Wolff-Denjoy theorems in geodesic spaces

We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit of Beardon’s framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ with respect to a large class of metrics including Hilbert’s and Kobayashi’s metrics. The results are generalized to $1$-Lipschitz compact mappings in infinite-dimesional Banach spaces.