  We provide a unifying approach which links results on algebraic actions by
Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that
relates entropy to the set of asymptotic pairs. In order to do this we
introduce a series of Markovian properties and, under the assumption that they
are satisfied, we prove several results that relate topological entropy and
asymptotic pairs (the homoclinic group in the algebraic case). As new
applications of our method, we give a characterization of the homoclinic group
of any finitely presented expansive algebraic action of (1) any elementary
amenable group with an upper bound on the orders of finite subgroups or (2) any
left orderable amenable group, using the language of independence entropy
pairs.
