Continuous Phase Foliations Create Analytic Caustic Collars

Proof. The pair plane and pair strip are simply connected, so the nonvanishing vector has a continuous real argument. For a single leaf normalize it to \pi/4 on \beta=\alpha+2\pi. For distinct leaves, both entries equal \beta-\alpha+O(1), uniformly in the pair variables, so normalize the argument to (0,\pi/2) on the connected region where \beta-\alpha is sufficiently large. Either normalization implies simultaneous periodicity: the difference after a simultaneous translation is a continuous integer multiple of 2\pi and vanishes in the normalizing region.

Consecutive vectors have the form (u,v),(v,w). A positive quarter-turn relation (v,w)=c(-v,u), c>0, would force v=0 and u,w to have the same sign, contrary to (2.11). Thus A_1-A_0 avoids \pi/2 modulo 2\pi. Connectedness, together with the edge normalization or the large-separation normalization, puts it in (-3\pi/2,\pi/2). Applying this to the iterated pair proves (2.12).

If A_j\le a_m\le A_{j+1}, that bound gives

On these two intervals \sin A_j is nonzero with sign (-1)^m, whereas \cos A_{j+1} is either zero or has the opposite sign. This is impossible: both have the sign, and the same zero condition, of x_{j+1}. This proves (2.13).

Finally, if x_i\ne0, then A_i lies strictly between two consecutive barriers. All subsequent arguments stay below the upper one. If x_k has the opposite sign, A_k cannot remain in the interval between those barriers, so lies strictly below its lower endpoint. All later arguments remain below that endpoint, a positive distance from A_i.□

Proof. Periodic leaves are graphs. Suppose first that G_b^q=\mathrm{id}+2\pi l on a whole leaf. For its pair strip, simultaneous periodicity gives A_q=A_0. If x_0=0, then A_0 is a barrier, and (2.13) makes A_q<A_0, a contradiction. Thus x_0 never vanishes on the connected strip. It equals 2\pi on its upper edge, so is everywhere positive. This says precisely that \theta_b is strictly increasing.

Constant-translation intervals also consist of graphs. Let b be interior to an interval on which \tau is constant. If \theta_b were not strictly increasing, its degree-one property and continuity would give an ordered same-leaf pair with x_0=0. By (2.11), x_{-1} and x_1 have opposite strict signs. Start instead with that preceding pair and perturb it slightly. We obtain a nonempty open set O of pairs on distinct leaves for which x_0x_2<0. We can take the two leaf parameters in disjoint relatively compact intervals I_1,I_2 inside the constant-translation interval, and take |\beta-\alpha| bounded on O.

Consider the cover of H(\T\times I_1)\times H(\T\times I_2) obtained from (\R\times I_1)\times(\R\times I_2) by identifying only

The pair dynamics K induced by (G_{b_1},G_{b_2}) preserves the measure lifted from product billiard area. This measure is defined by integrating counting measure on the covering fibers against product area; K bijects the fibers over the area-preserving base dynamics. It has full support, and any region of bounded |\beta-\alpha| has finite measure, since it meets uniformly finitely many sheets over each base point. This construction requires no differentiability or absolute continuity of H.

Equality of the two translation numbers and (2.7) imply, for every j\in\mathbb Z,

Consequently the saturation \bigcup_{j\in\mathbb Z}K^j(O) is an invariant finite-measure set. Poincaré recurrence gives recurrent points in O, indeed almost every point there is recurrent. One may obtain recurrence to a point from the usual measurable-set version by using a countable neighborhood base; the latter version follows because the forward images of a set of nonreturners are disjoint in finite measure.

For any such recurrent pair, the two leaf parameters remain fixed. Thus continuity of the argument for this fixed pair of leaves and simultaneous periodicity give A_{j_k}\to A_0 along a sequence j_k\to\infty. This contradicts Lemma 2.1, because x_0x_2<0. Hence \theta_b is strictly increasing throughout every constant-translation interval.

