Positive Entropy Implies Chaos Along Any Infinite Sequence

The paper proves Theorem 1.1 for amenable G-actions via a measure-theoretic route: variational principle to pick an ergodic positive-entropy measure, disintegration over the Pinsker σ-algebra, relatively independent self-joining, a relative independence (Sinai-type) theorem (Theorem 2.6), two carefully constructed partitions on X×X (Lemmas 3.2 and 3.3), and a Mycielski upgrade to obtain a dense Mycielski scrambled set in a typical fibre (Theorem 3.1), from which the topological statement follows . In contrast, the candidate solution’s Step 5 posits a two-atom partition {B0,B1} of X with dist(B0,B1)>4δ and with fibrewise measures 1/2, which generally does not exist on a connected compact metric space; the paper avoids this by working on X×X and by constructing product-space partitions tied to Δr, not separated partitions of X (see the construction of α and β in Lemmas 3.2 and 3.3) . Moreover, the model omits the crucial step of selecting finite sets Fm from the tail of the prescribed sequence {si} so that FF−1 avoids a finite warm-up set K (as guaranteed by Theorem 2.6), which is essential for establishing the needed conditional independence along the sequence . The rest of the model broadly follows the paper’s scheme (Pinsker disintegration, relative independence, Mycielski) but lacks these key technical details and uses a flawed separation assumption.