_i$ are even (case (i)), or $m_0$, $m_1$, $m_2$ are odd and $m_3$ is even (case (v)).
We prove (ii) and note that the rest of the cases follow from a similar argument depending upon the parity of the $m_i$. Suppose $n=m_0$ is even and $m_1$ is odd. Then
$$
\begin{equation*}
\begin{aligned} \, 0 &\sim m_0 (P_a-Q_+)+m_1(P_c-P_b) \\ &\sim 2k_0(P_a - Q_+)+2k_1(P_c-P_b)+P_c-P_b \\ &\sim k_0 (Q_-+Q_+) - k_0(2Q_+)+P_c - P_b \\ & \sim k_0(Q_- -Q_+)+P_c - P_b \\ &\sim k_0(Q_- -Q_+)+(Q_-+Q_+) - P_a - P_\nu \\ &\sim (k_0+1)Q_- - (k_0-1)Q_+- P_a - P_\nu, \end{aligned}
\end{equation*}
\notag
$$
which is equivalent to (ii). In particular, we have used the fact that $P_a$, $P_b$, $P_c$ and $P_\nu$ are branching points on (2.4), that is, $2P_a \sim 2P_b \sim 2P_c \sim 2P_\nu \sim Q_- + Q_+$. These cases are also further discussed in § 5.2 in the context of rotation numbers.
In the case of a transverse $\mathcal{H}$-ellipse, integrating differential equation (2.5) from $p_0$ to $p_1$ results in one of the following:
$$
\begin{equation*}
\begin{aligned} \, m_6(Q_+-P_b)+m_7(P_a-Q_+) &\sim 0, \\ m_8(Q_+-P_\nu)+m_9(Q_+-P_a) &\sim 0, \\ m_{10}(Q_+-P_b)+m_{11}(Q_+-P_\nu) &\sim 0. \end{aligned}
\end{equation*}
\notag
$$
The period is now $n = m_{2i} + m_{2i+1}$ for $i=2,3,4$. Just as before, case (vii) follows from the proof of Theorem 5.5 of [37] and the other cases are proved similarly to (ii) above.
Theorem 2.3 is proved. Remark 2.4. As noted in the above proof, divisorial conditions (i), (v) and (vii) are identical to the divisorial conditions for periodic billiard trajectories given in Theorem 5.5 of [37]. However, in the case of collared and transverse $\mathcal{H}$-ellipses conditions (i) and (vii) produce $2m$-periodic and $m$-elliptic periodic trajectories. And in the case of a collared $\mathcal{H}$-ellipse condition (v) produces trajectories which are $(4m+2)$-periodic and $(2m+1)$-elliptic periodic. See Remark 5.7 of [37] for further details. The above divisorial conditions lead to explicit Cayley-type conditions for elliptic periodicity. Theorem 2.5. A billiard trajectory within a collared $\mathcal{H}$-ellipse with caustic curve $\mathcal{C}_\nu$ is $n$-elliptic periodic but not $n$-periodic if and only if one of the following conditions is satisfied: (a) $\nu \in (-\infty, 0) \cup (b, c) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} B_{3} &B_{4} &\dots &B_{n+1} \\ B_{4} &B_{5} &\dots &B_{n+2} \\ \dots &\dots &\dots &\dots \\ B_{n+1} &B_{n+2} &\dots &B_{2n-1} \end{pmatrix}=0 \quad \textit{for } n \geqslant 2,
\end{equation*}
\notag
$$
where the entries $B_i$ are given in Theorem 2.1 above; (b) $\nu \in (-\infty, 0) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} F_{1} & F_{2} &\dots &F_{m} \\ F_{2} &F_{3} &\dots &F_{m+1} \\ \dots &\dots &\dots &\dots \\ F_m &F_{m+1} &\dots &F_{2m-1} \end{pmatrix}=0 \quad \textit{for } n=2m \geqslant 2
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
\det\begin{pmatrix} D_{2} &D_{3} &\dots &D_{m+1} \\ D_{3} &D_{4} &\dots &D_{m+2} \\ \dots &\dots&\dots&\dots \\ D_{m+1} & D_{m+2} &\dots &D_{2m} \end{pmatrix}=0 \quad \textit{for } n=2m+1 \geqslant 3,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-b)(X-c)}{(X-a)(X-\nu)}}=F_0+F_1X+F_2X^2+\dotsb
\end{equation*}
\notag
$$
is the Taylor expansion around $X=0$ and the entries $D_i$ are given in Theorem 2.1; (c) $\nu \in (b, c)$ and
$$
\begin{equation*}
\det\begin{pmatrix} G_{1} & G_{2} &\dots &G_{m} \\ G_{2} &G_{3} &\dots &G_{m+1} \\ \dots &\dots &\dots &\dots \\ G_{m} &G_{m+1} &\dots & G_{2m-1} \end{pmatrix}=0 \quad \textit{for } n=2m \geqslant 4
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
\det\begin{pmatrix} H_{2} & H_{3} &\dots &H_{m+1} \\ H_{3} &H_{4} &\dots &H_{m+2} \\ \dots &\dots &\dots &\dots \\ H_{m+1} & H_{m+2} &\dots &H_{2m} \end{pmatrix}=0 \quad \textit{for } n=2m+1 \geqslant 5,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-b)(X-\nu)}{(X-a)(X-c)}}=G_0+G_1X+G_2X^2+\dotsb
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-a)(X-b)(X-\nu)}{X-c}}=H_0+H_1X+H_2X^2+\dotsb
\end{equation*}
\notag
$$
are the Taylor expansions around $X=0$; (d) $\nu \in (-\infty, 0) \cup (b, c) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} I_{2} & I_{3} &\dots &I_{m+1} \\ I_{3} & I_{4} &\dots &I_{m+2} \\ \dots &\dots &\dots &\dots \\ I_{m+1} &I_{m+2} &\dots &I_{2m} \end{pmatrix}=0 \quad \textit{for } n=2m+1 \geqslant 3,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-b)(X-c)(X-\nu)}{X-a}}=I_0+I_1X+I_2X^2+\dotsb
\end{equation*}
\notag
$$
is the Taylor expansion around $X=0$. A billiard trajectory within a transverse $\mathcal{H}$-ellipse with caustic curve $\mathcal{C}_\nu$ is $n$-elliptic periodic but not $n$-periodic if and only if one of the following conditions is satisfied: (e) $\nu \in (-\infty, b) \cup (b, 0) \cup (0,a) \cup (a, c) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} B_{3} & B_{4} &\dots &B_{n+1} \\ B_{4} & B_{5} &\dots &B_{n+2} \\ \dots &\dots &\dots &\dots \\ B_{n+1} &B_{n+2} &\dots &B_{2n-1} \end{pmatrix}=0 \quad \textit{for } n \geqslant 2,
\end{equation*}
\notag
$$
where the entries $B_i$ are given in Theorem 2.1 above; (f) $\nu \in (-\infty,b) \cup (a, c) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} J_{1} & J_{2} &\dots &J_{m} \\ J_{2} & J_{3} &\dots &J_{m+1} \\ \dots &\dots &\dots &\dots \\ J_{m} &J_{m+1} &\dots &J_{2m-1} \end{pmatrix}=0 \quad \textit{for } n=2m \geqslant 2,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-c)(X-\nu)}{(X-a)(X-b)}}=J_0+J_1X+J_2X^2+\dotsb
\end{equation*}
\notag
$$
is the Taylor expansion around $X=0$; (g) $\nu \in (b,0)$ and
$$
\begin{equation*}
\det\begin{pmatrix} F_{1} &F_{2} &\dots &F_{m} \\ F_{2} & F_{3} &\dots &F_{m+1} \\ \dots &\dots &\dots &\dots \\ F_{m} &F_{m+1} &\dots &F_{2m-1} \end{pmatrix}=0 \quad \textit{for } n=2m \geqslant 2,
\end{equation*}
\notag
$$
where the entries $F_i$ are given in part (b) above; (h) $\nu \in (0,a)$ and
$$
\begin{equation*}
\det\begin{pmatrix} K_{1} & K_{2} &\dots &K_{m} \\ K_{2} & K_{3} &\dots &K_{m+1} \\ \dots &\dots &\dots &\dots \\ K_{m} & K_{m+1} &\dots &K_{2m-1} \end{pmatrix}=0 \quad \textit{for } n=2m \geqslant 4,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sqrt{\frac{\varepsilon(X-a)(X-c)}{(X-b)(X-\nu)}}=K_0+K_1X+K_2X^2+\dotsb
\end{equation*}
\notag
$$
is the Taylor expansion around $X=0$; (i) $\nu \in (-\infty, b) \cup (b, 0) \cup (a, c) \cup (c,\infty)$ and
$$
\begin{equation*}
\det\begin{pmatrix} I_{2} & I_{3} &\dots & I_{m+1} \\ I_{3} & I_{4}&\dots &I_{m+2} \\ \dots &\dots &\dots &\dots \\ I_{m+1} &I_{m+2} &\dots &I_{2m} \end{pmatrix}=0 \quad \textit{for } n=2m+1 \geqslant 3,
\end{equation*}
\notag
$$
where the entries $I_i$ are given in part (d) above; (j) $\nu \in (-\infty, b) \cup (0,a) \cup (a, c) \cup (c,\infty)$ and where is the Taylor expansion around $X=0$. Proof. We prove part (c) and note that the proofs for the remaining parts are similar. Part (c) uses divisorial conditions (iii) and (vi) from Theorem 2.3.