Graph leaves are dense, and then every leaf is a graph. Take any open parameter interval J. If it contains a subinterval where \tau is constant, the preceding argument supplies graph leaves. Otherwise choose a rational multiple r_*=2\pi l/q strictly between two values of the continuous function \tau on J. Travel along the parameter segment from the smaller value to the larger one, and let c be the first position approached by values \tau>r_*. Continuity gives \tau(c)=r_*. Values strictly below r_* also approach c: otherwise \tau=r_* on an interval immediately before c, contrary to our case assumption. Taking limits in (2.8) from these two strict sides gives G_c^q=\mathrm{id}+2\pi l. The periodic-leaf argument therefore supplies a graph in J. Notice that no monotonicity of \tau has been used.

Graph parameters are thus dense. Continuity of the lifted foliation makes every \theta_b nondecreasing. If a same-leaf ordered pair had x_0=0, its preceding and following pairs would still be ordered, because G_b is increasing. Nondecrease would give x_{-1},x_1\ge0, contradicting (2.11). Every \theta_b is therefore strictly increasing, proving the graph property.

Ordering and the grazing end. Inverting the increasing angular parametrizations shows that g_b is jointly continuous. Distinct graphs are disjoint, so their order cannot change with \theta. For each fixed angle, the continuous injective function b\mapsto g_b(\theta) is strictly monotone, in the same direction at every angle. Reverse b to make this direction decreasing. The upper-strip inclusion (2.6) then forces g_b(\theta)\to h(\theta) as b\downarrow0 at every angle. Monotonicity and compactness upgrade this to uniform convergence. Formula (2.2) now shows that F_b(\theta)-\theta is positive and tends uniformly to zero at that end. Its positive minimum on each fixed graph gives \tau(b)>0, and \tau(b)\to0 follows.

The forward twist and decreasing graph order give F_{b_1}<F_{b_2} when b_1<b_2. Since the maps are increasing, all their positive iterates have the same strict order. Passing to translation numbers shows that \tau(b) is nondecreasing.

Rational plateaus are impossible. If \tau(b)=2\pi l/q on an open parameter interval I, almost every point in its finite-area invariant annulus is recurrent for T^q. On each leaf, F_b^q-2\pi l has zero translation number and hence has fixed points: without a fixed point its periodic displacement would have a uniform strict sign. Every nonfixed point moves monotonically in a component of the complement of the fixed set towards an endpoint, so is not recurrent on the circle. Thus T^q is the identity almost everywhere in this annulus, and therefore everywhere, by continuity and full support of area. Its lift on each graph is necessarily F_b^q=\mathrm{id}+2\pi l, with the integer shift determined by the translation number. This contradicts strict ordering of the iterates on two distinct graphs.

The continuity, monotonicity, positivity, and zero limiting value of \tau now show that every sufficiently small positive rational l/q occurs. Its parameter is unique, since two parameters with that value would give a plateau. Values on either side are strictly below and above 2\pi l/q; taking limits in (2.8) proves (2.14) on the entire graph.□

Proof. Choose \kappa<2\pi small enough that (2.2) and (2.6) place every line with angular step in (0,\kappa) in U. Decrease it further so that Proposition 2.2 applies to every positive rational smaller than \kappa/(2\pi). The generating relation and (2.16) give every stated transition; for the last transition use precisely the definition of p_q in (2.17). No endpoint stationarity has been assumed.

The initial state belongs to an individually invariant graph in U, so every state of the chain belongs to that same graph. The equality \theta_q=\theta_0+2\pi l therefore forces p_q=p_0. Repeating the orbit now shows that its translation number is 2\pi l/q. Since 0<l/q<\kappa/(2\pi), uniqueness of the rational graph applies. In particular, all chains with the same q,l lie on the same graph, regardless of their starting angles.

