The horofunction boundary and Denjoy-Wolff type theorems

In this thesis we will study the horofunction boundary of metric spaces, in particular the Funk, reverse-Funk and Hilbert's metrics, and one of its applications, Denjoy-Wolff type theorems. In a Denjoy-Wolff type setting we will show that Beardon points are star points of the union of the ω-limit sets. We will also show that Beardon and Karlsson points are not unique in R2. In fact, we will show one can have a continuum of Karlsson points. We will establish two Denjoy-Wolff type theorem that confirm the Karlsson-Nussbaum conjecture

for classes of non-expanding maps on Hilbert' metric spaces. For unital Euclidean Jordan algebras we will give a description of the intersection of closed horoballs with the boundary of the cone as the radius tends to minus infinity.

We will expand on results by Walsh by establishing a general form for the Funk and reverse Funk horofunction boundaries of order-unit spaces. We will also give a full classification of the horofunctions of JH-algebras and the horofunctions and Busemann points of the spin factors for the Funk, reverse Funk and Hilbert metrics. Finally we will show that there exists a reverse-Funk non-Busemann horofunction for the cone of positive bounded self-adjoint operators on an infinite dimensional Hilbert space, the infinite dimensional spin factors and a space in which the pure states are weak* closed, answering a question raised by Walsh in [65].