First consider the case $n=2m$ and divisorial condition (iii): This divisorial condition is equivalent to the existence of a meromorphic function with a zero of order $m+1$ at $Q_-$, a pole of order $m-1$ at $Q_+$, and simple poles at $P_a$ and $P_b$. A basis of $\mathcal{L}((m-1)Q_{+} + P_a + P_c)$ is $\{1,f_1, \dots, f_m\}$, where The existence of such a function is equivalent to the indicated determinantal condition.Now consider $n=2m+1$ and divisorial condition (vi): This divisorial condition is equivalent to the existence of a meromorphic function with a zero of order $m+1$ at $Q_-$, a pole of order $m$ at $Q_+$, and a simple pole at $P_c$. A basis of the space of such functions $\mathcal{L}(mQ_{+} + P_c)$ is $\{1,g_1,\dots,g_m\}$ where The existence of such a function is equivalent to the second determinantal condition in part (c).Theorem 2.5 is proved. Example 2.6 (2-elliptic periodic trajectory). In a collared $\mathcal{H}$-ellipse a 2-elliptic periodic trajectory can be found by setting either $B_3=0$ or $F_1=0$, which are equivalent to and respectively. Clearly any solution to $F_1=0$ also satisfies $B_3=0$. In a transverse $\mathcal{H}$-ellipse a 2-elliptic periodic trajectory can be found by setting either $B_3=0$, or $F_1=0$, or $J_1=0$. The first two conditions are given above, and the third is equivalent to Any solution to $F_1=0$ or $J_1=0$ is also a solution to $B_3=0$.The pictures of such trajectories are shown in Figure 3. Example 2.7 (3-elliptic periodic trajectory). In a collared $\mathcal{H}$-ellipse a 3-elliptic periodic trajectory can be found by setting either In a transverse $\mathcal{H}$-ellipse a 3-elliptic periodic trajectory can be found by setting either
$$
\begin{equation*}
\det\begin{pmatrix} B_3 & B_4 \\ B_4 & B_5 \end{pmatrix}=0, \quad\text{or}\quad I_2=0, \quad\text{or}\quad L_2=0.
\end{equation*}
\notag
$$
The first two conditions are given above, while the third is equivalent to
$$
\begin{equation*}
0=(abc)^2 -2 a b c (a b-a c+b c)\nu+(b^2 c^2+2 a b c (c-b)+a^2(b^2+2 b c-3 c^2))\nu^2.
\end{equation*}
\notag
$$
Pictures of such 3-elliptic periodic trajectories are shown in Figure 4. § 3. Topological properties of billiards in confocal familiesAs described in § 2, billiards within confocal conics on $\mathcal{H}$ come in two distinct geometric types. In this section we present a topological description of billiards in each setting using Fomenko invariants (see [12], [11] and [10]). Such descriptions have been made for elliptic billiards in the Euclidean plane (see [20] and [21]), domains bounded by confocal parabolas [33], billiards with the Hooke potential [42], ones in the Minkowski plane and geodesics on ellipsoids in $\mathbf{M}^{3}$ [25], non-convex billiards [45], billiards with slipping [35], and broader classes of billiards and Hamiltonian impact systems (see [49], [34] and [41]). For a larger body of work on the topic, see also the references therein. 3.1. A transverse $\mathcal{H}$-ellipseIn the case of a transverse $\mathcal{H}$-ellipse the constants satisfy $b < 0 < a < c$, while the confocal curves are of elliptic type if $\lambda \in (b,a)\cup(a,c)$ and of hyperbolic type if $\lambda \in (-\infty, b] \cup [c,\infty)$; see § 2.2 for details. The billiard table $\mathcal{T}$ is the set of all points on and in the interior of the transverse $\mathcal{H}$-ellipse $\mathcal{C}_0$ with $z>0$. Topologically, $\mathcal{T}$ is homeomorphic to the closed planar disc. Consider a point $P \in \mathcal{C}_0$ and suppose $u,v \in T_P\mathcal{H}$ are unit vectors; the corresponding set is homeomorphic to $\mathbf{S}^{1}$. Let $\sim$ be the equivalence relation on the solid torus $\mathcal{T} \times \mathbf{S}^{1}$ defined by
$$
\begin{equation*}
(P,u) \sim (P,v) \quad \Longleftrightarrow\quad P \in \mathcal{C}_0 \text{ and } u,v \in \mathbf{S}^{1} \text{ reflect to each other off } \mathcal{C}_0.
\end{equation*}
\notag
$$
Every billiard trajectory inside $\mathcal{T}$ induces a trajectory in $\mathcal{T}\times\mathbf{S}^{1}/{\sim}$. Further, this correspondence between trajectories induces a projection of the billiard phase space onto $\mathcal{T}\times\mathbf{S}^{1}/{\sim}$, which preserves trajectories and leaves of the Liouville foliation. Theorem 3.1. The manifold $\mathcal{T}\times\mathbf{S}^{1}/{\sim}$ is represented by the Fomenko graph in Figure 5. Proof. Each leaf (level set) of the Liouville foliation on $\mathcal{T}\times\mathbf{S}^{1}/{\sim}$ corresponds to a billiard motion with a fixed confocal curve $\mathcal{C}_{\lambda}$, $\lambda \in \mathbb{R} \cup \{\infty\}$, as a caustic.
