Higher-order Lazutkin coefficient structure conjecture

Let $\Omega \subset \mathbb{R}^2$ be a smooth, bounded, strictly convex billiard table, and let $\Gamma \subset \Omega$ be a convex caustic of length $|\Gamma|$ with Lazutkin parameter $Q$. Let $\kappa$ denote the curvature of $\partial \Omega$, and let $\kappa_m$ be its $m$th derivative in arclength coordinates. Higher-order Lazutkin coefficient structure conjecture. There exists an asymptotic expansion

$$|\Gamma| \sim |\partial \Omega| + \sum_{k=1}^{\infty} \mathcal{I}_k Q^{2k/3},$$

and, for each $k \in \mathbb{N}$, the $\mathcal{I}_k$ are integral invariants of the form

$$\mathcal{I}_k = \int_0^{\ell_0} \kappa^{-r/3}\mathcal{P}_k(\kappa,\kappa_1,\ldots,\kappa_{k-1})\,ds,$$

where $r>0$ and $\mathcal{P}_k \in \mathbb{R}[\kappa,\kappa_1,\ldots,\kappa_{k-1}]$ has differential degree $2k-2$. Moreover, the highest derivatives in $\mathcal{P}_k$ appear quadratically as

$$\mathcal{P}_k=c_k\kappa_{k-1}^2+\kappa_{k-1}\mathcal{Q}_k+\mathcal{R}_k,$$

where $c_k\neq 0$, $\mathcal{Q}_k,\mathcal{R}_k\in\mathbb{R}[\kappa,\ldots,\kappa_{k-2}]$, $\mathcal{Q}_k$ has differential degree at most $k-1$, and $\mathcal{R}_k$ has differential degree at most $2k-2$. The coefficients $\mathcal{I}_k$ are nonzero combinatorial multiples of the Marvizi–Melrose integral invariants. The conjecture is intended to provide the higher-order coefficient structure needed to upgrade the paper's $C^2$ compactness result for marked length isospectral sets to compactness in the $C^\infty$ topology; proving it remains a work in progress.