2 Existence of smooth convex resonant caustics

To begin with, we introduce coordinates in the space of oriented lines, define the support function and the billiard map of a convex domain, and describe the Bialy–Mironov generating function following [Reference Bialy and Tabachnikov10]. We also recall the periodic version of the variational principle for twist maps following [Reference Meiss38]. Next, we combine all those elements to find necessary conditions for the existence of smooth convex resonant caustics in Theorem 2.3. This part is inspired by the computations in [Reference Bialy and Tabachnikov10, Theorem 2.2] and the Lagrangian approach to the existence of rotational invariant curves (RICs) of twist maps described in [Reference Kaloshin and Zhang31, Reference Levi and Moser34]. Finally, we discuss five simple examples: circles, ellipses, constant width curves, Gutkin billiard tables (also called constant angle curves), and centrally symmetric curves with $1/4$ -resonant caustics.

The billiard dynamics acts on the subset of oriented lines (rays) that intersect the boundary of the convex domain. An oriented line $\ell $ can be written as

$$ \begin{align*} \cos \varphi \cdot x + \sin \varphi \cdot y = \unicode{x3bb}, \end{align*} $$

where $\varphi \in \mathbb {T} := \mathbb {R}/2\pi \mathbb {Z}$ is the direction of the right normal to the oriented line and $\unicode{x3bb} \in \mathbb {R}$ is the signed distance to the origin. Thus, $(\varphi ,\unicode{x3bb} ) \in \mathbb {A} := \mathbb {T} \times \mathbb {R}$ are coordinates in the space of oriented lines, which is topologically a cylinder.

Let $\Gamma $ be a smooth strictly convex closed curve of $\mathbb {R}^2$ . We fix its counterclockwise orientation and assume its interior contains the origin O, so there is a positive smooth $2\pi $ -periodic function $h(\varphi )$ , called the support function of $\Gamma $ , such that $\{ \unicode{x3bb} = h(\varphi ) \}$ and $\{ \unicode{x3bb} = -h(\varphi +\pi ) \}$ are the 1-parameter families of oriented lines positively and negatively tangent to $\Gamma $ . Then,

$$ \begin{align*} z: \mathbb{T} \to \Gamma \subset \mathbb{R}^2 \simeq \mathbb{C}, \quad z(\varphi) = ( x(\varphi), y(\varphi) ) = h(\varphi) \mathit{e}^{\mskip2mu{i}\mskip1mu \varphi} + h'(\varphi) \mathit{e}^{\mskip2mu{i}\mskip1mu (\varphi+\pi/2)} \end{align*} $$

is a parameterization of $\Gamma $ , where $\varphi \in \mathbb {T}$ is the counterclockwise angle between the positive x-axis and the outer normal to $\Gamma $ at the point $z(\varphi )$ .

The space of the oriented lines that intersect the interior of $\Gamma $ is the open cylinder

$$ \begin{align*} \mathbb{A}_\Gamma = \{ (\varphi,\unicode{x3bb}) \in \mathbb{A} : -h(\varphi+\pi) < \unicode{x3bb} < h(\varphi) \} \end{align*} $$

and the billiard map $f: \mathbb {A}_\Gamma \to \mathbb {A}_\Gamma $ acts by the reflection law in $\Gamma $ . That is, $f(\varphi ,\unicode{x3bb} ) = (\varphi _1,\unicode{x3bb} _1)$ means that the oriented line $\ell _1$ with coordinates $(\varphi _1,\unicode{x3bb} _1)$ is the reflection of the oriented line $\ell $ with coordinates $(\varphi ,\unicode{x3bb} )$ with respect to the tangent to $\Gamma $ at the second intersection of $\ell $ with $\Gamma $ . See Figure 1. The shocking discovery by Bialy and Mironov was that the billiard map f is an exact twist map with generating function

$$ \begin{align*} S(\varphi,\varphi_1) = 2 h(\psi) \sin \theta, \quad \psi = \frac{\varphi_1 + \varphi}{2}, \quad \theta = \frac{\varphi_1 - \varphi}{2}. \end{align*} $$

To be precise, $\unicode{x3bb} _1 \,\mathit {d} \varphi _1 - \unicode{x3bb} \,\mathit {d} \varphi = f^* (\unicode{x3bb} \,\mathit {d} \varphi ) - \unicode{x3bb} \,\mathit {d} \varphi = \,\mathit {d} S$ , so

(5) $$ \begin{align} f(\varphi,\unicode{x3bb}) = (\varphi_1,\unicode{x3bb}_1) \Leftrightarrow \begin{cases} \unicode{x3bb} = -\partial_1 S(\varphi,\varphi_1) = h(\psi) \cos \theta - h'(\psi) \sin \theta, \\ \unicode{x3bb}_1 = \partial_2 S(\varphi,\varphi_1) = h(\psi) \cos \theta + h'(\psi) \sin \theta, \end{cases} \end{align} $$

and f preserves the standard area form $\mathit {d} \varphi \wedge \,\mathit {d} \unicode{x3bb} $ . See, for instance, [Reference Bialy and Tabachnikov10, Proposition 2.1]. Here, $\partial _i S$ denotes the derivative with respect to the ith variable. The strict convexity of $\Gamma $ implies the twist condition: $\partial _{12}S(\varphi ,\varphi _1) = \tfrac 12 \rho (\psi )\sin \theta> 0$ , where ${\rho (\psi ) = h"(\psi ) + h( \psi )}$ is the radius of curvature of $\Gamma $ at the point $z(\psi )$ .

Figure 1

The billiard map $f(\varphi ,\unicode{x3bb} ) = (\varphi _1,\unicode{x3bb} _1)$ , the normal angle $\psi = (\varphi _1 + \varphi )/2$ , and the incidence–reflection angle $\theta = (\varphi _1 - \varphi )/2$ .

We say that $\psi \in \mathbb {T}$ and $\theta \in (0,\pi )$ are the normal angle and the angle of incidence/reflection at each impact point, whereas $\varphi \in \mathbb {T}$ is the side angle. We consider $S(\varphi ,\varphi _1)$ defined on the universal cover $\{ (\varphi ,\varphi _1) \in \mathbb {R}^2 : \varphi < \varphi _1 < \varphi + 2\pi \}$ because ${S(\varphi +2\pi ,\varphi _1+2\pi ) = S(\varphi ,\varphi _1)}$ .

Let us briefly recall the classical variational principle for exact twists maps. The interested reader can find more details in [Reference Meiss38, §V]. We will introduce several operators which are not standard in the variational approach, but they simplify computations and shorten formulas.

Given any sequence $\{a_j\}$ , we consider the shift, sum, difference, and q-average operators

$$ \begin{align*} \tau\{ a_j \} = a_{j+1},\quad\!\! \sigma \{ a_j \} = a_{j+1} + a_j,\quad\!\! \delta \{ a_j \} = a_{j+1} - a_j, \quad\!\! \mu \{ a_j \} = \frac{a_j + \cdots + a_{j+q-1}}{q}. \end{align*} $$

For simplicity, we omit the dependence of $\mu $ on q and we just say that $\mu $ is the average operator.

Let $p/q \in (0,1)$ be a rational number with $\gcd (p,q) = 1$ . We call the elements of the space

$$ \begin{align*} X = \{ \{\varphi_j \} \in \mathbb{R}^{\mathbb{Z}} : \varphi_j < \varphi_{j+1} < \varphi_j + 2\pi, \ \varphi_{j+q} = \varphi_j + 2\pi p \text{ for all } j \in \mathbb{Z} \} \end{align*} $$

$p/q$ -periodic sequences. We define the $p/q$ -periodic action $A: X \to \mathbb {R}$ as

$$ \begin{align*} A\{ \varphi_j \} = \sum_{j=0}^{q-1} S(\varphi_j,\varphi_{j+1}) = 2 \sum_{j=0}^{q-1} h(\psi_j) \sin \theta_j, \end{align*} $$

where $\psi _j = \sigma \{ \varphi _j \}/2$ and $\theta _j = \delta \{ \varphi _j \}/2$ . Periodicities $\varphi _{j+q} = \varphi _j + 2\pi p$ , $\psi _{j+q} = \psi _j + 2\pi p$ , and $\theta _{j+q} = \theta _j$ imply that $\mu \{ S(\varphi _{j_0+j},\varphi _{j_0+j+1}) \} = 2\mu \{ h(\psi _{j_0+j}) \sin \theta _{j_0+j} \} = A\{ \varphi _j \}/q$ for all $j_0 \in \mathbb {Z}$ . Critical points of the action, which we call $p/q$ -periodic configurations, can be lifted to full $p/q$ -periodic orbits of the billiard map f by taking $\unicode{x3bb} _j = -\partial _1 S(\varphi _j,\varphi _{j+1}) = \partial _2 S(\varphi _{j-1},\varphi _j)$ . Thus, any $p/q$ -periodic configuration defines a $p/q$ -periodic billiard trajectory inside $\Gamma $ with side angles $\varphi _j$ , normal angles $\psi _j = \sigma \{ \varphi _j \}/2$ , and incidence–reflection angles $\theta _j = \delta \{ \varphi _j \}/2$ .