When $\lambda \notin \{a,b,c\}$, the level sets are nondegenerate. The level set is a single torus when the caustic is of hyperbolic type, that is, when $\lambda \in (-\infty, b) \cup (c,\infty) \cup \{\infty\}$. The level set is a union of two tori when the caustic is of elliptic type and intersects $\mathcal{C}_0$ in four distinct points (that is, when $\lambda \in (b,a)$). The level set is again a single torus when $\mathcal{C}_{\lambda}$ is of elliptic type and does not intersect $\mathcal{C}_0$ (that is, when $\lambda \in (a,c)$). If $\lambda = a$, then the level set contains a single closed trajectory, which is 2-periodic and contained in the plane $x=0$, and two homoclinic separatrices. A trajectory on any of the separatrices is placed on one side of the coordinate $yz$-plane and its segments contain alternately the foci $F_{-+}^x$ and $F_{++}^x$. For $\lambda \to a^-$ the caustic is of elliptic type and intersects $\mathcal{C}_0$. Trajectories can be in one of the two regions bounded by $\mathcal{C}_0$ and $\mathcal{C}_{\lambda}$, so the level set is the union of two tori. For $\lambda \to a^+$ the caustic is of elliptic type but does not intersect $\mathcal{C}_0$, hence the level set is a single torus. This collection of level sets is represented by the Fomenko atom $B$. A similar analysis at $\lambda =b$ provides an analogous result. For $\lambda = c$ segments of a trajectory have a simple geometric description. The plane that determines each billiard segment contains alternately the two antipodal pairs of foci (that is, the plane determining one segment contains $F_{++}^z$ and $F_{--}^z$, while the plane determining the next segment of the trajectory contains $F_{+-}^z$ and $F_{-+}^z$). Such trajectories do not limit to periodic trajectories as before. While $\lambda=c$ is a transition point for the caustic to change from the hyperbolic to elliptic type, no caustic curves intersect $\mathcal{C}_0$, and the billiard itself does not fundamentally change at $\lambda=c$. For $\lambda = \infty$ the trajectories are all light-like, that is, their segments are placed along generatrices of $\mathcal{H}$. Their behaviour is qualitatively identical to when $\lambda \in (-\infty, b) \cup (a, \infty)$. In a neighbourhood of $\lambda=b$ the analysis is similar to $\lambda=a$. Consider the limiting case $\lambda =0$. For $\lambda \to 0^-$ the billiard motion occurs in one of the two regions bounded by $\mathcal{C}_0$ and $\mathcal{C}_{\lambda}$, each lying on one side of the plane $y=0$. The limiting motion along the boundary is periodic: the trajectory moves along the time-like arc of the boundary. This periodic motion is represented by the two atoms $A$ in Figure 5. For $\lambda \to 0^+$ the same analysis is true except that the two regions between $\mathcal{C}_0$ and $\mathcal{C}_{\lambda}$ are on opposite sides of the plane $x=0$, and the limiting periodic trajectories are space-like arcs of $\mathcal{C}_0$. Theorem 3.1 is proved. Remark 3.2. We note that the trajectories with caustics $\mathcal{C}_{\lambda}$ such that $\lambda>0$ are space-like, while for $\lambda<0$ they are time-like. 3.2. A collared $\mathcal{H}$-ellipseIn the case of a collared $\mathcal{H}$-ellipse the constants satisfy $0<a<b<c$ and the confocal curves are of elliptic type if $\lambda \in (-\infty,a)$ and of hyperbolic type if $\lambda \in (b,c)$; see § 2.2 for details. Let $\mathcal{E}$ be the billiard table, that is, all points on and in the interior of the collared $\mathcal{H}$-ellipse $\mathcal{C}_0$. Topologically, $\mathcal{E}$ is homeomorphic to a closed annulus. Consider a point $P \in \mathcal{C}_0$ and suppose $u,v \in T_P\mathcal{H}$, where $u$ and $v$ are unit vectors; the corresponding set is homeomorphic to $\mathbf{S}^{1}$. Let $\sim$ be the equivalence relation on the hollow torus with thickened walls $\mathcal{E} \times \mathbf{S}^{1}$ defined by
$$
\begin{equation*}
(P,u) \sim (P,v) \quad\Longleftrightarrow\quad P \in \mathcal{C}_0 \text{ and } u,v \in \mathbf{S}^{1} \text{ reflect to one another off } \mathcal{C}_0.
\end{equation*}
\notag
$$
Every billiard trajectory inside $\mathcal{E}$ induces a trajectory in $\mathcal{E}\times\mathbf{S}^{1}/{\sim}$. Further, this correspondence between trajectories induces a projection of the billiard phase space onto $\mathcal{E}\times\mathbf{S}^{1}/{\sim}$, which preserves trajectories and leaves of the Liouville foliation. Theorem 3.3. The manifold $\mathcal{E}\times\mathbf{S}^{1}/{\sim}$ is represented by the Fomenko graph in Figure 6. Proof. Just as before, each leaf (level set) of the Liouville foliation on $\mathcal{E}\times\mathbf{S}^{1}/{\sim}$ corresponds to a billiard motion with a fixed confocal curve $\mathcal{C}_{\lambda}$, $\lambda \in (-\infty, a] \cup [b,\infty) \cup \{\infty\}$, as a caustic.
