Minimal self-joinings and positive topological entropy

We show that the properties of almost minimal self-joinings and strong almost minimal self-joinings, introduced by del Junco in Topological Dynamics, are compatible with positive topological entropy, as opposed to the stronger property of minimal self-joinings. This is done both by proving existence theorems and by explicitly constructing some symbolic systems having these properties, which are modifications of the Chacón system. It is shown furthermore that these systems have no non-trivial factors with completely positive topological entropy.