It is known that for expansive algebraic actions of amenable groups, the existence of nontrivial homoclinic pairs implies positive entropy. The converse was known only in the case the group ring of the amenable group is left Noetherian. I will discuss an approach establishing the converse not only for expansive algebraic actions of amenable groups with left Noetherian group rings, but also for expansive finitely presented algebraic actions of amenable groups whose group ring is a domain such as torsion-free elementary amenable groups and left-orderable amenable groups. The key ingredient is the various notions of topological Markov properties, which are weaker than the pseudo-orbit tracing property but strong enough to imply relation between independence entropy pairs and homoclinic pairs.
This is joint work with Sebastian Barbieri and Felipe Garcia-Ramos.