For $\lambda \notin \{a,b,c\}$ level sets are nondegenerate. The level set is a union of two tori when the caustic is of hyperbolic type, that is, when $\lambda \in (b, c)$. Then each torus corresponds to a billiard trajectory in one of the two regions symmetric about the plane $z=0$ and bounded by the confocal curves $\mathcal{C}_{\lambda}$ and the boundary $\mathcal{C}_0$. In the case $\lambda \in (c,+\infty)$ there are no confocal curves on $\mathcal{H}$, as the intersection of the cone (2.3) with the hyperboloid is empty. However, since the corresponding geodesics are the intersections of planes tangent to the cone with $\mathcal{H}$, we can still determine the corresponding billiard segments. In this case the level set is the union of two tori, one for each direction in which a billiard trajectory traverses $\mathcal{E}$ while winding around the $x$-axis. For $\lambda =\infty$ trajectories are light-like and each torus corresponds to a trajectory that winds about $\mathcal{E}$ in either the overhand or underhand direction. In the case $\lambda \in (-\infty,0) \cup (0,a)$ the level set remains the union of two tori, one for each direction in which a billiard can wind about $\mathcal{E}$. Each has a different geometric interpretation. If $\lambda \in (-\infty,0)$, then the caustic is of elliptic type and lies outside $\mathcal{E}$, while $\lambda \in (0,a)$ corresponds to a motion with an elliptic-type caustic inside $\mathcal{E}$. This motion is remarkable in the following sense: every torus corresponding to $\lambda \in (0,a)$ is simply the periodic geodesic flow on a space-like trajectory. The billiard motion does not reach the boundary, so each trajectory there is closed (see Figure 7). Thus, every torus in such a level set is resonant. In the limiting case as $\lambda \to 0^\pm$ the level set is again the union of two resonant tori, each of which corresponds again to a direction in which a trajectory can wind about $\mathcal{E}$.  Figure 7.A periodic billiard trajectory tangent to a confocal curve without reflection from the boundary $\mathcal{C}_0$ (in the language of § 2.4, this is a 1-elliptic periodic trajectory). Now consider the degenerate cases. For $\lambda =b$ the billiard trajectory is periodic and contained in the plane $y=0$, reflecting alternately off each boundary component of $\mathcal{E}$. There will be two $A$-atoms for this limiting motion, one for each of the two periodic trajectories. For $\lambda \to b^+$ the level set is the union of two tori, as described above for $\lambda \in (b,c)$. For $\lambda = c$ the billiard is periodic and contained in the plane $z=0$ and the level set is degenerate. For $\lambda \to c^-$ the level set is the union of two tori, one for each region of the table where trajectories are tangent to a caustic curve of hyperbolic type. For $\lambda \to c^+$ the level set is also a union of two tori, each corresponding to a direction in which the trajectory winds about $\mathcal{E}$. As there are four circles and four separatrices, there is a $C_2$-atom for $\lambda = c$. For $\lambda =a$ the billiard trajectory is periodic, space-like, wraps around $\mathcal{H}$ in the plane $x=0$, and the level set is degenerate. Each winding direction of this periodic motion corresponds to an $A$-atom. For $\lambda \to a^-$ the motion is the same as discussed above for $\lambda \in (0,a)$ and is the union of two tori. The level set is empty as $\lambda \to a^+$. Recall the notation for the gluing matrix and basis cycles on tori:
$$
\begin{equation*}
\begin{pmatrix} \lambda^+ \\ \mu^+ \end{pmatrix} =\begin{pmatrix} \alpha &\beta \\ \gamma &\delta \end{pmatrix} \begin{pmatrix} \lambda^- \\ \mu^- \end{pmatrix}
\end{equation*}
\notag
$$
(see [12] and [11]). The gluing matrix for the two left-hand edges is The new basis cycles $(\lambda^+,\mu^+)$ correspond geometrically to a motion along the circle in the intersection of the hyperboloid with the $yz$-plane and a 2-periodic trajectory in the plane $y=0$, respectively. For the two right-hand edges the gluing matrix is Geometrically, the noncontractible basis cycle $\lambda^+$ corresponds to the circular periodic trajectory around the collar in the plane $x=0$, and the other basis cycle $\mu^+$ is complementary to $\lambda^+$. Theorem 3.3 is proved. Remark 3.4. In this case space-like trajectories correspond to caustics $\mathcal{C}_{\lambda}$ with ${\lambda \!<\! a}$, while time-like ones correspond to $\lambda>b$. (Cf. Remark 3.2.) 3.3. Remarks on equivalent systemsWe note that the Fomenko graph in Figure 5 for the case of billiards in a transverse $\mathcal{H}$-ellipse is identical to the Fomenko graph corresponding to billiards inside an ellipse in the Minkowski plane (see [25]). Therefore, because their marked Fomenko graphs are identical, these two billiard systems are Liouville equivalent. In the case of a collared $\mathcal{H}$-ellipse the marked molecule is the same as for a particular instance of the Chaplygin case of integrable rigid-body dynamics (see [32]). Therefore, these integrable systems are Liouville equivalent. 3.4. Billiard booksA set of conjectures by Fomenko outlines a connection between integrable systems and elliptical billiards in the Euclidean plane. By considering domains that are subsets of an ellipse and are bounded by arcs of confocal hyperbolas and ellipses, it is conjectured that any integrable system can be represented by gluing copies of such domains together in a suitable fashion, thus obtaining a generalized billiard domain called a billiard book; see [49] for details. In the case of a transverse $\mathcal{H}$-ellipse the unmarked version of the Fomenko graph in Figure 5 is represented by a billiard book which is constructed by gluing together two copies of the billiard book representing the 3-atom $B$; see § 2 of [31] for a direct construction of this billiard book. In the case of a collared $\mathcal{H}$-ellipse the unmarked version of the Fomenko graph in Figure 6 is represented by a billiard book representing the 3-atom $C_2$. One can follow Algorithm 1 of [49] to create such a billiard book, and the necessary details, using the notation from that work, are as follows. This book is made of four sheets, $A_0'$. The permutations $\sigma_3=\sigma_4=\mathrm{id}$, $\sigma_1 = (12)(34)$ and $\sigma_2 = (14)(23)$ define the gluing of the edges; see Figure 8. § 4. Discriminantly factorizable and separable polynomialsDiscriminantly factorizable polynomials were introduced in [15] and defined by the property that their discriminants, as polynomials in several variables, admit a nontrivial factorization. In the Euclidean [26] and Minkowski planes [1] it has been shown that Cayley-type conditions contain a rich algebro-geometric structure related to discriminantly separable polynomials, which were introduced in [16]. Definition 4.1 (see [16]). A polynomial $F(x_1, \dots, x_n)$ is discriminantly separable if there exist polynomials $f_1(x_1), \dots, f_n(x_n)$ such that the discriminant $\mathcal{D}_{x_i}F$ of $F$ with respect to $x_i$ satisfies for each $1 \leqslant i \leqslant n$. Now we examine the periodicity conditions of Theorem 2.1 from that perspective and note that the elliptic-periodicity conditions can be treated similarly. Cayley-type conditions have numerators which are polynomials in the caustic parameter $\nu$ and whose coefficients are given in terms of the variables $a$, $b$ and $c$. For notational simplicity, in the formulae below we use sometimes the elementary symmetric polynomials in three variables, $p:=a+b+c$, $q:=ab+ac+bc$ and ${r:=abc}$. Example 4.2 (period 3). The condition $D_2=0$ is equivalent to finding roots of the $\nu$-polynomial and its discriminant with respect to $\nu$ is Example 4.3 (period 4). Solving the equation $B_3 =0$ is equivalent to finding roots of the $\nu$-polynomial and its discriminant with respect to $\nu$ is Example 4.4 (period 5). The formula $D_2D_4 - D_3^2 =0$ is equivalent to finding roots of the $\nu$-polynomial Example 4.5 (period 6). Finding the solutions of $B_3B_5 - B_4^2 =0$ is equivalent to finding roots of the $\nu$-polynomial Example 4.6 (period 7). The condition Each of the polynomials $G_i(a,b,c,\nu)$ in Examples 4.2–4.6 is discriminantly factorizable. But in contrast to the examples in [1] there is no obvious change of variable that leads to discriminantly separable polynomials for the above examples. However, some polynomials are nearly discriminantly separable in the variables $a$, $d = b/a$ and $e=c/a$. For example,
$$
\begin{equation*}
\mathcal{D}_\nu G_4=64 a^{30} (d-1)^2 d^8 (e-1)^2 e^8 (d-e)^2,
\end{equation*}
\notag
$$
where $(d-e)^2$ is the disqualifying factor. Similar calculations with the same change of variables can be made, which lead to expressions that are products of polynomials in the form
$$
\begin{equation*}
\mathcal{D}_\nu G_i (a,b,c,\nu)=f_1(a)f_2(d)f_3(e)f_4(d,e).
\end{equation*}
\notag
$$
Another possible change of variable is suggested by the similarity of the polynomial $G_3$ to $G_2(a,b,\gamma)$ in [1]. In terms of the elementary symmetric polynomials $p$, $q$ and $r$, first apply the transformation $(p,q,r) \mapsto (AB, A+B,1)$. This produces Applying the further transformation $(A,B) \mapsto(A,C := B/A)$ produces a discriminantly separable polynomial However, this double variable change does not produce discriminantly separable polynomials for any of the other examples computed above. For example, this double variable change as applied to Example 4.3 results in
$$
\begin{equation*}
\mathcal{D}_\nu G_4=2^6 (A^6 (-1+C)^2 C^2 - A^3 (4 - 6 C - 6 C^2+4 C^3)-27),
\end{equation*}
\notag
$$
which is discriminantly factorizable but not discriminantly separable.
