A combinatorial property of geodesic flow and isometric Z-actions

Fix a rotation $R$ of the circle, a point $p$ on the circle and some natural number $n$. Then the points $p,R(p),…,R^n(p)$ divide the circle into segments. A beautiful theorem from the 1960’s states that the lengths of these segments take on at most three distinct values. We will discuss generalizations of this Three Gap Theorem to higher dimensions, by replacing the circle with a Riemannian manifold and $R$ with an isometry or the time one map of the geodesic flow.

Joint with Benjamin Schmidt