Stability of Return Groups

In this presentation, we introduce the notion of eventual φ-stability in return groups. It provides a general framework that extends notions previously explored in a variety of studies. While it would be difficult to single out a main result, the significance of this property lies in its applications to known concepts. We have highlighted it relevance towards well-known families of shift spaces and showcased some closure properties as well as decidability and computability results. We hope this can lead to exiting, unifying and fresh research perspectives in the field.