Appears in collection : 2024 - T2 - WS1 - Low Dimensional Actions

A bifoliated plane is a topological plane equipped with two transverse (possibly singular) foliations. Given a group G, an Anosov-like action is an action of G on a bifoliated plane satisfying a few axioms, first among them is the fact that each point in the plane fixed by an element of the group is a hyperbolic fixed point.

Such actions were first introduced as an axiomatization, and generalization, of the natural action induced by a 3-dimensional (pseudo)-Anosov flow on its orbit space. It turns out that a lot of the dynamical behaviors that we see in Anosov flows also appears in this context. In this talk I will describe some of these features, such as recovering basic sets and the Smale order for non-transitive Anosov-like actions, as well as prove a rigidity result: An Anosov-like action is uniquely determined by its induced action on the circle at infinity of the bifoliated plane.

This is joint work with Christian Bonatti and Kathryn Mann.