Abstract:
Say that an action of a group G on the real line is hyperbolic-like if it commutes with the integer translations and every element either has no fixed points or exactly one pair of attractor/repeller in [0,1). I will explain how knowing which elements of a hyperbolic-like action admit fixed points is enough to characterize the action up to conjugacy.Our goal behind this theorem was its application to the problem of classifying (ℝ-covered) Anosov flows up to orbit equivalence, which I will explain. It also turns out that it yields an equivalence between contact Anosov flows up to orbit equivalence and contact structures up to isomorphism, allowing to directly use contact geometry to solve problems on Anosov flows and vice versa.
This is joint work with Kathryn Mann and Jonathan Bowden.