ОСОБЕННОСТИ СЛОЕНИЙ ЛИУВИЛЛЯ ДЛЯ БИЛЛИАРДОВ В НЕВЫПУКЛЫХ ОБЛАСТЯХ

The paper’s Theorem 5 states that, for a homogeneous hyperbolic billiard, if one removes all cutting 2-complexes Ti from the 3-atom U in a neighborhood of the saddle layer Λ=b, then: (1) each Ti slices into the one-dimensional boundary preimage graphs Gr<i, Gr=i, Gr>i (constructed by Algorithm 4); (2) the complement U\(T1∪...∪Tn) is a disjoint union of two copies of each 2-atom cylinder 2(Vj×I); and (3) the reattachment of all Ti is carried out layerwise along these graphs via the circle bundles (“rings”) of the 2-atoms, per Algorithm 5. This is stated explicitly in the paper’s Theorem 5 and surrounding text, with Ti defined as the π-preimage over a boundary arc, the graphs Gr constructed by Algorithm 4, and the two-cylinder (left/right) structure encoded in Algorithm 5 (two cylinders S^1×I per positive ring of each 2-atom) . The candidate solution’s three steps mirror this exactly: Step 1 identifies the Ti-slices with the preimage graphs; Step 2 proves the decomposition of the complement into two copies of Vj×I per elementary billiard; Step 3 describes the layerwise gluing along the graphs with 4-valent and degree-0 vertices matching right-angle and 3π/2 vertices. The only minor imprecision is that the model says an isolated degree-0 vertex arises on a “unique” Λ-level, whereas the paper augments Q3 by adding one point at each Λ-level where a trajectory hits a 3π/2 vertex . Aside from that, the proofs are substantially the same in structure and logic.