Finally, differentiating the action along a C^1 family cancels all interior terms by (2.16). The remaining terms are -p_0\dot\theta_0+p_q\dot\theta_q=0, because the endpoint advance is fixed and the momenta agree.□

Proof. Apply Proposition 3.2 on the real axis. Its steps are 2\pi/q+O(q^{-3}), hence positive and uniformly small. Applying \phi gives a stationary chain for \cS, with positive uniformly small angular increments and total advance 2\pi. Proposition 2.2 and Lemma 2.3 show that its final momentum equals its initial momentum. Thus the chain is stationary also at the identified endpoints. More explicitly, differentiation of its critical action cancels every interior term, and the two endpoint terms are

Here \phi'(x+2\pi)=\phi'(x), and periodicity of h cancels the two endpoint contributions from the subtracted integral. The action is constant in x, so its nonzero Fourier coefficients vanish.□

Proof. Contour displacement and m integrations by parts in (3.7) give

For positive \sigma, displace first to an interior closed strip and then take its width to the indicated limit; for \sigma=0, only integration by parts is needed. Coefficient differences satisfy the same bound multiplied by \|f-\widehat f\|_{r_0,\sigma}.

Increase N so that N^2\ge2K, and set \alpha=K/n^2\le1/2. The triangle inequality yields the useful identity and bound

Consequently the weighted matrix entry from k to n of R(f) is at most

At m=2 its column sums are O(N^{-1}), proving the stated operator bounds on weighted \ell^1. A Neumann series gives the inverse. The equality C_{-q}(f)=\overline{C_q(f)} for real inputs gives the real symmetry. Finally,

reduces the polynomially weighted estimate to the convergent sum of (1+|n-k|)^{a-m}.□

Proof of Theorem 3.1. Fix m=2, r_0=17, and

Choose N large enough for Lemmas 3.4 and 3.5, with \|R\|\le1/2 throughout this input ball. These choices precede all choices of approximation parameters.

Let f_0=\sum_{|k|\le P}[u]_ke^{\ii kx}, where P>N, and put \sigma_0=P^{-1}. For every fixed a\ge0,

We claim that, for every fixed positive integer a,

For N\le|n|\le P, the exponential weights are at most e. Integrate the real derivative along the segment from u to f_0, use C(u)=0, and apply Lemma 3.5 at width zero. This part is bounded by C|f_0-u|_0, which decays faster than any fixed power of P because u is smooth.

For |n|>P, instead integrate along \lambda f_0, 0\le\lambda\le1. The zero input has a uniform stationary mesh and a constant critical action, so C(0)=0. For the chosen a, use Proposition 3.3 with one fixed integer m>a+1. Formula (3.11) bounds all its required input derivatives uniformly in P. Once P exceeds this higher-order row threshold, real mesh uniqueness identifies these rows with the original ones. The final assertion of Lemma 3.5 gives

Division by (1+P)^a proves the tail bound. This establishes (3.12); its constants and threshold may depend on a, and no uniformity over all smoothness orders is asserted.

Keep the modes |k|<N fixed and set

At the j-th step define

The correction is real symmetric, has only high modes, and obeys |\eta_j|_{\sigma_j}\le2\eps_j. Since \sigma_j-\sigma_{j+1}=2^{-j-2}/P, the elementary Fourier smoothing estimate gives

If the correcting segment is in the fixed input ball on this smaller strip, integration of the derivative difference gives

The linear term cancels by (3.13): compatibility of the real row derivatives makes the matrix computed at width \sigma_j the same matrix on the smaller strip. Equivalently, one can integrate each smooth real row first and then take the smaller-strip norm, avoiding any derivative assertion for rough variations at the original strip boundary.

Choose P so large that \eps_0\le P^{-2r_0}, possible by (3.12), and enlarge it further as needed below. The simultaneous induction is

Indeed, before using (3.15), the known bound for \eps_j and (3.14) put the entire current segment in the ball: the initial norm is less than M/4, and the cumulative corrections are at most CP^{-r_0}\sum_{i\ge0}2^{-i}<M/4. Thus the quadratic estimate is applicable. Its ratio to the proposed bound for \eps_{j+1} is at most CP^{-r_0}2^{r_0+1-j}, which is less than one for one sufficiently large fixed P. This proves both the error bound and the input bound at every step.

