DIRECTIONAL STABLE SETS AND MEAN LI-YORKE CHAOS IN POSITIVE DIRECTIONAL ENTROPY SYSTEMS

The paper’s Theorem 1.1 asserts a Mycielski set C and, for every b>0 and every integer k≥2, a positive lower bound η_b on the limsup of the time-average of the minimal pairwise distance that depends only on b and is independent of k. This uniform-in-k claim directly contradicts the finite packing property of compact metric spaces: for any fixed η>0, a k-point subset with k exceeding the η-packing number of X must contain a pair within distance <η at every time, forcing the time-averaged minimum distance to be <η. The statement is thus impossible as written. The paper’s Step 2 builds the lower bound by importing a Z-action result (its Lemma 3.3) with a single η>0 and then transporting it to directional strips, ending with a bound η_b that depends only on b (see the display of (1.2) and the sequence of estimates around (3.11)–(3.12)), thereby inheriting the same uniform-in-k flaw . The candidate solution correctly identifies this obstruction and proposes the natural correction: allow the constant to depend on both b and k. That corrected statement is standard and can be proved by the same suspension method used in the paper (avoiding the need for any non-Følner subsequence arguments), together with the usual independence/weak-mixing machinery to produce, for each fixed k, k disjoint open sets with positive frequencies, yielding a positive η_{b,k}.