On the Furstenberg fixed point property

Given an abstract semigroup S, by the Furstenberg fixed point property, we refer to a fixed point property of the following type: Whenever (S, X) is a nonexpansive compact flow with a certain property (P) and nonempty convex phase space X in a Hausdorff locally convex space (E, Q), then there exists a point \(x\in X\) $x\in X$ such that \(s.x=x\) $s.x=x$ for each \(s\in S\) $s\in S$ . Motivated by the Furstenberg’s fixed point theorem on the existence of common fixed points for continuous affine compact flows, we are interested in investigating its natural nonlinear counterpart, and to this end we introduce and study a certain fixed point property for semigroups of nonexpansive mappings.