A PIECEWISE CONTRACTIVE MAP ON TRIANGLES

The paper defines the map T on X = L1 ∪ L2 ∪ L3 (three pairwise nonparallel, nonconcurrent lines) by sending a point to the farther of its two orthogonal projections or fixing it in case of a tie (Definition 2.1), and claims that every orbit converges to a fixed point or a periodic orbit (Corollary 4.4), via the construction of trapping intervals and an induced contraction map (Theorem 3.3, Corollary 3.4, Theorem 3.5). However, the proof of existence of a “trapping interval” in the strong sense of Definition 2.7 (i.e., a uniform minimal return time for all points in the interval) is not actually established in Lemma 2.8: the proof only shows there is Ii with T^n(Ii) ⊂ Ii, but does not prove uniform minimality across Ii, which is subsequently used to define the induced map and its period conclusions (this breaks the chain to Theorem 3.5) . By contrast, the model’s solution establishes convergence by selecting an infinitely-often return pattern to a line and invoking Banach’s contraction on the corresponding return map; it then builds a forward-invariant tube that forces that itinerary and yields exponential convergence. That argument correctly proves every ω-limit set is a fixed point or a finite periodic orbit. One caveat: the model’s addendum asserting that only periods 1, 2, or 3 can occur is not justified and is inconsistent with the paper’s Section 4 phenomena; removing that remark leaves the model’s core proof intact .