The summable corrections converge on the positive-width strip \mathcal A_{\sigma_\infty}, with all derivatives through r_0, to a real-symmetric holomorphic function f_*. Its low Fourier modes agree with those of u, and continuity of each real mesh gives C_n(f_*)=0 for every |n|\ge N. The real segment between u and f_* lies in the same input ball. Put v=f_*-u=\Pi_Hv. Integrating the real derivative,

The bound \|R\|\le1/2 implies |v|_0\le|v|_0/2, hence v=0. The original smooth u is therefore analytic.

Finally \phi'=e^u>0, so \phi and its inverse are analytic. Equation (3.1) makes r analytic in the normal angle, and the equation h''+h=r makes h analytic. Thus h n_\theta+h'e_\theta is an analytic regular boundary parametrization, proving Theorem 3.1.□

Proof. The midpoint implicit function theorem, positivity of \phi' on the real axis, and compactness modulo the period give a uniform holomorphic domain for (3.2). Reversing its two endpoints gives antisymmetry. The calculation of (3.3) applies on this holomorphic domain. Differentiating at fixed second endpoint is the operator \partial_x-\partial_b, which gives the second expansion.□

Proof. Antisymmetry gives \mathcal R=D_1(V,T_tV)-D_1(V,T_{-t}V). Put

By (4.2),

The remaining divided difference has the nonsingular expression

These formulas prove regularity and the first identity in (4.5). Since D_{12}(x,x+b)=b/4+O(b^3), the second follows as well. They also show that the first two variations of E, on domains with room for the shifts, have bounds in terms of a fixed number of derivatives through order two in y. No negative power of t is needed in these bounds. Antisymmetry gives the stated parity.

For real t, the mean action

is independent of \eta. Differentiating at zero and shifting one integral gives \langle V'\mathcal R\rangle=0. The mean-zero identity for complex t follows by holomorphic continuation.

For (4.6), separate each off-diagonal variation into its value at y and its difference from that value. With

the difference terms are

since T_{\pm t/2}a=c_\pm/t. The common-value terms, including the variation of V', are (Ew)'.

At an exact solution, differentiating the factor t^{-3} contributes zero. Differentiation of the explicit shifts contributes (c_+-c_-)/t_0^3=\partial a/t_0, whereas the assigned jet contributes \partial(a\partial z). This proves (4.7).□

Proof. We begin with t_0>0. Write U_j(y)=V(y+jt_0,t_0) and seek interior-stationary nodes

The endpoint increment is 2\pi l. Multiply the interior stationarity equation at node j by the fixed factor t_0^{-3}U'_j. At \xi=0 the residual is E(y+jt_0,t_0). By (4.6), its derivative in \xi is the Dirichlet flux matrix

plus the diagonal potential

The prefactor in the finite-dimensional equations is frozen; subtracting its variation explains the second term in this potential.

The inverse of A_0 has a uniform complex bound. For A_0\xi=f, let J_{j+1/2}=a_{j+1/2}(\xi_{j+1}-\xi_j). Then

The condition \sum_{j=0}^{q-1}(\xi_{j+1}-\xi_j)=0 determines J_{1/2}. Since each a_{j+1/2}^{-1} is close to four, the modulus of their sum is at least a fixed positive multiple of q. Summing the resulting increments gives

On Y_1, the potential is bounded by CQ\eps. Thus a Neumann series gives the same bound for the full inverse if CQ^3\eps is small.

Reducedness supplies the useful lower bound

The nonlinear equations involve only three neighboring nodes, bounded fixed derivatives of D, and the factor t_0^{-3}. Their Hessian in the supremum norm therefore has size at most CQ^3, independently of the number of nodes. Applying the linearized inverse to the nonlinear remainder defines a contraction on a ball of radius CQ^2\eps when CQ^7\eps is small. Hence \xi=O_{\mathrm p}(\eps) on Y_1, holomorphically in y. Uniqueness preserves periodicity and real symmetry. Fixed derivatives are controlled on smaller strips by Cauchy’s estimate.