We characterize such configurations in the next proposition, where we also recall a formula for the length of their corresponding periodic billiard trajectories.

Proposition 2.1. A $p/q$ -periodic sequence $\{ \varphi _j \}$ is a $p/q$ -periodic configuration if and only if

$$ \begin{align*} \sigma \{ h'(\psi_j) \sin \theta_j \} - \delta \{ h(\psi_j) \cos \theta_j \} = 0 \quad \text{for all } j \in \mathbb{Z}, \end{align*} $$

where $\psi _j = \sigma \{ \varphi _j \}/2$ and $\theta _j = \delta \{ \varphi _j \}/2$ , in which case,

$$ \begin{align*} 2q \mu \{ h(\psi_j) \sin \theta_j \} = A\{ \varphi_j\} = L,\quad \mu \{ h'(\psi_j) \sin \theta_j \} = 0, \quad \mu \{\theta_j\} = \pi p/q, \end{align*} $$

where L is the length of the $p/q$ -periodic configuration $\{ \varphi _j \}$ .

Proof. A $p/q$ -periodic sequence $\{ \varphi _j \}$ is a critical point of the action if and only if

$$ \begin{align*} \sigma \{ h'(\psi_j) \sin \theta_j \} - \delta \{ h(\psi_j) \cos \theta_j \} = \partial_2 S(\varphi_j,\varphi_{j+1}) + \partial_1 S(\varphi_{j+1},\varphi_{j+2}) = 0 \end{align*} $$

for all $j \in \mathbb {Z}$ . The first equality above follows from $S(\varphi _j,\varphi _{j+1}) = 2 h(\psi _j) \sin \theta _j$ , and relations $\psi _j = \sigma \{ \varphi _j \}/2 = (\varphi _{j+1} + \varphi _j)/2$ and $\theta _j = \delta \{ \varphi _j \}/2 = (\varphi _{j+1} - \varphi _j)/2$ .

Identities $2q \mu \big \{ h(\psi _j) \sin \theta _j \big \} = L$ and $\mu \{ h'(\psi _j) \sin \theta _j \} = 0$ are proved in [Reference Bialy and Tabachnikov10, Theorem 2.2]. By q-periodicity of $\{\theta _j\}$ , we get

$$ \begin{align*} \mu \{ \theta_j \} &= \frac{1}{q} \sum_{i=0}^{q-1} \theta_{j+i} = \frac{1}{q} \sum_{i=0}^{q-1} \frac{\theta_{j+i} + \theta_{j+i+1}}{2} = \frac{1}{q} \sum_{i=0}^{q-1} \frac{\psi_{j+i+1} - \psi_{j+i}}{2}\\& = \frac{\psi_{j+q} - \psi_j}{2q} = \frac{\pi p}{q}.\\[-41pt] \end{align*} $$

We look for necessary conditions for the existence of convex resonant caustics inside $\Gamma $ . Tangent lines to such a caustic can be counterclockwise or clockwise oriented. We fix the counterclockwise orientation, so we assume that $p/q < 1/2$ from now on. Any $p/q$ -resonant convex caustic gives rise to a 1-parameter family of $p/q$ -periodic orbits of the billiard map. We want to parameterize this family using a dynamical parameter $t \in \mathbb {R}$ in which the billiard map acts as the constant shift $t \mapsto t + \omega $ with angular frequency $\omega = 2\pi p/q$ . The dynamical parameter is not unique. If $a(t)$ is any smooth $\omega $ -periodic function such that $1 + a'(t)> 0$ , then $s = t + a(t)$ is another dynamical parameter.

Let us stress the three main differences between this setting, where we deal with functions of a continuous variable $t \in \mathbb {R}$ , and the previous setting, where we had sequences whose elements are labeled by a discrete index $k \in \mathbb {Z}$ .

First, we define the shift, sum, difference, and average operators as

$$ \begin{align*} \tau\{ a(t) \} = a(t + \omega),\quad \sigma\{ a(t) \} = a(t + \omega) + a(t), \quad \delta\{ a(t) \} = a(t + \omega) - a(t), \end{align*} $$

and $\mu \{ a(t) \} = ({1}/{q}) \sum _{j=0}^{q-1} a(t + j \omega )$ . These operators diagonalize in the Fourier basis. Operator $\mu $ is the projection onto the resonant $q\mathbb {Z}$ -harmonics: $\mu = \mu _{q\mathbb {Z}}$ , but we omit the $q\mathbb {Z}$ subscript for simplicity. Both claims are proved in Appendix A.

Second, we define the $p/q$ -periodic action of a side function $\varphi (t)$ as

$$ \begin{align*} A\{ \varphi(t) \} = q \mu \{ S(\varphi(t),\varphi(t+\omega) ) \} = 2q \mu \{ h(\psi(t)) \cdot \sin \theta(t) \}, \end{align*} $$

where $\varphi (t)$ , the normal function $\psi (t)$ , and the incidence-reflection function $\theta (t)$ are related by

(6) $$ \begin{align} \psi = \varphi + \theta = \sigma\{ \varphi \}/2, \quad 2\theta = \delta \{ \varphi \}, \quad \mu\{ \psi - t \} = 0, \quad \mu\{ t - \varphi \} = \omega/2 =\mu\{ \theta \}. \end{align} $$

We ask functions $\varphi (t) - t$ , $\psi (t) - t$ , and $\theta (t)$ to be $2\pi $ -periodic, as a continuous analogue of the discrete periodicity conditions $\varphi _{j+q} = \varphi _j + 2\pi p$ , $\psi _{j+q} = \psi _j + 2\pi p$ , and ${\theta _{j+q} = \theta _j}$ . Hence, they are lifts of some functions $\varphi ,\psi : \mathbb {T} \to \mathbb {T}$ and $\theta : \mathbb {T} \to \mathbb {R}$ . To simplify the exposition, we sometimes abuse the notation, and use the same symbol for an object and its lift. Other times, we denote lifts with a tilde. We also ask that $\varphi '(t)> 0$ , so that $\varphi :\mathbb {R} \to \mathbb {R}$ can be inverted.

Third, condition $\sigma \{ h'(\psi _j) \sin \theta _j \} - \delta \{ h(\psi _j) \cos \theta _j \} = 0$ becomes the difference equation

(7) $$ \begin{align} \sigma \{ h'\circ \psi \cdot \sin \theta \} - \delta \{ h \circ \psi \cdot \cos \theta \} = 0. \end{align} $$

Proof. (f) Let $g(\varphi )$ be the smooth support function of the $p/q$ -resonant caustic. Set

$$ \begin{align*} G = \mathrm{graph}(g) := \{ (\varphi,\unicode{x3bb}) \in \mathbb{A} : \unicode{x3bb} = g (\varphi) \}. \end{align*} $$

The caustic is inside $\Gamma $ , so $G \subset \mathbb {A}_\Gamma $ . Clearly, G is f-invariant, so $f|_G$ defines a smooth preserving orientation circle diffeomorphism $r: \mathbb {T} \to \mathbb {T}$ such that

$$ \begin{align*} \tilde{f}( \varphi,\tilde{g}(\varphi) ) = ( \tilde{r}(\varphi),\tilde{g}(\tilde{r}(\varphi)) ), \quad \tilde{r}^q(\varphi) = \varphi + 2\pi p = \varphi + q\omega, \end{align*} $$

where $\tilde {f}:\tilde {\mathbb {A}}_\Gamma \to \tilde {\mathbb {A}}_\Gamma $ , $\tilde {g}:\mathbb {R} \to \mathbb {R}$ , and $\tilde {r}:\mathbb {R} \to \mathbb {R}$ are the corresponding lifts. Identity $\tilde {r}^q(\varphi ) = \varphi + q\omega $ is the key. It follows from the definition of $p/q$ -resonant caustic. It implies that $\varphi \mapsto \tilde {r}(\varphi )$ becomes the constant shift $t \mapsto t +\omega $ in the smooth parameter

$$ \begin{align*} t = \tilde{s}(\varphi) := \frac{1}{q} \sum_{j=0}^{q-1} ( \tilde{r}^j(\varphi) - j \omega ). \end{align*} $$

Inversion of $\tilde {s}(\varphi (t)) = t$ defines a smooth function such that $\varphi '(t)> 0$ , $\varphi (t) - t$ is $2\pi $ -periodic, and any sequence $\{ \varphi (t + j\omega ) \}$ , $t \in \mathbb {R}$ , is a $p/q$ -periodic configuration of f. Then, Proposition 2.1 implies that the normal function $\psi (t)$ and the incidence–reflection function $\theta (t)$ obtained from $\varphi (t)$ by relations (6) satisfy the difference equation (7).

