Chaos for foliated spaces and pseudogroups

We generalize “sensitivity to initial conditions” to foliated spaces and pseudogroups, offering a definition of Devaney chaos in this setting. In contrast to the case of group actions, where sensitivity follows from the other two conditions of Devaney chaos, we show that this is true only for compact foliated spaces, exhibiting a counterexample in the non-compact case. Finally, we obtain an analogue of the Auslander–Yorke dichotomy for compact foliated spaces and compactly generated pseudogroups.