Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

We show that if S={ft:Y→Y}t≥0$S=\lbrace f_{t}:Y\rightarrow Y\rbrace _{t\ge 0}$ is a one‐parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space (Y,d)$(Y,d)$ that satisfies some geometric properties, then there exists ξ∈∂Y$\xi \in \partial Y$ such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in Rn$\mathbb {R}^{n}$ or Cn$\mathbb {C}^{n}$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.