(p) We define

$$ \begin{align*} \unicode{x3bb}(t) = -\partial_1 S(\varphi(t),\varphi(t+\omega)) = \partial_2 S(\varphi(t-\omega),\varphi(t)). \end{align*} $$

Since $\varphi (t) - t$ and $\unicode{x3bb} (t)$ are $2\pi $ -periodic, the map $c = (\varphi ,\unicode{x3bb} ) : \mathbb {R} \to \mathbb {R}^2$ can be projected to a map from $\mathbb {T}$ to $\mathbb {A}$ . Since $g(\varphi )$ is the support function of the caustic, we get that $\unicode{x3bb} (t) = g(\varphi (t))$ and $c(\mathbb {T}) = G \subset \mathbb {A}_\Gamma $ . Condition $\varphi '(t)> 0$ implies that $c'(t) \neq (0,0)$ , so $c:\mathbb {T} \to G$ is a parameterization. Implicit equations (5) imply $f(c(t)) = c(t+\omega )$ .

(a) The $p/q$ -periodic sequences $\{ \varphi (t + j\omega ) \}_{j \in \mathbb {Z}}$ form a $1$ -parameter family of critical points of the $p/q$ -periodic action, with $t \in \mathbb {R}$ a smooth parameter. Therefore, the action is constant on this $1$ -parameter family. That is, all periodic billiard trajectories tangent to the $p/q$ -resonant caustic have the same length.

To provide a first insight into the usefulness of condition (f) in Theorem 2.3, let us give some information about functions $\varphi (t)$ , $\psi (t)$ , and $\theta (t)$ in five examples.

Example 2.8. Gutkin billiard tables [Reference Bialy4, Reference Gutkin22], also called constant angle curves, are another classic example. We claim that circles are the only convex billiard tables with a $p/q$ -resonant caustic, with $p/q \in (0,1/2)$ , whose incidence–reflection function $\theta (t)$ is constant. In that case, $\theta (t) \equiv \omega /2$ with $\omega = 2\pi p/q$ , so $\varphi (t) = t - \omega /2$ and $\psi (t) = t$ are the functions determined from $\theta (t)$ by relations (6). Therefore, the difference equation (7) becomes

$$ \begin{align*} \tan(\omega/2) ( h'(t + \omega) + h'(t) ) = h(t + \omega) - h(t). \end{align*} $$

If $h(t) = \sum _{l \in \mathbb {Z}} \hat {h}_l \mathit {e}^{\mskip 2mu {i}\mskip 1mu l t}$ satisfies this equation, then $\hat {h}_l = 0$ for any index $l \in \mathbb {Z}$ such that

$$ \begin{align*} \tan(l \pi p/q) \neq l \tan (\pi p /q). \end{align*} $$

Cyr [Reference Cyr16] proved that given any integer $l \not \in \{-1,0,1\}$ , equation $\tan (l \pi p/q) = l \tan (\pi p /q)$ has no rational solution $p/q \in (0,1/2)$ . This proves the claim, because circles are the only convex curves whose support function is a trigonometric polynomial of degree one.

Example 2.9. A smooth centrally symmetric convex curve $\Gamma $ with support function $h(\psi )$ has a convex $1/4$ -resonant caustic if and only if

$$ \begin{align*} h^2(\psi) = \hat{c}_0 + \sum_{l \in 2 + 4\mathbb{Z}} \hat{c}_l \mathit{e}^{\mskip2mu{i}\mskip1mu l \psi} \quad \text{and} \quad h + h"> 0. \end{align*} $$

This claim is proved in [Reference Bialy and Tsodikovich11, Proposition 3.1]. Along the proof, the authors check three facts. First, all such curves are centrally symmetric: $h(\psi + \pi ) = h(\psi )$ . Second, once one of such curves is fixed, there is a constant $R> 0$ such that $h^2(\psi ) + h^2(\psi + \pi /2) = R^2$ . Third, then we can take $\psi (t) = t$ and determine the incidence–reflection function $\theta (t)$ by means of

$$ \begin{align*} h(t) = R \sin \theta(t), \quad h(t+\pi/2) = R \cos \theta(t). \end{align*} $$

Note that $\theta (t + \pi /2) = \pi /2 - \theta (t)$ and $\psi (t + \pi /2) = \psi (t) + \pi /2$ . The last relation means that the tangent lines to $\Gamma $ at the impacts of any $1/4$ -periodic trajectory form a rectangle. This fact plays a key role in the Bialy–Mironov proof of a strong version of the Birkhoff conjecture [Reference Bialy and Mironov8]. If we set $\varphi = \psi - \theta $ and $\omega = \pi /2$ , then functions $\varphi (t)$ , $\psi (t)$ , and $\theta (t)$ satisfy condition (f).

3 Persistence of smooth convex resonant caustics

Necessary conditions in Theorem 2.3 are not sufficient for the existence of smooth convex resonant caustics, see Example 2.7. However, if the envelope of the 1-parameter family of lines from $z(\psi (t))$ to $z(\psi (t+\omega ))$ , where t is the dynamical parameter, is a smooth convex curve, then those conditions are sufficient too. That is the case when we consider small enough smooth deformations $\Gamma _\epsilon = \Gamma _0 + \mathrm {O} (\epsilon )$ of a smooth strictly convex curve $\Gamma _0$ , not necessarily a circle, with a smooth strictly convex resonant caustic.

We consider that setting. To be precise, we assume the following hypothesis from here on.

1. (H) Let $p/q \in (0,1/2)$ be a rational rotational number such that $\gcd (p,q) = 1$ . Let $\Gamma _0$ be a smooth strictly convex curve with support function $h_0(\varphi )$ . We assume that there is a smooth convex $p/q$ -resonant caustic with support function $g_0(\varphi )$ inside $\Gamma _0$ . We also assume that the origin is in the interior of the caustic, so $0 < g_0(\varphi ) < h_0(\varphi )$ . Let $\Gamma _\epsilon = \Gamma _0 + \mathrm {O} (\epsilon )$ , with $\epsilon \in [-\epsilon _0,\epsilon _0]$ , be a deformation of the unperturbed curve with smooth support function $h(\varphi ;\epsilon )$ . Let $(\varphi _1,\unicode{x3bb} _1) = f_\epsilon (\varphi ,\unicode{x3bb} )$ , $S_\epsilon (\varphi ,\varphi _1) = 2 h(\psi ;\epsilon ) \sin \theta $ , and $A_\epsilon \{\varphi \} = q \mu \{ S_\epsilon (\varphi ,\tau \{\varphi \}) \} = 2 q \mu \{ h \circ \psi \cdot \sin \theta \}$ be the perturbed billiard map in $\Gamma _\epsilon $ , the perturbed generating function, and the perturbed action, respectively.

We need two parameters in that perturbed setting. The dynamical parameter t parameterizes the invariant objects. The perturbative parameter $\epsilon \in [-\epsilon _0,\epsilon _0]$ labels the ovals. Then, the shift, sum, difference, and average operators are applied to functions that depend on t and $\epsilon $ , although they only act on t. For instance, $\tau \{ a(t;\epsilon ) \} = a(t + \omega ;\epsilon )$ . We will denote the derivatives of the support function $h(\psi ;\epsilon ) = h_\epsilon (\psi )$ as $h' = {\mathit {d} h}/{\mathit {d} \psi }$ and $\dot {h} = {\mathit {d} h}/{\mathit {d} \epsilon }$ . Analogously, we will denote the derivatives of any function $a(t;\epsilon ) = a_\epsilon (t)$ as $a' = {\mathit {d} a}/{\mathit {d} t}$ and $\dot {a} = {\mathit {d} a}/{\mathit {d} \epsilon }$ .

Next, we state an immediate extension of Theorem 2.3.

Remark 3.2. Similarly to Remark 2.4, if these necessary conditions hold, then

$$ \begin{align*} 2q \mu\{ h \circ \psi \cdot \sin \theta \} = L,\quad \mu \{ h' \circ \psi \cdot \sin \theta \} = 0, \quad 2q \mu\{ \dot{h} \circ \psi \cdot \sin \theta \} = \dot{L}. \end{align*} $$

Only the last formula is new. Let us prove it. If we derive the first relation with respect to $\epsilon $ , use the summation by parts formula, and take advantage of (7), we get

$$ \begin{align*} \dot{L} &= 2q\mu \{ \dot{h} \circ \psi \cdot \sin \theta + h'\circ \psi \cdot \sin \theta \cdot \dot{\psi} + h \circ \psi \cdot \cos \theta \cdot \dot{\theta} \} \\ &= 2q\mu\{ \dot{h} \circ \psi \cdot \sin \theta \} + q\mu\{ h'\circ \psi \cdot \sin \theta \cdot \sigma\{ \dot{\varphi} \} + h \circ \psi \cdot \cos \theta \cdot \delta\{ \dot{\varphi} \} \} \\ &= 2q\mu\{ \dot{h} \circ \psi \cdot \sin \theta \} + q\mu\{ [ \sigma\{ h'\circ \psi \cdot \sin \theta \} - \delta\{ h\circ \psi \cdot \cos\theta \} ] \cdot \tau\{\dot{\varphi}\} \} \\ &= 2q\mu\{ \dot{h} \circ \psi \cdot \sin \theta \}. \end{align*} $$

We can normalize $\Gamma _\epsilon $ by a scaling in such a way that $\dot {L} = 0$ , but we do not need it.

