  Shapley operators of undiscounted zero-sum two-player games are
order-preserving maps that commute with the addition of a constant. We
characterize the fixed point sets of Shapley operators, in finite dimension
(i.e., for games with a finite state space). Some of these characterizations
are of a lattice theoretical nature, whereas some other rely on metric or
tropical geometry. More precisely, we show that fixed point sets of Shapley
operators are special instances of hyperconvex spaces: they are sup-norm
non-expansive retracts of $\R^n$, and also lattices in the induced partial
order. Moreover, they retain properties of convex sets, with a notion of
``convex hull'' defined only up to isomorphism. This provides an effective
construction of the injective hull or tight span, in the case of additive
cones. For deterministic games with finite action spaces, these fixed point
sets are supports of polyhedral complexes, with a cell decomposition attached
to stationary strategies of the players, in which each cell is an alcoved
polyhedron of $A_n$ type. We finally provide an explicit local representation
of the latter fixed point sets, as polyhedral fans canonically associated to
lattices included in the Boolean hypercube.