§ 5. Periodic trajectories and extremal polynomials5.1. Polynomial equations as periodicity conditionsWe can formulate the periodicity conditions of Theorem 2.1 in terms of the existence of solutions to certain polynomial equations. Theorem 5.1. Billiard trajectories with caustic $\mathcal{C}_\nu$ in a collared or a transverse $\mathcal{H}$-ellipse are $n$-periodic if and only if there exists a pair of polynomials of degrees $d_1$ and $d_2$, respectively, such that for $k = e_{d_1}^2 - \varepsilon f_{d_2}^2$: (a) if $n=2m$, then $d_1=m$, $d_2 = m-2$ and
$$
\begin{equation}
p_m^2(s) - \biggl(\frac{1}{a}-s\biggr) \biggl(\frac{1}{b}-s\biggr) \biggl(\frac{1}{c}-s\biggr) \biggl(\frac{1}{\nu}-s\biggr) q_{m-2}^2(s)=\operatorname{sign}{k};
\end{equation}
\tag{5.1}
$$
(b) if $n=2m+1$, then $d_1=m$, $d_2 = m-1$ and Proof. First consider the case $n=2m$. There are real polynomials $p_m^*(X)$ and $q_{m-2}^*(X)$ of degrees $m$ and $m-2$, respectively, such that the expression
$$
\begin{equation*}
p_m^*(X) - q_{m-2}^*(X)\sqrt{\varepsilon(X-a)(X-b)(X-c)(X-\nu)}
\end{equation*}
\notag
$$
has a zero of order $2m$ at $X=0$. Multiplying this expression by its algebraic conjugate
$$
\begin{equation*}
p_m^*(X)+q_{m-2}^*(X)\sqrt{\varepsilon(X-a)(X-b)(X-c)(X-\nu)}
\end{equation*}
\notag
$$
we arrive at a polynomial of degree $2m$ of the form
$$
\begin{equation*}
[p_m^*(X)]^2 - \varepsilon[q_{m-2}^*(X)]^2 (X-a)(X-b)(X-c)(X-\nu),
\end{equation*}
\notag
$$
which has a zero of order $2m$ at $X=0$. It follows that
$$
\begin{equation*}
[p_m^*(X)]^2 - \varepsilon[q_{m-2}^*(X)]^2 (X-a)(X-b)(X-c)(X-\nu)=k X^{2m}
\end{equation*}
\notag
$$
for some nonzero constant $k$. Using the property that $x = |x|\operatorname{sign}{x}$ and dividing both sides of this equation by $|k| X^{2m}$, we get that
$$
\begin{equation*}
\frac{[p_m^*(X)]^2}{|k| X^{2m}} - \frac{\varepsilon[q_{m-2}^*(X)]^2 (X-a)(X-b)(X-c)(X-\nu)}{|k| X^{2m}}=\operatorname{sign}{k}.
\end{equation*}
\notag
$$
Let $s=1/X$ and define
$$
\begin{equation*}
p_m(s)=\frac{s^m p_m^*(1/s)}{\sqrt{|k|}} \quad\text{and}\quad q_{m-2}(s)=\frac{s^{m-2}q_{m-2}^*(1/s)\sqrt{ac|b\nu|}}{\sqrt{|k|}}.
\end{equation*}
\notag
$$
Then these polynomials $p_m(s)$ and $q_{m-2}(s)$ satisfy equation (5.1), which proves part (a) above.
For $n=2m+1$ there are polynomials $p_m^*(X)$ and $q_{m-1}^*(X)$ of degrees $m$ and ${m-1}$, respectively, such that the expression
$$
\begin{equation*}
p_m^*(X) - q_{m-1}^*(X)\sqrt{\frac{\varepsilon(X-a)(X-b)(X-c)}{(X-\nu)}}
\end{equation*}
\notag
$$
has a zero of order $2m+1$ at $X=0$. Multiplying by
$$
\begin{equation*}
(X-\nu) \biggl(p_m^*(X)+q_{m-1}^*(X)\sqrt{\frac{\varepsilon(X-a)(X-b)(X-c)}{(X-\nu)}} \biggr)
\end{equation*}
\notag
$$
we obtain a polynomial of the form
$$
\begin{equation*}
(X-\nu)[p_m^*(X)]^2 - \varepsilon[q_{m-1}^*(X)]^2(X-a)(X-b)(X-c)
\end{equation*}
\notag
$$
which has a zero of order $2m+1$ at $X=0$. Since the degree of this expression is ${2m+1}$, it follows that
$$
\begin{equation*}
(X-\nu)[p_m^*(X)]^2 - \varepsilon[q_{m-1}^*(X)]^2(X-a)(X-b)(X-c)=k X^{2m+1}
\end{equation*}
\notag
$$
for some nonzero constant $k$. Again, rewriting $k = |k|\operatorname{sign}{k}$ and dividing both sides by $|k| X^{2m+1}$ we get that
$$
\begin{equation*}
\frac{(X-\nu)[p_m^*(X)]^2}{|k| X^{2m+1}} - \frac{\varepsilon[q_{m-1}^*(X)]^2(X-a)(X-b)(X-c)}{|k| X^{2m+1}}=\operatorname{sign}{k}.
\end{equation*}
\notag
$$
Again, set $s=1/X$ and define
$$
\begin{equation*}
p_m(s)=\frac{s^m p_m^*(1/s)\sqrt{|\nu|}}{\sqrt{|k|}} \quad\text{and}\quad q_{m-1}(s)=\frac{s^{m-1}q_{m-1}^*(1/s)\sqrt{a|b|c}}{\sqrt{|k|}}.
\end{equation*}
\notag
$$
Then these polynomials $p_m(s)$ and $q_{m-1}(s)$ satisfy equation (5.2) above, which proves part (b).
Theorem 5.1 is proved. Corollary 5.2. If the billiard trajectories inside a collared or a transverse $\mathcal{H}$-ellipse are $n$-periodic with caustic $\mathcal{C}_\nu$, then there exist real polynomials $\widehat{p}_n$ and $\widehat{q}_{n-2}$ of degrees $n$ and $n-2$, respectively, that satisfy Pell’s equation Proof. For $n=2m$ set and for $n=2m+1$ set The Pell-type equations above arise as a functional polynomial condition for periodicity. These solutions of Pell’s equations have further connections with the geometric properties of billiard trajectories. One can compare these results with [1]. 5.2. Rotation numbersSuppose $c_0<c_1<c_2<c_3$ are fixed constants and Then there exist polynomials $\widehat{p}_n$ and $\widehat{q}_{n-2}$ of degrees $n$ and $n-2$, respectively, such that if and only if there is an integer $n_1>0$ such that
$$
\begin{equation*}
n_1\int_{c_1}^{c_2}\frac{ds}{\sqrt{T(s)}} =n\int_{c_3}^{+\infty}\frac{ds}{\sqrt{T(s)}}.