For real y, the uncorrected increments are bounded below by a fixed multiple of t_0. The correction, using (4.16) and a stronger fixed polynomial smallness condition, preserves positivity and keeps every step below 3d. After applying \phi, these are actual stationary billiard chains: subtracting the endpoint primitive in the definition of D telescopes, and the coordinate change has nonzero derivative. Their initial states lie in the graph collar. Lemma 2.3 therefore gives momentum return as well as angle return. Each chain is periodic with rotation l/q, and Proposition 2.2 identifies the same unique invariant graph for every starting y.

Invert V(\,\cdot\,,t_0) locally around the uncorrected nodes to write

on Y_2. This inverse is obtained by contraction for the small displacement from y+jt_0; it requires no assertion about a global complex inverse. On the real axis, uniqueness of the rational graph implies \widetilde y_j=\widetilde y_1^{\,j} and \widetilde y_q=y+2\pi l. Consequently

Its displacement and, on Y_3, its derivative displacement are O_{\mathrm p}(\eps). Contraction gives an inverse on Y_4. Thus

is an exact solution at t_0, with V_*-V(\,\cdot\,,t_0)=O_{\mathrm p}(\eps). The stationarity identity follows first on the real axis from the graph dynamics and then throughout the strip by holomorphy.

We must also give V_* an appropriate first parameter jet. Initially assign \dot V_*=V_t(\,\cdot\,,t_0). Its residual jet is O_{\mathrm p}(\eps) on Y_6. Indeed Cauchy’s estimate gives E_t=O_{\mathrm p}(\eps) there: the distance of t_0 from the parameter boundary exceeds 2\delta. A polydisk of radius \delta/4 in each variable about (y,t_0), with y\in Y_6, lies in \mathcal D_0^1, so this estimate costs only C\delta^{-1}. Replacing V by V_*, with the same assigned jet, changes that derivative by O_{\mathrm p}(\eps). To make its polynomial character explicit, if J is any assigned jet, direct differentiation gives

Only fixed derivatives occur, V_t and its needed y-derivatives have Cauchy bounds, and (4.16) controls every denominator.

Let \Pi=q^{-1}\sum_{j=0}^{q-1}T_{jt_0}. This is a projection on periodic functions, and \Pi\partial=\partial\Pi=0. Equation (4.7) at the exact solution shows that \Pi\dot E_*=0. Put a_*=a(V_*); after the fixed derivative losses, it is still close to 1/4. On \ker\Pi, the operator \partial(a_*\partial) has an inverse with polynomial norm. For an explicit verification, put T=T_{t_0} and

The identity (T-1)R_q=1-\Pi gives \partial^{-1}=t_0R_qT_{t_0/2} on \ker\Pi, of norm at most Cq|t_0|. Projected multiplication (1-\Pi)a_*:\ker\Pi\to\ker\Pi differs from multiplication by 1/4 by at most 2\|a_*-1/4\|, so is invertible by a Neumann series. Composing these three inverses yields

The projection commutes with all translations; no trivial action of the half-shift on its range is needed. We may therefore correct \dot V_* by V_*'z, where z=O_{\mathrm p}(\eps), to make its total residual jet zero.

