A Braided View of a Knotty Story

Abstract:
Periodic orbits of 3-d dynamical systems admitting a Poincaré section can

be described as braids. This characterisation can be transported to the

Poincaré section and Poincaré map, resulting in the braid type.

Information from braid types allows to estimate bounds for the topological

entropy of the map while revealing detailed orbit information from the

original system, such as the orbits that are necessarily present along with

the given one(s) and their organisation. We review this characterisation

with some examples --from a user-friendly perspective--,

focusing on systems whose Poincaré section is homotopic to a disc.