\end{equation*}
\notag
$$
Here $n_1$ is the number of zeros of $\widehat{p}_n$ in $(c_0,c_1)$; see [38]. Thus, we can define the rotation number by
$$
\begin{equation*}
\rho:=\frac{n_1}{n} =\int_{c_3}^{+\infty}\frac{ds}{\sqrt{T(s)}}\bigg/\int_{c_1}^{c_2}\frac{ds}{\sqrt{T(s)}}.
\end{equation*}
\notag
$$
We consider the case when the boundary is a collared $\mathcal{H}$-ellipse. In this case there are three possibilities for the types of trajectories. (i) The caustic is of elliptic type and lies outside $\mathcal{E}$, and the billiard is within $\mathcal{E}$. Then $\nu < 0$ and $(\lambda_1,\lambda_2) \in [0,a]\times [b,c]$. The condition for $n$-periodicity is
$$
\begin{equation*}
m_0\int_0^{a}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} +m_1\int_{b}^{c}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} =0.
\end{equation*}
\notag
$$
We make the change $s=1/\lambda$ and set $c_0=1/\nu$, $c_1=1/c$, $c_2=1/b$ and $c_3=1/a$. Then we obtain
$$
\begin{equation*}
m_0\int_{\infty}^{c_3}\frac{ds}{\sqrt{T(s)}} +m_1\int_{c_2}^{c_1}\frac{ds}{\sqrt{T(s)}}=0.
\end{equation*}
\notag
$$
From
$$
\begin{equation*}
-m_1\int_{c_1}^{c_2}\frac{ds}{\sqrt{T(s)}}=m_0\int_{c_3}^{\infty}\frac{ds}{\sqrt{T(s)}}
\end{equation*}
\notag
$$
and Lemma 5.3 we conclude that
$$
\begin{equation*}
m_1<0, \qquad -m_1=n_1<m_0=n \quad\text{and}\quad \rho=-\frac{m_1}{m_0}.
\end{equation*}
\notag
$$
(ii) The caustic is of elliptic type and lies outside $\mathcal{E}$, and the billiard is outside $\mathcal{E}$. Then, in accordance with the discussion in § 3 of [37], billiard trajectories are space-like and are reflected off one component of $\mathcal{C}_0$. Then $\nu<0$ and $(\lambda_1,\lambda_2) \in [\nu, 0]\times[b,c]$. The billiard moves between one component of $\mathcal{C}_0$ and the caustic, does not cross the coordinate plane $x=0$, but must cross the coordinate planes $y=0$ and $z=0$ an even number of times. This is the only case that can have an odd period. The condition for $n$-periodicity is
$$
\begin{equation*}
m_2\int_0^{\nu}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} +m_3\int_{b}^{c}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} =0.
\end{equation*}
\notag
$$
We add and subtract
$$
\begin{equation*}
m_2\int_a^{0}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}},
\end{equation*}
\notag
$$
and obtain
$$
\begin{equation*}
m_2\int_a^{\nu}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} +m_3\int_{b}^{c}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} =m_2\int_a^{0}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}}.
\end{equation*}
\notag
$$
Since cycles around $[\nu, a]$ and $[b, c]$ are homologous, we get that
$$
\begin{equation*}
(m_3-m_2)\int_{b}^{c}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} =m_2\int_a^{0}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}}.
\end{equation*}
\notag
$$
We make the change $s=1/\lambda$, set $c_0=1/\nu$, $c_1=1/c$, $c_2=1/b$ and $c_3=1/a$ and obtain
$$
\begin{equation*}
(m_2-m_3)\int_{c_1}^{c_2}\frac{ds}{\sqrt{T(s)}}=m_2\int_{c_3}^{\infty}\frac{ds}{\sqrt{T(s)}}.
\end{equation*}
\notag
$$
From Lemma 5.3 we conclude that
$$
\begin{equation*}
m_3<m_2, \qquad m_2-m_3=n_1<m_2=n \quad\text{and}\quad \rho=1-\frac{m_3}{m_2}.
\end{equation*}
\notag
$$
(iii) The caustic is of hyperbolic type and the billiard is inside $\mathcal{E}$. Then the caustic is symmetric about the plane $z=0$ and $\nu \in [b, c]$, so that $(\lambda_1,\lambda_2)\in [0,a]\times[b,\nu]$. The trajectory must become tangent to the caustic at some point inside $\mathcal{E}$. The condition for $n$-periodicity is
$$
\begin{equation*}
m_4\int_0^{a}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} +m_5\int_{b}^{\nu}\frac{d\lambda}{\sqrt{\mathcal{P}(\lambda)}} =0.
\end{equation*}
\notag
$$
We make the change $s=1/\lambda$, set $c_0=1/c$, $c_1=1/\nu$, $c_2=1/b$ and $c_3=1/a$ and obtain
$$
\begin{equation*}
m_4\int_{\infty}^{c_3}\frac{ds}{\sqrt{T(s)}} +m_5\int_{c_2}^{c_1}\frac{ds}{\sqrt{T(s)}} =0.
\end{equation*}
\notag
$$
From
$$
\begin{equation*}
-m_5\int_{c_1}^{c_2}\frac{ds}{\sqrt{T(s)}}=m_4\int_{c_3}^{\infty}\frac{ds}{\sqrt{T(s)}}
\end{equation*}
\notag
$$
and Lemma 5.3 we conclude that
$$
\begin{equation*}
m_5<0, \qquad -m_5=n_1<m_4=n \quad\text{and}\quad \rho=-\frac{m_5}{m_4}.
\end{equation*}
\notag
$$
5.3. Zolotarev polynomials and periodic trajectoriesIt is well known that the Chebyshev polynomials can be defined recursively by $T_0(x)=1$, $T_1(x) =x$ and for $n=1,2,\dots$ . There are other parametrizations like see [4], for example. The reciprocal values of the leading coefficients are $L_n=2^{n-1}$ for $n=1, 2, \dots$ and $L_0=1$. Chebyshev proved that the polynomials $L_nT_n(x)$ are the solutions of the following minmax problem:As also shown by Chebyshev, Chebyshev polynomials satisfy a polynomial Pell equation, that is, there exists a polynomial $Q_{n-1}$ of degree $n-1$ such that The previous sections show that Pell’s equation plays a fundamental role in a polynomial formulation of the periodicity conditions. Solutions of Pell’s equation are, up to rescaling, extremal polynomials in the uniform norm on the union of two intervals defined by Pell’s equation. We call these generalized Chebyshev polynomials on two intervals Zolotarev polynomials, since they were introduced in works of Zolotarev [51], a prominent student of Chebyshev. These polynomials were studied further by Akhiezer (see [2] and also [3] and [4]). For some more recent results about Zolotarev polynomials, see [28]. Following the classics, consider the union of two intervals where
$$
\begin{equation*}
\alpha_{n,m}=1-2\operatorname{sn}^2\biggl(\frac{m}{n}K\biggr) \quad\text{and}\quad \beta_{n,m}=2\operatorname{sn}^2\biggl(\frac{n-m}{n}K\biggr)-1.
