Flux normalization and clustering tests for billiard strip targets

Definitions and mathematical statement

Problem 25 asks for hitting-time and extreme-value laws in billiards when a shrinking phase-space target is a strip induced by a configuration-space set rather than a phase-space ball. The source does not identify the billiard class, target geometry, collision versus flow observation, invariant measure or singularity assumptions. The appended "Open Problems and Questions: Thursday" is the next heading, not part of the target definition.

The aid treats planar collision strips. For a bounded planar billiard configuration domain $Q$, let $L$ be its total boundary length, $s$ boundary arclength, and $\phi\in(-\pi/2,\pi/2)$ the outgoing angle from the inward normal. With unit-speed collision dynamics $T$, normalize invariant collision measure by \[ {\,\mathrm{d}}\mu(s,\phi)=\frac{\cos\phi}{2L}\,{\,\mathrm{d}} s\,{\,\mathrm{d}}\phi. \] A full-angle strip $A_I=\{(s,\phi):s\in I\}$ over a boundary arc has exact mass $\mu(A_I)=|I|/L$. A two-sided arclength strip $|s-s_0|<\varepsilon$, away from boundary-coordinate endpoints, therefore has mass $2\varepsilon/L$.

For a compact strictly convex target $K$ lying inside the free configuration domain, let $B_K$ be the collision states whose next free-flight segment enters $K$. In the flux setting described by the aid, transporting Liouville flux to the target boundary gives \[ \mu(B_K)=\frac{\operatorname{Per}(K)}L. \] Convexity ensures that a segment does not make multiple separate entries into $K$; the collision/flux correspondence must be well defined up to null grazing or singular states. For homothetic targets $K_\varepsilon=q_0+\varepsilon K_0$ contained in the free domain, $\mu(B_{K_\varepsilon})=\varepsilon\operatorname{Per}(K_0)/L$. These are collision-event masses, not the configuration-area mass or a physical-time entry rate.

Conditionally, if a target family $A_\varepsilon$ of mass $a_\varepsilon\sim C\varepsilon$, $C>0$, has an extremal-index-one Poisson visit law on the collision scale, then for $\tau_{A_\varepsilon}=\inf\{j\ge1:T^jx\in A_\varepsilon\}$, \[ \mu\{a_\varepsilon\tau_{A_\varepsilon}>t\}\longrightarrow e^{-t}\quad(t\ge0). \] For an observable whose exceedance set at level $u$ is $A_{e^{-u}}$, the associated maximum $M_n=\max_{0\le j<n}\psi\circ T^j$ has the Gumbel normalization \[ u_n(y)=\log(Cn)+y,\qquad\mu\{M_n\le u_n(y)\}\longrightarrow\exp(-e^{-y}), \] under the same Poisson hypothesis. The scale used in $e^{-u}$ fixes the arclength reference units. These limit laws are conditional conclusions, not consequences of the mass calculation alone.

The aid's fixed-lag clustering diagnostic is restricted to finitely many uniformly regular branches. For fixed $k$, restrict the returning boundary coordinate $s_k(s,\phi)$ to the central fiber $s=s_0$. If its returns to $s_0$ are isolated of finite contact order $m$, with a uniform local bound $|s_k(s_0,\phi)-s_0|\ge c|\phi-\phi_0|^m$, bounded derivatives in the transverse coordinate, and the required finite-branch control, then \[ \mu(A_\varepsilon\cap T^{-k}A_\varepsilon)=O(\varepsilon^{1+1/m})=o(\mu(A_\varepsilon)). \] For several contact orders, the estimate is taken branchwise. In contrast, an interval of persistent fiber overlap, with regularity and positive-density control, has intersection of order $\varepsilon$ and can cause clustering. Isolated finite-order contact and persistent overlap are therefore distinct mechanisms. This fixed-lag diagnostic does not control arbitrarily many branches, grazing singularities, growing lags or all the mixing/short-return requirements for a Poisson theorem; persistent overlap alone is not a complete formula for the extremal index.

The aid reports that the boundary-strip version is substantially answered by Bunimovich and Su's 2024 Poisson theorem for billiard-boundary holes. Their precise billiard hypotheses and theorem citation are not supplied. The registry labels the entry partially solved; attempt 1 is a partial result, problem partially solved at the run, internal rigor proved and novelty confidence low. The machine-generated aid is not peer reviewed and requires independent verification before citation.

Short English statement

The question concerns strip targets determined by position rather than phase-space balls. The aid gives exact collision-measure normalizations and conditional exponential/Gumbel laws, then distinguishes negligible isolated finite-order returns from persistent fiber overlap that can create clustering. These calculations do not replace the regularity and mixing hypotheses of a billiard Poisson theorem.

Sources

• Local UnsolvedMath registry, permanent ID $20001410$, source code catalogue entry; aim-dynamical-systems-notes.json, notes[67].
• AIM, Stochastic methods for non-equilibrium dynamical systems, Problem 25, with the ensuing Thursday heading: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• Bunimovich and Su (2024), Poisson theorem for billiard-boundary holes, as reported in the research summary; no article title, publication, theorem number or exact model hypotheses supplied. The original literature field is empty.
• Supplied research attempt 1 and registry update August 14, 2026; no independent source visit or verification of the literature status is claimed.