Full persistence of resonant caustics is an extremely rare phenomenon, so we introduce the more common concept of $\mathrm {O} (\epsilon ^m)$ -persistence for some order $m \in \mathbb {N} \cup \{ 0 \}$ .

Since $\mathrm {O} (\epsilon ^m)$ -persistence is the main concept of this work, some comments are in order. If $\varphi _0(t)$ , $\psi _0(t)$ , and $\theta _0(t)$ are the unperturbed side, normal, and incidence–reflection functions, then

$$ \begin{align*} \sigma \{ h'\circ \psi_0 \cdot \sin \theta_0 \} - \delta \{ h\circ \psi_0 \cdot \cos \theta_0 \} = \mathrm{O}(\epsilon) \quad \text{as } \epsilon \to 0, \end{align*} $$

since $h = h_0 + \mathrm {O} (\epsilon )$ . Therefore, the unperturbed caustic always $\mathrm {O} (\epsilon ^0)$ -persists. We do not need to check all three conditions (F) ${}_m$ , (P) ${}_m$ , and (A) ${}_m$ , because there are logical dependencies among them. We prove in Proposition 3.4 that conditions (F) ${}_m$ and (P) ${}_m$ are equivalent and both imply condition (A) ${}_m$ . We will only check condition (F) ${}_m$ in our computations. We have included the other conditions as part of our definition to present a broader view of the problem.

The three conditions look similar, but they have different characteristics. On the one hand, conditions (F) ${}_m$ and (P) ${}_m$ are stated in terms of a single iteration of the perturbed map $f_\epsilon $ . On the other hand, condition (A) ${}_m$ requires to consider all the shifts

$$ \begin{align*} \bar{\varphi}_j = \bar{\varphi}_j(t;\epsilon) = \tau^j\{ \varphi(t;\epsilon) \} = \varphi(t + j\omega;\epsilon), \quad j=0,\ldots,q. \end{align*} $$

Hence, conditions (F) ${}_m$ and (P) ${}_m$ are easier to deal with from a computational point of view.

Condition (F) ${}_m$ means that there is a reparameterization $\varphi _\epsilon (t) = \varphi (t;\epsilon )$ of the original angle $\varphi $ in terms of a new dynamical parameter t such that

$$ \begin{align*} \partial_2 S_\epsilon(\varphi_\epsilon(t-\omega),\varphi_\epsilon(t)) + \partial_1 S_\epsilon(\varphi_\epsilon(t),\varphi_\epsilon(t+\omega)) = \mathrm{O}(\epsilon^{m+1}) \quad \text{as } \epsilon \to 0. \end{align*} $$

Thus, condition (F) ${}_m$ is related to the Lagrangian formulation as a second-order difference equation of the invariance condition for (non-resonant and resonant) rotational curves of exact twist maps. It was inspired by ideas contained in [Reference Kaloshin and Zhang31, Reference Levi and Moser34]. Compare with [Reference Levi and Moser34, (3)] in the non-resonant setting and with [Reference Kaloshin and Zhang31, (1)] in the resonant setting. Finally, condition (A) ${}_m$ follows the variational approach in [Reference Damasceno, Dias Carneiro and Ramírez-Ros17].

Let us prove the logical dependencies among these $\mathrm {O} (\epsilon ^m)$ -persistence conditions and how the dominant terms in their $\mathrm {O} (\epsilon ^{m+1})$ -errors are related.

Proposition 3.4. Conditions $(\mathbf{F})_m$ and $(\mathbf{P})_m$ are equivalent and both imply condition $(\mathbf{A})_m$ . Let $\varphi _0(t) = \varphi (t;0)$ be the unperturbed side function. If the $p/q$ -resonant caustic $\mathrm {O} (\epsilon ^m)$ -persists and we follow the notation introduced in Definition 3.3, then there is a smooth $2\pi $ -periodic function $\mathcal {E}_{m+1}(t)$ and a smooth $2\pi /q$ -periodic function $\mathcal {L}_{m+1}(t)$ such that

(9a) $$ \begin{align} \sigma \{ h'\circ \psi \cdot \sin \theta \} - \delta \{ h\circ \psi \cdot \cos \theta \} &= \epsilon^{m+1} \tau\{\mathcal{E}_{m+1}\} + \mathrm{O}(\epsilon^{m+2}), \end{align} $$

(9b) $$ \begin{align} f_\epsilon \circ c_\epsilon - \tau\{ c_\epsilon \} &= ( 0, \epsilon^{m+1} \tau\{\mathcal{E}_{m+1}\} + \mathrm{O}(\epsilon^{m+2}) ), \end{align} $$

(9c) $$ \begin{align} A_\epsilon \{ \varphi \} - L(\epsilon) &= \epsilon^{m+1} q \mathcal{L}_{m+1} + \mathrm{O}(\epsilon^{m+2}), \end{align} $$

as $\epsilon \to 0$ . In addition, $\mu \{ \mathcal {E}_{m+1} \varphi ^{\prime }_0 \} = \mathcal {L}^{\prime }_{m+1}$ and $\int _{\mathbb {T}} \mathcal {E}_{m+1} \varphi ^{\prime }_0 = 0$ .

Proof. First, we check that $(\mathbf{F})_m \Rightarrow (\mathbf{P})_m$ and $(\mathbf{A})_m$ , $\mathcal {L}^{\prime }_{m+1} = \mu \{ \mathcal {E}_{m+1} \varphi ^{\prime }_0 \}$ , and $\int _{\mathbb {T}} \mathcal {E}_{m+1} \varphi ^{\prime }_0 = 0$ .

Let $\varphi (t;\epsilon )$ , $\psi (t;\epsilon )$ , and $\theta (t;\epsilon )$ be the functions described in condition (F) ${}_m$ and $\mathcal {E}_{m+1}(t)$ be the function determined by (9a). Set $\varphi _1(t;\epsilon ) = \varphi (t+\omega ;\epsilon )$ , so $\varphi _1 = \tau \{ \varphi \}$ . We consider the smooth $2\pi $ -periodic functions $\unicode{x3bb} (t;\epsilon )$ and $\unicode{x3bb} _1(t;\epsilon )$ given by

$$ \begin{align*} \unicode{x3bb} = h \circ \psi \cdot \cos \theta - h' \circ \psi \cdot \sin \theta, \quad \unicode{x3bb}_1 = h \circ \psi \cdot \cos \theta + h' \circ \psi \cdot \sin \theta. \end{align*} $$

Set $c_\epsilon (t) = \big ( \varphi (t;\epsilon ),\unicode{x3bb} (t;\epsilon ) \big )$ and $d_\epsilon (t) = \big ( \varphi _1(t;\epsilon ),\unicode{x3bb} _1(t;\epsilon ) \big )$ . Implicit equations (5) imply that $f \circ c_\epsilon = d_\epsilon $ . In addition, $\unicode{x3bb} _1 - \tau \{ \unicode{x3bb} \} = \sigma \{ h' \circ \psi \cdot \sin \theta \} - \delta \{ h \circ \psi \cdot \cos \theta \} = \epsilon ^{m+1} \tau \{ \mathcal {E}_{m+1} \} + \mathrm {O} (\epsilon ^{m+2})$ , which is equivalent to estimate (9b). This proves condition (P) ${}_m$ .