\end{equation*}
\notag
$$
Define
$$
\begin{equation*}
TA_{n}(x,m,\kappa)=L\biggl(v_{n,m}^{n}(u)+\frac{1}{v_{n,m}^{n}(u)}\biggr),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
v_{n,m}(u)=\frac{\theta_1(u-\frac{m}{n}K)}{\theta_1(u+\frac{m}{n}K)}, \qquad x_{n,m}=\frac{\operatorname{sn}^2(u)\operatorname{cn}^2(\frac{m}{n}K) +\operatorname{cn}^2(u)\operatorname{sn}^2(\frac{m}{n}K)}{\operatorname{sn}^2(u) -\operatorname{sn}^2(\frac{m}{n}K)}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
L_{n,m}=\frac{1}{2^{n-1}} \biggl(\frac{\theta_0(0)\theta_3(0)}{\theta_0(\frac{m}{n}K)\theta_3(\frac{m}{n}K)}\biggr)^{2n}, \qquad \kappa_{n,m}^2=\frac{2(\beta_{n,m} -\alpha_{n,m})}{(1 -\alpha_{n,m})(1+\beta_{n,m})}.
\end{equation*}
\notag
$$
Akhiezer [2] proved the following result. Theorem 5.4 (Akhiezer). (a) The function $TA_{n}(x,m,\kappa)$ is a polynomial of degree $n$ in $x$, with leading coefficient $1$ and second coefficient $-n\tau_{1}^{(n,m)}$, where (b) The maximum of the modulus of $TA_{n}$ on the union of the two intervals $[-1,\alpha_{n,m}]\cup [\beta_{n,m}, 1]$ is $L_{n,m}$. (c) The function $TA_{n}$ takes the values $\pm L_{n,m}$ with alternating signs at $\mu=n-m+1$ consecutive points of the interval $[-1,\alpha]$ and at $\nu=m+1$ consecutive points of the interval $[\beta, 1]$. In addition
$$
\begin{equation*}
TA_{n}(\alpha_{n,m},m,\kappa_{n,m})=TA_{n}(\beta_{n,m},m,\kappa_{n,m})=(-1)^{m}L_{n,m}
\end{equation*}
\notag
$$
and for any $x\in (\alpha_{n,m}, \beta_{n,m})$ the inequality holds. (d) The polynomials $TA_{n}(x,m,\kappa_{n,m})$ are the Zolotarev polynomials for $E_{n, m}= [-1,\alpha_{n,m}]\cup [\beta_{n,m}, 1]$ with norm $L_{n,m}=||TA_{n}(x,m,\kappa_{n,m})||_{E_{n,m}}$, and (e) Outside $E_{n,m}$ the derivative of the polynomial $TA_{n}(x,m,\kappa_{n,m})$ with respect to $x$ has only one zero $c_{n,m}$. It belongs to $[\alpha_{n,m}, \beta_{n,m}]$; moreover,
$$
\begin{equation}
c_{n,m}=\frac{\alpha_{n,m}+\beta_{n,m}}{2}-\tau_1^{(n,m)}.
\end{equation}
\tag{5.5}
$$
(f) Let $F$ be a polynomial of degree $n$ in $x$ with leading coefficient $1$ such that $\qquad$i) $\max|F(x)|=L_{n,m}$ for $x\in [-1,\alpha_{n,m}]\cup [\beta_{n,m}, 1]$; $\qquad$ii) $F(x)$ takes values $\pm L_{n,m}$ with alternating signs at $n-m+1$ consecutive points of the interval $[-1, \alpha_{n,m}]$ and at $m+1$ consecutive points of the interval $[\beta_{n,m}, 1]$. Then $F(x)=TA_{n}(x,m,\kappa_{n,m})$. The above formulae for $TA_n$ and $E_{n,m}$ provide a complete parametrizations for the Zolotarev polynomials and their supports in the case of two intervals. They can be used in our study of periodic trajectories. We consider one case of $3$-periodic trajectories for example, but the same consideration can be applied to any case of any period. Consider the transversal case of $3$-periodic trajectories when $b<\nu<0<a<c$. We want to construct an affine transformation
$$
\begin{equation*}
h(s)=\widehat ls+\widehat m\colon E_{3, m}=[-1,\alpha_{3,m}]\cup [\beta_{3,m}, 1]\to [c_0, c_1]\cup[c_2, c_3].
\end{equation*}
\notag
$$
One of the questions is to determine $m$. We set $Y=\operatorname{sn}(K/3)$. We are going to calculate $\operatorname{sn}(2K/3)$ in two different ways. The first is to express $\operatorname{sn}(K-u)$ in terms of $\operatorname{sn}(u)$, $\operatorname{cn}(u)$ and $\operatorname{dn}(u)$. The second is to express $\operatorname{sn}(2\cdot u)$ in terms of $\operatorname{sn}(u)$, $\operatorname{cn}(u)$ and $\operatorname{dn}(u)$. We obtain
$$
\begin{equation*}
\operatorname{sn}\biggl(\frac{2K}{3}\biggr) =\frac{\operatorname{cn}(K/3)}{\operatorname{dn}(K/3)}\quad\text{and}\quad \operatorname{sn}\biggl(\frac{2K}{3}\biggr)=\frac{2\operatorname{sn}(K/3) \operatorname{cn}(K/3)\operatorname{dn}(K/3)} {1-\kappa^2\operatorname{sn}^4(K/3)}.
\end{equation*}
\notag
$$
By taking squares and using the formulae expressing $\operatorname{cn}^2(u)$ and $\operatorname{dn}^2(u)$ in terms of $\operatorname{sn}(u)$ and $\kappa$ we have
$$
\begin{equation*}
\kappa^2=\frac{2Y-1}{Y^3(2-Y)}\quad\text{and}\quad \operatorname{sn}\biggl(\frac{2K}{3}\biggr)=Y(2-Y).
\end{equation*}
\notag
$$
We want to calculate the affine transformation
$$
\begin{equation*}
h(s)=\widehat ls+\widehat m\colon E_{3, m}=[-1,\alpha_{3,m}]\cup [\beta_{3,m}, 1]\xrightarrow[]{} [c_0, c_1]\cup[c_2, c_3].
\end{equation*}
\notag
$$
From $\widehat l+\widehat m=1/a$, $\widehat l\beta_{3m}+\widehat m=1/b$ and $\widehat l\alpha_{3m} +\widehat m = 1/b$ we obtain
$$
\begin{equation}
\frac{a-\beta_{3,m} c}{c(1-\beta_{3,m})}=\frac{a-\alpha_{3,m} b}{1-\alpha_{3,m}}.
