Chaos for foliated spaces and pseudogroups

TL;DR

This paper extends the concept of chaos to foliated spaces and pseudogroups, establishing conditions under which sensitivity to initial conditions implies chaos, and presents a dichotomy result for compact cases.

Contribution

It introduces a generalized definition of Devaney chaos for foliated spaces and pseudogroups, and analyzes the relationship between sensitivity and chaos in these settings.

Findings

Sensitivity implies chaos only in compact foliated spaces.

Counterexample shows this implication fails in non-compact cases.

An analogue of the Auslander-Yorke dichotomy is established for compact spaces.

Abstract

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.