Set $\bar {\varphi }_j = \tau ^j \{ \varphi \}$ , $\bar {\psi }_j = \tau ^j \{ \psi \}$ and $\bar {\theta }_j = \tau ^j \{ \theta \}$ for all $j \in \mathbb {Z}$ . Note that $\bar {\varphi }_{j+q}(t;\epsilon ) = \bar {\varphi }_j(t;\epsilon ) + 2\pi p$ , $\bar {\psi }_{j+q}(t;\epsilon ) = \bar {\psi }_j(t;\epsilon ) + 2\pi p$ , and $\bar {\theta }_{j+q}(t;\epsilon ) = \bar {\theta }_j(t;\epsilon )$ . If we derive the expression that defines the action and we recall that $\varphi = \varphi _0 + \mathrm {O} (\epsilon )$ , then we obtain the estimate

$$ \begin{align*} \frac{\mathit{d}}{\mathit{d} t}[ A_\epsilon \{ \varphi \} ] &= \sum_{j=0}^{q-1} [ \partial_1 S_\epsilon (\bar{\varphi}_j,\bar{\varphi}_{j+1}) \bar{\varphi}^{\prime}_j + \partial_2 S_\epsilon (\bar{\varphi}_j,\bar{\varphi}_{j+1}) \bar{\varphi}^{\prime}_{j+1} ] \\ &= \sum_{j=0}^{q-1} [ \partial_2 S_\epsilon (\bar{\varphi}_{j-1},\bar{\varphi}_j) + \partial_1 S_\epsilon (\bar{\varphi}_j,\bar{\varphi}_{j+1}) ]\bar{\varphi}^{\prime}_j \\ &= \sum_{j=0}^{q-1} [ \sigma \{ h'\circ \bar{\psi}_{j-1} \cdot \sin \bar{\theta}_{j-1} \} - \delta \{ h\circ \bar{\psi}_{j-1} \cdot \cos \bar{\theta}_{j-1} \} ] \bar{\varphi}^{\prime}_j \\ &= \epsilon^{m+1} \sum_{j=0}^{q-1} \tau^j\{ \mathcal{E}_{m+1} \varphi' \} + \mathrm{O}(\epsilon^{m+2}) \\ &= \epsilon^{m+1} q \mu\{ \mathcal{E}_{m+1} \varphi'\} + \mathrm{O}(\epsilon^{m+2}) \\ &= \epsilon^{m+1} q \mu\{ \mathcal{E}_{m+1} \varphi^{\prime}_0 \} + \mathrm{O}(\epsilon^{m+2}), \end{align*} $$

which, by integration, is equivalent to estimate (9c) for any smooth $2\pi /q$ -periodic function $\mathcal {L}_{m+1}$ such that $\mathcal {L}^{\prime }_{m+1} = \mu \{ \mathcal {E}_{m+1} \varphi ^{\prime }_0 \}$ . This proves condition (A) ${}_m$ , and the relation between $\mathcal {L}_{m+1}$ and $\mathcal {E}_{m+1}$ . The operator $\mu $ is the projection onto the $q\mathbb {Z}$ -harmonics, so the zeroth harmonics of $\mathcal {E}_{m+1} \varphi ^{\prime }_0$ and $\mathcal {L}^{\prime }_{m+1}$ coincide, so $\int _{\mathbb {T}} \mathcal {E}_{m+1} \varphi ^{\prime }_0 = \int _{\mathbb {T}} \mathcal {L}^{\prime }_{m+1} = 0$ .

Second, we check that (P) ${}_m \Rightarrow $ (F) ${}_m$ . Let $c_\epsilon (t) = ( \varphi (t;\epsilon ),\unicode{x3bb} (t;\epsilon ) )$ be the parameterization described in condition (P) ${}_m$ and $\mathcal {E}_{m+1}(t)$ be the error function given in (9b). Property $\varphi '> 0$ holds because $c_\epsilon : \mathbb {T} \to G_\epsilon \subset \mathbb {A}_{\Gamma _\epsilon }$ is a parameterization and $G_\epsilon $ is a graph. Functions $\varphi $ , $\psi = \sigma \{\varphi \}/2$ , and $\theta = \delta \{ \varphi \}/2$ satisfy relations (6). Set $( \varphi _1(t;\epsilon ),\unicode{x3bb} _1(t;\epsilon ) ) = f_\epsilon (c_\epsilon (t))$ . We deduce from implicit equations (5) that

$$ \begin{align*} \sigma \{ h'\circ \psi \cdot \sin \theta \} - \delta \{ h\circ \psi \cdot \cos \theta \} = \unicode{x3bb}_1 - \tau\{ \unicode{x3bb} \} = \epsilon^{m+1} \tau\{ \mathcal{E}_{m+1} \} + \mathrm{O}(\epsilon^{m+2}), \end{align*} $$

which is exactly estimate (9a). This proves condition (F) ${}_m$ .

Remark 3.5. We can prove that $\int _{\mathbb {T}} \mathcal {E}_{m+1} \varphi ^{\prime }_0 = 0$ in another way. The flux of the exact twist map $f_\epsilon $ across the graph $G_\epsilon = c_\epsilon (\mathbb {T})$ , which is a rotational curve, is zero [Reference Meiss38, §V]. This flux is

$$ \begin{align*} \int_{f_\epsilon(G_\epsilon)} \unicode{x3bb} \,\mathit{d} \varphi - \int_{G_\epsilon} \unicode{x3bb} \,\mathit{d} \varphi &= \int_{\mathbb{T}} (\unicode{x3bb}_1 \varphi^{\prime}_1 - \unicode{x3bb} \varphi') \\ &= \int_{\mathbb{T}} \delta \{ \unicode{x3bb} \varphi'\} + \epsilon^{m+1} \int_{\mathbb{T}} \tau\{ \mathcal{E}_{m+1} \varphi' \} + \mathrm{O}(\epsilon^{m+2}) \\ &= \epsilon^{m+1} \int_{\mathbb{T}} \mathcal{E}_{m+1} \varphi^{\prime}_0 + \mathrm{O}(\epsilon^{m+2}) \quad \text{as } \epsilon \to 0. \end{align*} $$

We have used that if $c_\epsilon (t) = ( \varphi (t;\epsilon ),\unicode{x3bb} (t;\epsilon ) )$ and $(\varphi _1,\unicode{x3bb} _1) = f(\varphi ,\unicode{x3bb} )$ , then $\varphi _1 = \tau \{ \varphi \}$ and $\unicode{x3bb} _1 = \tau \{ \unicode{x3bb} \} + \epsilon ^{m+1} \tau \{ \mathcal {E}_{m+1} \} + \mathrm {O} (\epsilon ^{m+2})$ . We have also used that $\varphi = \varphi _0 + \mathrm {O} (\epsilon )$ .

We will check in the next section that if the $p/q$ -resonant caustic $\mathrm {O} (\epsilon ^m)$ -persists under a deformation of the unit circle, then the Melnikov potential $\mathcal {L}_{m+1}(t)$ is uniquely determined from previously computed objects. See the second item in Proposition 4.2.

4 High-order persistence in deformed circles

Let us apply the previous high-order persistence theory to smooth deformations of circles. The main goal is to check that the smooth $2\pi $ -periodic coefficients of the Taylor expansions in powers of the perturbative parameter $\epsilon $ of the perturbed side, normal, and incidence–reflection functions can be computed recursively order by order as long as some compatibility conditions hold. Such compatibility conditions have to do with the inversion of the difference operator $\delta $ , so they boil down to the fact that certain smooth $2\pi $ -periodic functions have no $q\mathbb {Z}$ -resonant harmonics. We look for a practical way to find the exact order at which a given resonant caustic is destroyed, so we write down explicit formulas for all recursive computations.

To begin with, we assume the following hypothesis from here on.

1. (H′) Let $p/q \in (0,1/2)$ be a rational rotational number such that $\gcd (p,q) = 1$ . Let $\Gamma _0$ be the unit circle centered at the origin, see Example 2.5. The unperturbed side, normal, and incidence–reflection functions are

   $$ \begin{align*} \varphi_0(t) = t - \omega/2, \quad \psi_0(t) = t, \quad \theta_0(t) = \omega/2, \quad \omega = 2\pi p/q. \end{align*} $$

   Let $\Gamma _\epsilon $ , with $\epsilon \in [-\epsilon _0,\epsilon _0]$ , be a deformation of $\Gamma _0$ with smooth support function (1).

We look for some perturbed side, normal, and incidence–reflection functions

(10) $$ \begin{align} \varphi(t;\epsilon) \asymp \sum_{k \ge 0} \epsilon^k \varphi_k(t), \quad \psi(t;\epsilon) \asymp \sum_{k \ge 0} \epsilon^k \psi_k(t), \quad \theta(t;\epsilon) \asymp \sum_{k \ge 0} \epsilon^k \theta_k(t) \end{align} $$

that satisfy condition (F) ${}_m$ for an order $m \in \mathbb {N} \cup \{0\}$ as high as possible.

The study of the approximate difference equation (8) requires to recursively compute the asymptotic expansions of $\mathcal {R} := h \circ \psi \cdot \cos \theta $ and $\mathcal {Q} := h' \circ \psi \cdot \sin \theta $ as $\epsilon \to 0$ .