\end{equation}
\tag{5.6}
$$
We have two potential cases: (a) $m=1$ and (b) $m=2$. (a) $m=1$. Then $\alpha_{3,1}=1-2Y^2$ and $\beta_{3,1}=-1+4Y-2Y^2$. Equation (5.6) leads to Using $-\widehat l+\widehat m=1/\nu$ we also get that However, the last two equations, (5.7) and (5.8), are not compatible with the equation
$$
\begin{equation}
3(abc)^2-2(abc)(ab+bc+ac)\nu+(4abc(a+b+c)-(ab+ac+bc)^2)\nu^2=0
\end{equation}
\tag{5.9}
$$
for the caustic in the $3$-periodic case. (b) $m=2$. Then $\beta_{3,2}=1-2Y^2$ and $\alpha_{3,2}=1-4Y+2Y^2$. Equation (5.6) leads to Using $-\widehat l+\widehat m=1/\nu$ we also obtain The last two equations (5.10) and (5.11) are compatible with equation (5.9) for the caustic in the $3$-periodic case, which takes now the form for
$$
\begin{equation*}
\begin{gathered} \, P=(a-b)c-2b(c-a)Y+a(b-c)Y^2, \\ Q=a^2b^2(3c(a-b)+2(ab-2ac-bc)Y -a(b-c)Y^2) \end{gathered}
\end{equation*}
\notag
$$
and We have proved the following. Proposition 5.5. In the transverse $3$-periodic case for $b<\nu<0<a<c$ the following relations take place, where $Y=\operatorname{sn}(K/3)$: and Here $\sim$ indicates that two polynomials are equal up to a scalar factor. The polynomial $\widehat p_3$ that appears in Proposition 5.5 is presented in Figure 9, a. The polynomial presented in Figure 9, b, cannot arise in the case under consideration. This contrasts the case of the Euclidean plane, where the situation is exactly opposite: see [26]. 5.4. Periodic light-like trajectories and Akhiezer polynomials on two symmetric intervalsBy definition, light-like trajectories have a velocity $v$ satisfying $\langle{v},{v}\rangle=0$ and their caustic is $\mathcal{C}_\infty$, the caustic at infinity. As noted in Theorem 2.1, closed light-like trajectories can only be of even period. We can adjust the above polynomial-based results to the setting of light-like trajectories by considering the limit as $\nu \to \infty$. Proposition 5.6. A light-like trajectory in a collared or a transverse $\mathcal{H}$-ellipse is periodic with period $n=2m$ if and only if there exist real polynomials $\widehat{p}_n$ and $\widehat{q}_{n-2}$ of degrees $n$ and $n-2$, respectively, that satisfy Pell’s equation Pell’s equation in the above proposition describes extremal polynomials on two intervals, $[0,1/c] \cup [1/b,1/a]$ or $[1/b,0]\cup[1/c,1/a]$, for a collared or a transverse $\mathcal{H}$-ellipse, respectively. We can express each of these in terms of certain polynomials of even degree composed with an affine transformation $[c_1,c_2]\cup[c_3,c_4] \mapsto [-1,-\alpha]\cup[\alpha,1]$ for $0 < \alpha < 1$, which are a simplification of the Zolotarev polynomials we discussed previously. In this case these polynomials are called the Akhiezer polynomials; they are denoted by $A_{2m}$ and obtained by means of a quadratic substitution from the Chebyshev polynomials $T_m$:
$$
\begin{equation}
A_{2m}(x;\alpha)=\frac{(1-\alpha^2)^m}{2^{2m-1}}T_m\biggl( \frac{2x^2-1-\alpha^2}{1-\alpha^2} \biggr).
\end{equation}
\tag{5.12}
$$
We illustrate this idea in the example of light-like trajectories of period 4. As noted in [37], a collared $\mathcal{H}$-ellipse can contain a light-like period 4 orbit when $c = ab/(b-a)$ and $a<b<2a$; a transverse $\mathcal{H}$-ellipse can contain a light-like period 4 orbit when $b = ac/(a-c)$. Under these restrictions on $a$, $b$ and $c$ in each case, we can produce a functional solution in terms of Zolotarev polynomials, which depends only on two of the parameters $a$, $b$ and $c$. Proposition 5.7. Consider a light-like period $4$ trajectory in a collared $\mathcal{H}$-ellipse. The polynomial $\widehat{p}_4$ is, up to a constant factor, equal to where $T_2(x) = 2x^2-1$ and $x=2as-1$. Proof. First we seek to find an affine transformation
$$
\begin{equation*}
g\colon [-1,-\alpha] \cup [\alpha,1] \mapsto \biggl[0,\frac1c\biggr]\cup\biggl[\frac1b,\frac1a\biggr].
\end{equation*}
\notag
$$
Writing $g(x) = Ax+B$ we see that
$$
\begin{equation*}
-A+B=g(-1)=0 \quad\text{and}\quad A+B=g(1)=\frac{1}{a}.
\end{equation*}
\notag
$$
Solving this system implies that $A=B=1/(2a)$. The other two endpoints produce the equations
$$
\begin{equation*}
-A\alpha+B=g(-\alpha)=\frac{1}{c} \quad\text{and}\quad A\alpha+B=g(\alpha)=\frac{1}{b}.
\end{equation*}
\notag
$$
Solving each equation gives $\alpha={2a}/{b}-1$ and $\alpha=-{2a}/{c}+1$, which are equal due to the assumption that $c=ab/(b-a)$. Set $x=g^{-1}(s)$. A simple calculation of the composition of $g$ and (5.12) proves the proposition. Repeating the above proof in the case of a transverse $\mathcal{H}$-ellipse produces a similar result. Proposition 5.8. Consider a light-like period $4$ trajectory in a transverse $\mathcal{H}$-ellipse. The polynomial $\widehat{p}_4$ is, up to a constant factor, equal to where $T_2(x) = 2x^2-1$ and $x=(2acs-a)/(2c-a)$. 5.5. Degenerate cases and classical Chebyshev polynomialsIn this subsection we consider the cases when the caustic $\mathcal{C}_{\nu}$ is degenerate, in particular, $a<\nu=b<c$. We derive conditions for the $2m$-periodicity of the corresponding trajectories. Note that the discussion in § 3 implies that there are two closed trajectories on this level set, both of which are $2$-periodic. However, in the limit as ${\nu\to b}$, the rotation number can approach another value. The next proposition gives a condition for resonance in this limit. Proposition 5.9. A trajectory is periodic with period $n=2m$ in the case ${a<\nu=b<c}$ if and only if there exist real polynomials $\widehat{p}_m(s)$ and $\widehat{q}_{m-1}(s)$ of degrees $m$ and $m-1$, respectively, such that (a) $\widehat{p}_m^{\,2}(s)-\biggl(s-\dfrac1a\biggr)\biggl(s-\dfrac1c\biggr) \widehat{q}_{m-1}^{\,2}(s)=1$; (b) $\widehat{q}_{m-1}(1/b)=0$. Condition (a) in Proposition 5.9 is the standard Pell equation describing extremal polynomials on one interval $[1/c,1/a]$, thus the polynomials $\widehat{p}_m$ can be obtained as Chebyshev polynomials composed with an affine transformation $[1/c,1/a]\to[-1,1]$. Condition (b) implies an additional constraint on the parameters $a$, $b$ and $c$. We have the following. Proposition 5.10. The polynomials $\widehat {p}_m$ and the parameters $a$, $b$ and $c$ have the following properties: (a) $\widehat{p}_m(s) =T_m\biggl(\dfrac{2ac}{c-a}\biggl(s-\dfrac{a+c}{2ac}\biggr)\biggr)$, where $T_m$ is defined by (5.4); (b) the condition $\widehat{q}_{m-1}(1/b)=0$ is equivalent to for Proof. The increasing affine transformation is given by the formula $h(s)=\widehat ls + \widehat m$, where
$$
\begin{equation*}
\widehat m=\frac{a+c}{2ac} \quad\text{and}\quad \widehat l=\frac{c-a}{2ac}.
\end{equation*}
\notag
$$
We apply Corollary 5.2. The interior extremal points of the Chebyshev polynomial $T_m$ of degree $m$ on the interval $[-1, 1]$ are given by
$$
\begin{equation*}
x_k=\cos \biggl(\frac{k}{m}\pi\biggr), \qquad k=1, \dots, m-1,
\end{equation*}
\notag
$$
according to formula (5.4). The proof of part (a) is complete.
Part (b) follows from $h^{-1}(1/b)=x_k$. The proposition is proved.
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\Bibitem{DraGasRad22} |
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