Tree-Like Spaces with the Fixed-Point Property, Realized as Inverse Limits of Special Finite Trees

In this paper, we explore some properties of inverse limit sequences on sub-spaces of Euclidean n-space. We address some well-known examples, in particular the example by David Bellamy of the "tree-like" continuum that does not have the fixed-point property. We highlight some spaces with the fixed-point property that are between snake-like continua and Bellamy's example in their level of complexity. Specifically, we prove the fixed-point property for inverse limits of limit sequences on the unit interval and on the n-ad (in two configurations), and for inverse limits that can be mapped via a continuous function with small point pre-images to a generalized relative of the n-ad, which we call the (m, n)-ad. We conclude with a sufficient condition on a tree-like space for the space to have the fixed-point property.