Lemma 4.1. Set $c = \cos (\omega /2)$ and $s = \sin (\omega /2)$ . The coefficients of the asymptotic expansions

$$ \begin{align*} \mathcal{R} = h \circ \psi \cdot \cos \theta \asymp c + \sum_{k \ge 1} \mathcal{R}_k \epsilon^k, \quad \mathcal{Q} = h' \circ \psi \cdot \sin \theta \asymp \sum_{k \ge 1} \mathcal{Q}_k \epsilon^k \end{align*} $$

have the form

$$ \begin{align*} \mathcal{Q}_k &= \mathcal{Q}_k [{\kern-1.8pt}[ h^{\prime}_{\le k},\psi_{<k},\theta_{<k} ]{\kern-1.8pt}] = s h^{\prime}_k + \tilde{\mathcal{Q}}_k [{\kern-1.8pt}[ h^{\prime}_{<k},\psi_{<k},\theta_{<k} ]{\kern-1.8pt}], \\\mathcal{R}_k &= \mathcal{R}_k [{\kern-1.8pt}[ h_{\le k},\psi_{<k},\theta_{\le k} ]{\kern-1.8pt}] = \tilde{\mathcal{R}}_k [{\kern-1.8pt}[ h_{\le k},\psi_{<k},\theta_{<k} ]{\kern-1.8pt}] - s\theta_k. \end{align*} $$

In addition, $\tilde {\mathcal {Q}}_1 = 0$ , $\tilde {\mathcal {R}}_1 = ch_1$ , $\tilde {\mathcal {Q}}_2 = s h^{\prime \prime }_1 \psi _1 + c h^{\prime }_1 \theta _1$ , and $\tilde {\mathcal {R}}_2 = c h_2 + c h^{\prime }_1 \psi _1 - s h_1 \theta _1 - c \theta ^2_1/2$ .

The proof of Lemma 4.1 is postponed to Appendix B. The first coefficients are obtained from the explicit recurrences for $\tilde {\mathcal {Q}}_k$ and $\tilde {\mathcal {R}}_k$ given in Lemma B.4.

Once we know that $\mathcal {Q}_{< k}$ and $\mathcal {R}_{< k}$ only depend on $\psi _{<k}$ and $\theta _{< k}$ , we deduce that if $\varphi _{<k}(t;\epsilon )$ , $\psi _{<k}(t;\epsilon )$ and $\theta _{<k}(t;\epsilon )$ satisfy condition (F) $_{k-1}$ , then $\varphi _{\le k}(t;\epsilon )$ , $\psi _{\le k}(t;\epsilon )$ and $\theta _{\le k}(t;\epsilon )$ satisfy condition (F) ${}_k$ provided coefficients $\varphi _k(t)$ , $\psi _k(t)$ and $\theta _k(t)$ are chosen in such a way that $\sigma \{ \mathcal {Q}_k \} = \delta \{ \mathcal {R}_k \}$ . We prove below that these kth coefficients can be found if and only if $\mathcal {Q}_k$ has no $q\mathbb {Z}$ -resonant harmonics, in which case, all three kth coefficients are uniquely determined provided that they have no $q\mathbb {Z}$ -resonant harmonics either.

Proof. We use several times along this proof that $\mu \circ \tau = \mu $ , $\mu \circ \sigma = 2\mu $ , and $\mu \circ \delta = 0$ on the space of smooth $2\pi $ -periodic functions. We also use that if $b(t)$ is a smooth $2\pi $ -periodic function, $\delta \{ a \} = b$ has a smooth $2\pi $ -periodic solution $a(t)$ if and only if $\mu \{ b \} = 0$ , in which case, the solution is unique under the additional condition $\mu \{ a \} = 0$ . See Lemma A.2.

(a) The asymptotic expansions in Lemma 4.1 imply that

$$ \begin{align*} \sigma \{ h'\circ \psi \cdot \sin \theta \} - \delta \{ h\circ \psi \cdot \cos \theta \} = \sum_{j = 1}^k ( \sigma\{ \mathcal{Q}_j \} - \delta\{ \mathcal{R}_j \} ) \epsilon^j + \mathrm{O}(\epsilon^{k+1}). \end{align*} $$

The $\mathrm {O} (\epsilon ^{k-1})$ -persistence hypothesis means that $\sigma \{ \mathcal {Q}_j \} = \delta \{ \mathcal {R}_j \}$ for $j=1,\ldots , k-1$ . If $\mathcal {E}_k$ and $\mathcal {L}_k$ are the $p/q$ -resonant error and potential of order k, respectively, then $\tau \{ \mathcal {E}_k \} = \sigma \{ \mathcal {Q}_k \} - \delta \{ \mathcal {R}_k \}$ and $\mathcal {L}^{\prime }_k = \mu \{ \mathcal {E}_k \} = 2\mu \{ \mathcal {Q}_k \}$ , since $\varphi ^{\prime }_0(t) = 1$ . Finally, $\bar {\mathcal {Q}}_k = ({1}/{2\pi }) \int _{\mathbb {T}} \mathcal {Q}_k = ({1}/{4\pi }) \int _{\mathbb {T}} \mathcal {E}_k = 0$ . (See the last claims in Proposition 3.4.)

(b) It is a direct consequence of properties $\mathcal {L}^{\prime }_k = 2\mu \{ \mathcal {Q}_k \}$ and $\mathcal {Q}_k = \mathcal {Q}_k [{\kern-1.8pt}[ h^{\prime }_{\le k},\psi _{<k},\theta _{<k} ]{\kern-1.8pt}]$ .

(c) Condition (11) is necessary: $\mathrm {O} (\epsilon ^k)$ -persistence means $\sigma \{ \mathcal {Q}_k \} = \delta \{ \mathcal {R}_k \}$ , so ${\mu \{ \mathcal {Q}_k \} = 0}$ . Condition (11) is sufficient: if $\mu \{ \mathcal {Q}_k \} = 0$ , then

$$ \begin{align*} \mu \{ \delta\{ \tilde{\mathcal{R}}_k \} - \sigma\{ \mathcal{Q}_k \} \} = -2 \mu \{ \mathcal{Q}_k \} = 0, \end{align*} $$

so (12) has a unique smooth $2\pi $ -periodic solution $\theta _k(t)$ . Then, we find $\varphi _k$ and $\psi _k$ by means of relations (13). Finally, the $\mathrm {O} (\epsilon ^k)$ -terms of $\sigma \{ h'\circ \psi \cdot \sin \theta \}$ and $\delta \{ h\circ \psi \cdot \cos \theta \}$ are $\sigma \{ \mathcal {Q}_k \}$ and $\delta \{ \tilde {\mathcal {R}}_k - s \theta _k \}$ , respectively. See Lemma 4.1. Both terms coincide when the first relation in (12) holds. Property $\varphi '(t;\epsilon ) = 1 + \mathrm {O} (\epsilon )> 0$ holds for $\epsilon \in [-\epsilon _0,\epsilon _0]$ if $0 < \epsilon _0 \ll 1$ . Relations (6) follow from condition $\mu \{ \theta _k \} = 0$ and relations (13), since $\psi _k = \varphi _k + \theta _k$ implies that $\mu \{ \psi _k \} = 0$ and $\psi _k = \sigma \{ \varphi _k \}/2$ . This proves that condition (F) ${}_k$ holds.

(d) We deduce from Lemma 4.1 that if $\mathcal {Q}^\star _k$ is the $\mathrm {O} (\epsilon ^k)$ -coefficient of $(h^\star )' \circ \psi _{<k} \cdot \sin \theta _{<k}$ , then $\mathcal {Q}^\star _k = s \eta ^{\prime }_k + \mathcal {Q}_k$ . The existence of the correction $\eta _k$ follows from property $\bar {\mathcal {Q}}_k = 0$ .

Finally, we prove Theorem 1.1 by recursively applying Proposition 4.2 for ${k=1},\ldots ,m$ .

Proof of Theorem 1.1

We denote the Fourier coefficients of smooth $2\pi $ -periodic functions with a hat, so we write $h_1(t) = \sum _{l \in \mathbb {Z}} \hat {h}_{1,l} \mathit {e}^{\mskip 2mu {i}\mskip 1mu l t}$ and $\theta _1(t) = \sum _{l \in \mathbb {Z}} \hat {\theta }_{1,l} \mathit {e}^{\mskip 2mu{i}\mskip 1mu l t}$ .

(a) We know that $\mathcal {Q}_1 = s h^{\prime }_1$ from Lemma 4.1. Hence, the necessary and sufficient condition (11) for $\mathrm {O} (\epsilon )$ -persistence becomes $\mu \{ h^{\prime }_1 \} = 0$ or, equivalently, $\mu _{q\mathbb {Z}^\star } \{ h_1 \} = 0$ .

(b) We know that $\mathcal {Q}_2 = s h^{\prime }_2 + \tilde {\mathcal {Q}}_2$ and $\tilde {\mathcal {Q}}_2 = s h^{\prime \prime }_1 \psi _1 + c h^{\prime }_1 \theta _1$ from Lemma 4.1. Let us check that $\mu \{ \tilde {\mathcal {Q}}_2 \} = s \mu \{ \theta _1 \theta ^{\prime }_1 \}$ . To do that, we need three properties.

