A GENERALIZATION OF THE BROUWER PLANE TRANSLATION THEOREM

The paper’s Theorem 3 states exactly the target claim: for an orientation‑preserving homeomorphism f: R^2 → R^2, the existence of a topologically chain recurrent point implies a fixed point . Its proof proceeds by contrapositive: build an open neighborhood N of the diagonal with N(p) ∩ f(N(p)) = ∅ when Fix(f)=∅, convert an N-chain into a finite family of (eventually pairwise disjoint) free topological disks, and apply Franks’ periodic disk‑chain criterion to force a fixed point . The candidate solution follows the same route: (i) constructs such an N (explicitly, via radii tied to d(x)=∥f(x)−x∥), (ii) turns an N‑chain into free disks, (iii) shrinks to a periodic disk chain, and (iv) invokes Franks to conclude a fixed point. Two technical differences are minor: the paper ensures disk fibers by replacing N with B_δ using a standard continuous selection δ (citing Ziemer’s theorem) , while the model builds N directly with ball fibers; and the paper’s “clean up” of intersections is via a case analysis that either shrinks N, trims the chain, or immediately finds a cycle, whereas the model shrinks along arcs to produce disjoint tubes . The model leaves one small gap (lower semicontinuity of its chosen σ(x)); however, this can be fixed by adopting the paper’s B_δ refinement. Net: both arguments are correct and essentially the same method (periodic disk‑chain + Franks), with the paper giving a slightly more polished handling of the technical selection.