New notions of recurrence in a multidimensional setting

In one dimension, an infinite word is said to be recurrent if every prefix occurs at least twice. A straightforward extension of this definition in higher dimensions turns out to be rather unsatisfying. In this talk, we present several notions of recurrence in the multidimensional case. In particular, we are interested in words having the property to be strongly uniformly recurrent: for each direction q, every prefix occurs in that direction (i.e. in positions iq) with bounded gaps. We will provide several constructions of such words and focus on the strongly uniform recurrence in the case of square morphisms.
Joint work with Émilie Charlier and Svetlana Puzynina.