First, once we know that the resonant caustic $\mathrm {O} (\epsilon )$ -persists, we determine the first-order coefficient $\theta _1(t)$ by solving (12) with $k=1$ . That is,

(15) $$ \begin{align} s \delta\{ \theta_1 \} = \delta\{ \tilde{\mathcal{R}}_1 \} - \sigma\{ \mathcal{Q}_1 \} = \delta\{ ch_1 \} - \sigma\{ sh^{\prime}_1 \}, \quad \mu\{ \theta_1 \} = 0. \end{align} $$

Second, we determine $\varphi _1(t)$ and $\psi _1(t)$ by solving (13) with $k=1$ . In particular,

$$ \begin{align*} \theta_1 = \delta\{ \varphi_1 \}/2, \quad \psi_1 = \sigma\{ \varphi_1 \}/2. \end{align*} $$

Third, we recall that if $a(t)$ and $b(t)$ are $2\pi $ -periodic functions, then

$$ \begin{align*} \mu \{ a \delta\{b\} \} = -\mu \{ \delta\{ a \} \tau \{ b \} \}, \quad \mu \{ a \sigma\{ b \} \} = \mu \{ \sigma\{ a \} \tau\{ b \} \}. \end{align*} $$

Next, we use these three properties to get the formula we are looking for:

$$ \begin{align*} \mu\{ \tilde{\mathcal{Q}}_2 \} &= \mu\{ s h^{\prime\prime}_1 \psi_1 + c h^{\prime}_1 \theta_1 \} = \mu\{ s h^{\prime\prime}_1 \sigma\{ \varphi_1\} + c h^{\prime}_1 \delta\{ \varphi_1 \} \}/2 \\ &= \mu\{ [\sigma\{ s h^{\prime\prime}_1\} - \delta\{ c h^{\prime}_1\} ] \tau\{ \varphi_1 \} \}/2 = \mu\{ [\sigma\{ s h^{\prime}_1\} - \delta\{ c h_1\} ]' \tau\{ \varphi_1 \} \}/2 \\ &= -s \mu\{ \delta\{ \theta^{\prime}_1\} \tau\{\varphi_1 \} \}/2 = s \mu\{ \delta\{ \varphi_1\} \theta^{\prime}_1 \}/2 = s \mu\{ \theta_1 \theta^{\prime}_1 \}. \end{align*} $$

Thus, condition (11) for $\mathrm {O} (\epsilon ^2)$ -persistence becomes $\mu \{ h^{\prime }_2 + \theta _1 \theta ^{\prime }_1 \} = 0$ or, equivalently,

$$ \begin{align*} \mu_{q \mathbb{Z}^\star} \{ h_2 + \theta_1^2/2 \} = 0. \end{align*} $$

Set $e^\pm _l = \mathit {e}^{\mskip 2mu {i}\mskip 1mu l \omega } \pm 1$ . The Fourier coefficients of the unique smooth $2\pi $ -periodic solution of (15) are easily determined: $\hat {\theta }_{1,l} = 0$ for all $l \in q\mathbb {Z}$ and

$$ \begin{align*} s e^-_{l} \hat{\theta}_{1,l} = ( c e^-_l - \mskip2mu{i}\mskip1mu l s e^+_l ) \hat{h}_{1,l} \quad \text{for all } l \not \in q\mathbb{Z}. \end{align*} $$

The right-hand side of this identity vanishes, and $\hat {\theta }_{1,l}$ too, when $l = \pm 1$ . If $l \not \in q\mathbb {Z} \cup \{-1,1\}$ , then

$$ \begin{align*} \hat{\theta}_{1,l} = \bigg( \frac{c}{s} - \mskip2mu{i}\mskip1mu l \frac{e^+_l}{e^-_l} \bigg) \hat{h}_{1,l} = \begin{cases} \dfrac{\tan(l\omega/2) - l \tan(\omega/2)} {\tan(\omega/2) \tan(l\omega/2)} \hat{h}_{1,l} & \text{if }2l \not \in q\mathbb{Z}, \\[6pt] \dfrac{\hat{h}_{1,l}}{\tan(\omega/2)} & \text{if }2l \in q\mathbb{Z}. \end{cases} \end{align*} $$

We have used that $e^+_l = 0$ when $2l \in q\mathbb {Z}$ and $l \not \in q\mathbb {Z}$ , whereas $e^-_l \neq 0$ for all $l \not \in q\mathbb {Z}$ . The formulas above are equivalent to the expression of $\theta _1(t)$ given in Theorem 1.1.

(c) We know from Lemma 4.1 that $\mathcal {Q}_m = s h^{\prime }_m + \tilde {\mathcal {Q}}_m$ , where $\tilde {\mathcal {Q}}_m$ is a smooth $2\pi $ -periodic function, only depending on $h_1,\ldots ,h_{m-1}$ —since approximations $\varphi _{<k}$ , $\psi _{<k}$ and $\theta _{<k}$ are uniquely determined from $h_{<k}$ for any $k = 1, \ldots , m-1$ —that can be explicitly computed from recurrences given in Appendix B. We also know that $\int _{\mathbb {T}} \mathcal {Q}_m = 0$ and so $\int _{\mathbb {T}} \tilde {\mathcal {Q}}_m = 0$ . This means that there is a smooth $2\pi $ -periodic function $\zeta _m$ such that $s \zeta ^{\prime }_m = \tilde {\mathcal {Q}}_m$ . Finally, the necessary and sufficient condition for $\mathrm {O} (\epsilon ^m)$ -persistence becomes $\mu \{ h^{\prime }_m + \zeta ^{\prime }_m \} = 0$ or, equivalently, $\mu _{q\mathbb {Z}^\star } \{ h_m + \zeta _m \} = 0$ .

Functions $\zeta _m$ can be recursively computed from the formulas listed in Appendix B. For instance, we have already seen that $\zeta _1 = 0$ and $\zeta _2 = \theta ^2_1/2$ along the proof of Theorem 1.1. After some tedious computations by hand that we do not include here, we get that

$$ \begin{align*} \zeta_3 = \theta_1 \theta_2 + h_1 \theta_1^2/2 - h^{\prime\prime}_1 \psi_1^2/2 + c\theta_1^3/3s - ch^{\prime}_1 \theta_1 \psi_1/s, \end{align*} $$

which will allow us to analyze some $\mathrm {O} (\epsilon ^3)$ -persistence problems in the future. See §6. We stress that further expressions for $\zeta _4,\zeta _5,\ldots $ can be obtained using a symbolic algebra system, but we have doubts about their practical usefulness.

5 Polynomial deformations of the unit circle

We tackle two problems in this section, both related with polynomial deformations of the unit circle. First, to prove Theorem 1.3. Second, to prove that the support function (1) of any polynomial deformation (4) with $P(x,y;\epsilon ) = 1 + \epsilon P_1(x,y)$ and $P_1(x,y) \in \mathbb {R}_n[x,y]$ satisfies condition (3); see Lemma 5.2. This second result allows us to explain with Theorem 1.3 some of the numerical experiments about the polynomial deformations (4) with $P(x,y;\epsilon ) = 1 - \epsilon y^n$ performed in [Reference Martín, Tamarit-Sariol and Ramírez-Ros37], which was the original motivation of this work.

To begin with, we check that, once the order $m \in \mathbb {N}$ and the degree $n \ge 3$ are fixed, all resonant caustics of high enough period $\mathrm {O} (\epsilon ^m)$ -persists under polynomial deformations of degree $\le n$ . We explicitly quantify how high this period q should be. Some resonant caustics become more persistent in the presence of symmetries.

We have postponed the proofs of Theorem 5.1 and Lemma 5.2 to Appendix C, since they are technically similar. The main difficulty is to check that some recursively computed $2\pi $ -periodic trigonometric polynomials have degrees $\le mn$ by induction on a recursive index $m \ge 1$ .

Theorem 1.3 is a direct consequence of Theorem 5.1.

The experiments described in [Reference Martín, Tamarit-Sariol and Ramírez-Ros37, Numerical Result 5] suggest that the $p/q$ -resonant caustic does not $\mathrm {O} (\epsilon ^\chi )$ -persist under the polynomial deformation (4) with $P(x,y;\epsilon ) = 1 - \epsilon y^n$ . To prove it, we should check that $h_\chi + \zeta _\chi $ has some non-zero $(q\mathbb {Z} \setminus \{0\})$ -harmonic.

6 Open problems

We describe three open problems that have arisen during the development of our high-order perturbation theory. We have not addressed them here. Each of them is a non-trivial research challenge. They are works in progress.

As a general principle, we claim that many billiard computations are greatly simplified when working with the Bialy–Mironov generating function, so its discovery opens the door to the resolution of many billiard problems that seemed almost intractable.

We also stress that both our high-order perturbation theory and our list of open problems can be extended to the setting of dual, symplectic, wire, pensive, and coin billiards, provided we deal with those billiards in deformed circles.

6.1 Co-preservation of resonant caustics

Tabachnikov asked whether there are convex domains, other than circles and ellipses, that possess resonant caustics with different rotation numbers. See [Reference Bolsinov, Matveev, Miranda and Tabachnikov13, Question 4.7]. Theorem 1.1 and Cyr’s result on Gutkin’s equation provide an excellent framework to study the co-preservation of resonant caustics under deformations of the unit circle. We mention two problems about such co-preservation.

