The dynamics of area-preserving flows on closed orientable surfaces is a well-understood topic, and there exists a canonical invariant decomposition of the phase space into a region (a collection of topological annuli) where the dynamics is integrable, and a finite number of pieces of positive genus where the dynamics is quasi-minimal (closely resembling the dynamics of an irrational flow on a torus).

In this work, we show that a very similar canonical decomposition remains valid when dealing with conservative homeomorphisms with zero topological entropy. We present this decomposition while also exhibiting examples of different phenomena that may arise, as well as several properties of the “quasi-minimal” regions.

Part of the work involves proving a Thurston–Nielsen–type reduction result, showing that maps homotopic to Dehn twists and with zero entropy actually possess invariant “Dehn-like” annuli. Time permitting, we will also discuss some applications to Reeb flows on three-dimensional manifolds.