"Chaotic dynamics in foliated spaces"
Barral Lijo, RamonDevaney characterized chaos for a continuous map using three conditions: existence of a dense orbit, density of periodic orbits, and sensitivity to initial conditions. For group and semigroup actions on Polish spaces, the first two conditions imply the third. Motivated by this result, previous definitions of chaos for foliated spaces have only consider generalizations of the first two conditions, with no mention of sensitivity. In this talk, we will study whether and when this omission is warranted. We present a definition of sensitivity for foliated spaces using the holonomy pseudogroup, showing that most results for groups and semigroups generalize to compact foliated spaces; in particular, existence of a dense leaf and density of compact leaves imply sensitivity. On the other hand, we exhibit a counterexample in the non-compact setting: a smooth and transversely affine foliation by surfaces of an open 4-manifold such that there is a dense leaf and the set of compact leaves is dense, but the holonomy pseudogroup is not sensitive to initial conditions.