Finally set w=(V_*-V)/V' and differentiate this quotient using the corrected jet of V_*. Both w and \dot w, with the fixed number of derivatives needed below, are O_{\mathrm p}(\eps) after the losses up to Y_{10}. Taylor’s formula and (4.6) give

The same formula differentiated in the assigned jets has a quadratic remainder. To see this directly, write the remainder as the integral of the second variation along the segment from V to V_*. A first parameter derivative either replaces one of its two increments by its assigned jet, or differentiates a coefficient. The first case still contains two O_{\mathrm p}(\eps) factors. The second costs only fixed derivatives in (4.18), with denominators at worst t_0^{-4}, and also retains both factors. Cauchy estimates on the remaining fixed buffers absorb the required derivatives. Moreover (Ew)' and its jet are quadratic since E,E_t,w,\dot w=O_{\mathrm p}(\eps). This proves both (4.12) and (4.13). All costs are fixed powers of Q: the mesh inverse costs Q^2, its Hessian Q^3, the cyclic inverse at most Q^2, and all other steps use fixed derivatives and fixed Cauchy losses. Negative t_0 follow by evenness, with the sign of the first jet reversed.□

Proof of Proposition 4.3. The first division takes F=-E. Its mean vanishes by Lemma 4.2. Every nonzero divisor zero for a mode |n|\le N has reduced denominator at most N. At those centers with |t_j|<d-3\delta, Equation (4.12) and d_n(t_j)=0 give

Apply Lemma 4.5 with

Denote the result by H. Then

The extra polynomial factor multiplying \eps^2 in the bound for H is absorbed by the smallness requirement. Define

The denominator stays away from zero, so b has mean zero and is O_{\mathrm p}(\eps) on \mathcal D_{12}^{1/2}.

Before dividing again, we need the compatibility of b at its resonances. Fix a retained rational parameter t_0 with |t_0|<d-14\delta, and let g=a\partial w use the value and assigned jet from Lemma 4.4. For every retained nonzero mode, the exact first-division identity gives, as an identity of values and first jets at t_0,

The right-hand side and its first jet are bounded by CQ^B\eps^2e^{-|n|(s-10\delta)}, using both compatibility equations and (4.23).

If d_n(t_0)\ne0, reducedness of t_0/(2\pi)=l/q gives

If d_n(t_0)=0, then |d_n'(t_0)|=|n|/|t_0|, and differentiating (4.26) bounds the same difference H_n-g_n. Thus in both cases

This is the reason for the first-jet compatibility: the value equation alone would give no estimate for a resonant coefficient of g.

The function g has norm O_{\mathrm p}(\eps) on Y_{10}. Its omitted Fourier tail on Y_{13} is bounded by CQ^B\eps e^{-3N\delta}=O_{\mathrm p}(\eps^2). Summing the retained estimates on that smaller strip therefore gives

Since \langle a^{-1}g\rangle=\langle\partial w\rangle=0, the mean correction in (4.25) satisfies \lambda=g_0+O_{\mathrm p}(\eps^2). Hence

At a zero of d_n, the coefficient (\partial w)_n vanishes. Equation (4.27) supplies the second division with its required Fourier values.

Apply Lemma 4.5 to F=b, with

All its centers satisfy the comparison range |t_0|<d-14\delta. Its output v satisfies

Both divisions, multiplication by a^{-1}, and the mean adjustment preserve parity and real symmetry.

Since ab=H+\lambda(t) and \partial\lambda=0, (4.24) and (4.28) yield

Here the remaining difference costs only a derivative and a domain loss, by \partial f=\int_{-1/2}^{1/2}f'(y+ut,t)\,\dd u. Take \Delta V=V'v. Cauchy estimates between indices 17 and 20 bound its first three y-derivatives by O_{\mathrm p}(\eps). Equation (4.6) makes its linearized residual quadratic, because (Ev)'=O_{\mathrm p}(\eps^2). The nonsingular Formula (4.8) gives

including t=0.

