Welded braids on 3D dynamics

ongoing

The connection between braid group theory and topological dynamics arises from the remarkable applications of braid group theory to the study of the periodic orbit structure of iterated surface homeomorphisms. It began the early 1980s and has since grown into a significant area within the theory of low-dimensional dynamical systems. Braid theory, a very powerful and important tool, has been used extensively, among others, by Boyland, Ghrist, Guaschi, Hall, Matsuoka and Vandervorst for proving that the existence of certain orbits forces the existence of several other orbits and for showing how the orbits are interacting among them. A central objective of topological dynamics on 3-dimensional manifolds is to comprehend the structure of the orbits of a self-map. While there has been done a lot of research on the complexity of orbit structures in 3-dimensional dynamics, there are not known results that are analogous to the impressive known results about the orbit structure on surfaces that use braid theory. One of the main reasons is that the classical braid group theory is trivial on 3-dimensional manifolds. To counter this obstacle, WEB3 proposes to use the theory of welded braid groups, which is the three-dimensional analogue of classical braids. The aim of WEB3 is to extend the existing literature concerning the orbit structure of 3-dimensional dynamics and in particular to establish a new machinery for exploring the behavior of periodic points of homeomorphisms of 3-dimensional manifolds. This proposal will have significant and wide impact in the field of low-dimensional topology and topological dynamics by fostering innovation and creating new knowledge. WEB3 is the most fitting project for enriching the current literature both on braid group theory and on topological dynamics, since it combines knowledge from both of these two fields of mathematics and features the necessary tools for emerging them successfully together.