EXISTENCE OF ASYMPTOTIC PAIRS IN POSITIVE ENTROPY GROUP ACTIONS

The paper proves the main theorem by passing to a successor Z-action on X×Õ, showing it has positive (unconditional) entropy via the multiorder factor, and then invoking the BHR lemma to extract S-asymptotic pairs and transport them to ≺-asymptotic pairs in X. The candidate solution instead identifies positive relative entropy of the same successor skew-product over the order factor using the multiorder entropy identity, and then applies a relative version of the asymptotic-pair theorem to produce in-fiber asymptotic pairs, which translate to ≺-asymptotic pairs in X. The conclusions match; the arguments differ (unconditional vs relative route).