Fixed-point-free maps on tree-like continua induced by commuting diagrams on trees

It is a simple and well-known fact that any continuous function f from a (finite) tree to itself has a fixed point, i.e. a point x such that f(x) = x.  The same is true for some generalizations of trees, such as dendroids.

For the broader class of tree-like continua, the situation is different.  A tree-like continuum is a space that can be represented as an inverse limit of trees.

David Bellamy constructed the first example of a tree-like continuum with a fixed-point-free self-map in 1980, which was a landmark result in continuum theory and fixed-point theory.  Since then, several variants of this example have been presented, e.g. by Oversteegen & Rogers, Minc, and Fearnley & Wright.  However, in all of these works it seems that simple, accurate visualizations of these examples are not easily attainable.  I will present a recent, simplified construction of a tree-like continuum and a fixed-point-free self-map, with pictures. (This is joint work with Rodrigo Hernández-Gutiérrez.)