Wolff-Denjoy theorems in geodesic spaces

TL;DR

This paper extends Wolff-Denjoy theorems to complete geodesic spaces, unifying various results and applying to convex domains with different metrics, including in infinite-dimensional Banach spaces.

Contribution

It generalizes Wolff-Denjoy theorems to a broad class of geodesic spaces and metrics, including infinite-dimensional Banach spaces.

Findings

Theorem applies to convex domains in $ ^n$ and $C^n$ with various metrics.

Results include infinite-dimensional Banach space mappings.

Unifies multiple existing Wolff-Denjoy results.

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 $\mathbb{R}^{n}$ 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.