The construction uses fixed numbers of matrix inversions, contractions, products and derivatives, two applications of Lemma 4.5, and fixed Cauchy losses. All dimension and resonance costs are bounded by powers of N, and all shrinking-domain costs by powers of \delta^{-1}. The action bounds are fixed before d is shrunk. A single exponent B_0 and constant C_0 thus give (4.10)–(4.11) and absorb every smallness condition. This proves the Proposition with the asserted uniformity.□

Proof of Theorem 4.1. Fix a sufficiently small y-width s_0; all constants in Proposition 4.3 are now fixed. Begin with (4.29) on radius d_0<s_0/4, choosing m and then d_0 below. At step i, take

and reduce both current radii by 20\delta_i. Their limits satisfy d_\infty\ge4d_0/5 and s_\infty\ge s_0-d_0/5. If the current residual norm is \eps_i>0, choose

If it vanishes, an exact solution has already been obtained. Every fixed power of 1+\log(1/\eps_i) is bounded by a constant times \eps_i^{-1/4}. Thus, with one fixed enlarged exponent B, Proposition 4.3 gives

on the next domains, provided the polynomial smallness requirement holds.

Choose m so that 2m+2>4B+4, and only then shrink d_0. The finite constant C_m is thereby absorbed, giving the initial bound in the induction

Indeed substituting this bound in (4.30) and dividing by the desired next bound gives at most

which is at most one for small d_0. The correction bound becomes

These bounds both imply the polynomial smallness condition at every step and keep the candidates in their fixed smallness class: the initializer has slack and the total correction is arbitrarily small as d_0\to0. The induction is therefore justified before each application of the Newton estimate. The order of choices is B, then m, then d_0.

The corrections converge uniformly, with their first three y-derivatives, on the limiting domains. Their limit is holomorphic, real-symmetric, even in t, and periodic modulo y, with V_y close to one. The residual tends to zero, so (4.1) holds wherever the two shifts remain in the limit domain. At t=0, Equation (4.5) gives V''=0; periodicity gives V(y,0)=y+c, with real c. Replacing y by y-c removes the constant and preserves the equation and all symmetries. Finally choose a smaller product neighborhood with enough y-margin for both shifts. This gives precisely the domain and conclusions stated in the Theorem.□

Proof. Let V be the function supplied by Theorem 4.1, and set

For real y,t sufficiently close to the zero-parameter circle, \Theta_y>0 and

The normalization in (3.2) changes a chain action only by a nonzero constant factor and a telescoping endpoint term. Consequently, the equation in Theorem 4.1 gives

For sufficiently small t>0, successive angles have positive small increments. The generating relation for the actual billiard map and (5.2), shifted once in y, therefore give

In particular, these line states form an invariant graph, and the invariance is onto: the parameter map is the circle translation y\mapsto y+t.

Both functions in (5.1) are even in t. For \Theta this follows from the parity of V. For p, use antisymmetry of \cS and (5.2):

At t=0, the generating relation gives p(y,0)=h(\Theta(y,0)). There is also a quantitative inward displacement. Put

The line-coordinate formula of Section 2, expanded at fixed \theta=\Theta(y,t), reads

Since a(y,t)=\frac12\Theta_y(y,0)t+O(t^2), and the displacement h(\Theta)-p is even and analytic in t, this implies

The expansion and its derivatives are uniform on the real circle.

An even holomorphic function of t is holomorphic in \lambda=t^2 near zero, as is seen from its convergent power series. Write the resulting functions as \widehat\Theta(y,\lambda) and \widehat p(y,\lambda). The analytic inverse function theorem, positivity of \widehat\Theta_y, and degree one give a jointly real-analytic angular inverse y=Y(\theta,\lambda), with its natural 2\pi-equivariance. Compactness of the circle makes the parameter neighborhood uniform. Define the periodic function

It is the support candidate for the graph in (5.3). In particular, g(\theta,0)=h(\theta). Substituting the angular inverse in (5.4) gives, now at fixed \theta,

Thus no change of angle can remove the strict inward derivative. Choose \lambda_0>0 strictly inside the common analytic parameter neighborhood. Decreasing it if necessary, we may arrange on the entire closed cylinder 0\leq\lambda\leq\lambda_0 that

where c_0 is a constant. The first two inequalities follow from h>0 and r=h+h''>0, and the third from (5.5).

For each \lambda define

Figure 1 illustrates this construction: a point of the line graph specifies a support line, and the formula for \Gamma_\lambda gives its contact point. We next verify that these formal envelopes are the boundaries of the bodies K_\lambda.

