Wolff–Denjoy theorems in geodesic spaces

Abstract

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 R n or 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 infinitedimensional Banach spaces.

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

“…Note that in particular the intersection of horoballs' closures $$\bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\zeta ,r)}$$ consists of a single point if $$(Y,d)$$ satisfies Axiom 3 (see [8, Lemma 3.4]).…”

“…Thus, $$(D,d)$$ is locally compact and it follows from the Hopf–Rinow theorem that $$(D,d)$$ is proper. It is not difficult to show that $$(D,d)$$ satisfies Axiom 1 with respect to the norm closure in V (see [8, Lemma 4.2]).…”

“…Recall that $$D\subset V$$ is strictly convex if for any $$z,w\in \overline{D}$$ the open segment $$$$\begin{equation*} (z,w)=\lbrace sz+(1-s)w:s\in (0,1)\rbrace \end{equation*}$$$$lies in D . The following lemma was proved in [8, Lemma 4.4]. Lemma Suppose that D is a bounded strictly convex domain of V and $$ (D,d)$$ is a complete geodesic space whose topology coincides with the norm topology with the condition $$$$\begin{equation} d(sx+(1-s)y,z)\le \max \lbrace d(x,z),d(y,z)\rbrace \text{ for all }x,y,z\in D\text{ and }s\in [0,1].$$…”

“…It follows from [8, Lemma 3.4] that, in the above theorem, the assumption about the single‐point intersection of big horoballs' closures $$\bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\zeta ,r)}$$ can be replaced by Axiom 3. Then, Theorem 3.4 immediately lead to the following result.…”

“…Beardon showed in [4] that a Wolff–Denjoy‐type theorem holds for any proper metric space $$(Y,d)$$ satisfying $$$$\begin{equation} d(x_{n},y_{n})-\max \lbrace d(x_{n},w),d(y_{n},w)\rbrace \rightarrow \infty \end{equation}$$$$for any sequences $$\lbrace x_{n}\rbrace$$ and $$\lbrace y_{n}\rbrace$$ in Y converging to distinct points in $$\partial Y$$ and any $$w\in Y$$ (see Section 2 for more details). Beardon's framework was extended in [8] by showing that in proper geodesic spaces condition (B) can be relaxed to the following: $$$$\begin{equation} d(x_{n},y_{n})-d(y_{n},w)\nrightarrow -\infty \end{equation}$$$$for any sequences $$\lbrace x_{n}\rbrace$$ and $$\lbrace y_{n}\rbrace$$ in Y converging to distinct points in $$\partial Y$$ and any …”