First, we consider the co-preservation of caustics with different rotation numbers but equal periods. Let $q \neq 2, 3, 4, 6$ be a fixed period, so

$$ \begin{align*} \mathcal{P}_q = \{ p \in \mathbb{N} : p/q < 1/2, \ \gcd(p,q) = 1 \} \end{align*} $$

has several elements. If the deformation of the unit circle with support function (1) preserves all $p/q$ -resonant caustics with $p \in \mathcal {P}_q$ , then none of the smooth $2\pi $ -periodic functions $h_1$ and $h_2 + \big (\theta _1^{[p/q]}\big )^2/2$ with $p \in \mathcal {P}_q$ have resonant $(q\mathbb {Z} \setminus \{ 0 \})$ -harmonics, where

$$ \begin{align*} \theta_1^{[p/q]}(t) = \sum_{l \not \in q\mathbb{Z} \cup \{-1,1\}} \hat{\theta}_{1,l}^{[p/q]} \mathit{e}^{\mskip2mu{i}\mskip1mu l t}, \quad \hat{\theta}_{1,l}^{[p/q]} = \nu_l(p/q) \hat{h}_{1,l} \end{align*} $$

for some functions $\nu _l:(0,1/2) \to \mathbb {R}$ defined in (2), which have no rational roots when $|l| \ge 2$ . The above necessary conditions for $\mathrm {O} (\epsilon ^2)$ -persistence impose rather stringent restrictions on the first-order coefficient $h_1$ . We do not state any specific result because we have not yet found one satisfactory enough.

Second, we consider the co-preservation of caustics with rotation numbers $1/2$ and $1/q$ for some fixed period $q \ge 3$ under deformations of the unit circle. The preservation of the $1/2$ -resonant caustic implies that all $(2\mathbb {Z} \setminus \{0\})$ -harmonics of the support function (1) are zero, which greatly simplifies the problem. The case $q=3$ for centrally symmetric deformations was studied by J. Zhang [Reference Zhang44]. His method consists in a careful analysis of the obstructions for $\mathrm {O} (\epsilon ^2)$ -persistence of the $1/3$ -resonant caustic. His computations are more involved than those outlined here because he uses the standard generating function for billiards (that is, the minus distance between consecutive impacts) and he considers deformations of the unit circle written in polar coordinates. We plan to extend Zhang’s results to some periods $q>3$ , starting with periods $q=4,5$ for which it is necessary to study the obstructions for $\mathrm {O} (\epsilon ^3)$ -persistence.

6.2 A convergence problem

We recall that the function $\zeta _m$ in Theorem 1.1 only depends on $h_1,\ldots ,h_{m-1}$ . Hence, if the $p/q$ -resonant caustic $\mathrm {O} (\epsilon ^m)$ -persists, but not $\mathrm {O} (\epsilon ^{m+1})$ -persists, and we fix any higher order $r> m$ , then there is a smooth $2\pi $ -periodic function $\eta ^{[r]}_\epsilon = \mathrm {O} (\epsilon ^{m+1})$ such that the resonant caustic $\mathrm {O} (\epsilon ^r)$ -persists under the new deformation $\Gamma ^{[r]}_\epsilon = \Gamma _\epsilon + \mathrm {O} (\epsilon ^{m+1})$ with support function $h^{[r]}_\epsilon = h_\epsilon + \eta ^{[r]}_\epsilon $ . This function $\eta ^{[r]}_\epsilon $ can also be recursively computed and only contains $(q\mathbb {Z} \setminus \{0\})$ -harmonics. See the last item in Proposition 4.2.

A natural question is the following. Does $h^{[\infty ]}_\epsilon := \lim _{r \to +\infty } h^{[r]}_\epsilon $ exist? That is, we look for a corrected support function $h^{[\infty ]}_\epsilon = h_\epsilon + \mathrm {O} (\epsilon ^{m+1})$ such that the $p/q$ -resonant caustic persists at all orders under the deformation $\Gamma ^{[\infty ]}_\epsilon $ with support function $h^{[\infty ]}_\epsilon $ . This problem is closely related to the density property established in [Reference Kaloshin and Zhang31].

The first author, in collaboration with V. Kaloshin and K. Zhang, is working on a more ambitious version of this problem. Namely, once a rational rotation number $1/q$ is fixed, the goal is to construct a functional

$$ \begin{align*} \mathcal{H}_{\mathbb{Z} \setminus q\mathbb{Z}} \ni h \stackrel{\mathcal{F}_{1/q}}{\mapsto} \eta \in \mathcal{H}_{q\mathbb{Z} \setminus\{0\}} \end{align*} $$

such that: (1) $\mathcal {H}_A$ is a neighborhood of zero in a suitable space of $2\pi $ -periodic functions with only A-harmonics; (2) $\mathcal {F}_{1/q}(0) = 0$ ; (3) $\mathcal {F}_{1/q}$ is as regular as possible at $h = 0$ ; and (4) the convex domain with support function $1 + h + \mathcal {F}_{1/q}[h]$ has a $1/q$ -resonant caustic for any $h \in \mathcal {H}_{\mathbb {Z} \setminus q\mathbb {Z}}$ .

6.3 Exponentially small phenomena

Let $\Gamma _\epsilon $ be a polynomial deformation of the unit circle of degree $\le n$ in the sense of Definition 1.2. Let $q \ge 3$ be a fixed period. Theorem 1.3 implies that the $p/q$ -resonant caustics $\mathrm {O} (\epsilon ^{\chi - 1})$ -persist under $\Gamma _\epsilon $ for all $p \in \mathcal {P}_q$ , so there are perturbed normal functions $\psi ^{[p/q]}_\epsilon (t)$ and perturbed incidence–reflection functions $\theta ^{[p/q]}_\epsilon (t)$ such that the error

$$ \begin{align*} \mathcal{E}^{[p/q]}_\epsilon(t) := \sigma \{ h^{\prime}_\epsilon(\psi^{[p/q]}_\epsilon(t)) \cdot \sin \theta^{[p/q]}_\epsilon(t) \} - \delta \{ h_\epsilon(\psi^{[p/q]}_\epsilon(t)) \cdot \cos \theta^{[p/q]}_\epsilon(t) \} \end{align*} $$

is $\mathrm {O} (\epsilon ^\chi )$ as $\epsilon \to 0$ . We recall that $\chi \asymp 2q/n$ as odd $q \to +\infty $ for centrally symmetric deformations, and $\chi \asymp q/n$ as $q \to +\infty $ otherwise. Let us focus on the second case, which is the generic one. It is natural to ask whether, in this generic case,

$$ \begin{align*} \mathcal{E}^{[p/q]}_\epsilon(t) = \mathrm{O}( \epsilon^\chi ) \simeq \mathrm{O}( \epsilon^{q/n} ) = \mathrm{O}( \mathit{e}^{-q |\!\log \epsilon|/n} ) \end{align*} $$

as $q \to +\infty $ for fixed $\epsilon \in [-\epsilon _0,\epsilon _0]$ . This is a hard problem. It was partially addressed in [Reference Martín, Tamarit-Sariol and Ramírez-Ros36], where the authors establish an exponentially small upper bound on the difference of lengths of $1/q$ -periodic billiard trajectories in analytic strictly convex domains as $q \to +\infty $ . The numerical experiments described in [Reference Martín, Tamarit-Sariol and Ramírez-Ros37] suggest that these upper bounds can be improved to exponentially small asymptotic formulas for some deformations. To be precise, we deduce from the computations about the polynomial deformations (4) with $P(x,y;\epsilon ) = 1 - \epsilon y^n$ that there are constants $\xi (\Gamma _\epsilon )$ with a finite limit as $\epsilon \to 0$ such that the error $\mathcal {E}^{[1/q]}_\epsilon (t)$ has ‘size’ $q^{-3} \mathit {e}^{-q [|\!\log \epsilon |/n)+ \xi (\Gamma _\epsilon )]}$ as $q \to +\infty $ for any fixed $\epsilon \in [-\epsilon _0,\epsilon _0]$ .

This problem can be addressed by the direct approach used by Wang in [Reference Wang43] or the approach used by Kaloshin and Zhang in [Reference Kaloshin and Zhang31]. However, we feel that exponentially small asymptotic formulas can only be obtained with more refined techniques, like the extension of $\psi ^{[p/q]}_\epsilon (t)$ and $\theta ^{[p/q]}_\epsilon (t)$ to complex values of t, the analysis of their complex singularities, resurgence theory, and complex matching. See [Reference Delshams and Ramírez-Ros18, Reference Gelfreich21, Reference Martín, Sauzin and Seara35] for examples of how these refined techniques are applied in the setting of discrete systems (maps).