Each K_\lambda is a compact convex body with nonempty interior: it contains a disk about the origin of radius \min_\theta g(\theta,\lambda)>0, and the constraint with normal z/|z| bounds |z| by \max_\theta g(\theta,\lambda). Moreover K_0=\overline\Omega, by the definition of h.

We verify directly that \Gamma_\lambda is the whole boundary and has support function g(\cdot,\lambda). Differentiation gives

For a fixed angle \alpha,

This projection has its unique maximum, modulo 2\pi, at \theta=\alpha, where its value is g(\alpha,\lambda). Hence every point \Gamma_\lambda(\theta) satisfies all the constraints defining K_\lambda, with equality for its own normal, and belongs to \partial K_\lambda.

Conversely, let z\in\partial K_\lambda. The continuous nonnegative function g(\theta,\lambda)-z\cdot n_\theta must vanish somewhere: a positive minimum would place a disk about z inside K_\lambda. At a zero \alpha, its derivative also vanishes. Thus

which gives z=\Gamma_\lambda(\alpha). The unique-maximum property proves that this parametrization is one-to-one on \T; (5.8) makes it regular. Each support line has exactly one contact point, so the boundary contains no nontrivial line segment and the body is strictly convex. In fact its curvature is 1/\rho>0. All these curves are real analytic.

The last inequality in (5.6) gives, for 0\leq\lambda<\mu\leq\lambda_0,

It follows that K_\mu is contained in the interior of K_\lambda, with a uniform positive gap between them. In particular the boundaries are pairwise disjoint, and \Gamma_\lambda\subset\Omega for \lambda>0.

The positively oriented tangent at \Gamma_\lambda(\theta) has velocity e_\theta and line equation z\cdot n_\theta=g(\theta,\lambda). For \lambda>0 these are precisely the line states in (5.3), with t=\sqrt\lambda. They meet \Omega, because their contact points lie in its interior. The onto invariance in (5.3) therefore preserves tangency under both T and T^{-1}. It supplies the tangent successor segment at the forward endpoint and the tangent predecessor segment at the back endpoint. Reflection is a linear involution: if v' is the reflection of v, then -v is the reflection of -v'. Reversing each such pair of segments proves tangency preservation for the opposite orientation as well. Thus every positive leaf is a smooth closed convex caustic, with the required reflection property at either endpoint.

It remains to show that these curves fill a physical collar. The map

is continuous and injective, by simplicity of the curves and strict nesting. Since its domain is compact, it is a homeomorphism onto its image. Set

This is relatively open in K_0=\overline\Omega and contains \partial\Omega. All leaves with \lambda<\lambda_0 lie in A. For the converse, fix z\in A and consider its minimum support slack

This function is continuous, satisfies m_z(0)\geq0>m_z(\lambda_0), and obeys

It has exactly one zero in [0,\lambda_0). At that zero, z\in\partial K_\lambda, hence z=\Gamma_\lambda(\theta) for exactly one \theta. The restriction of F to \T\times[0,\lambda_0) is therefore a homeomorphism onto A. Rescaling \lambda=b\lambda_0 completes the proof.□

Proof of Theorem 1.3. Theorem 3.1 proves that the billiard boundary is real analytic. Theorem 4.1 and Proposition 5.1 give the asserted jointly real-analytic family of strictly convex caustics and its full physical collar.□

Proof of Theorem 1.1. Proposition 5.1 gives exactly the continuous caustic collar of [14, Definition 1.1]: the boundary is the zero leaf, the positive leaves are smooth closed convex caustics, and the family parametrizes a relatively open neighborhood of the boundary in the closed table. The original assumptions supply the remaining smoothness, strict convexity, and positive curvature hypotheses. Theorem 1.2, namely [14, Theorem 1.2], therefore implies that \Omega is an ellipse. Undoing the initial translation gives a center c\in\R^2 and a real symmetric positive-definite matrix Q with

as required.□