Stabilization of monomial maps in higher codimension

A monomial self-map f on a complex toric variety is said to be κ-stable if the action induced on the 2κ-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of f, we can find a toric model with at worst quotient singularities where f is κ-stable. If f is replaced by an iterate one can find a k-stable model as soon as the dynamical degrees λ_κ of f satisfy λ2 _κ² > λ_κ-1λ_κ+1. On the other hand, we give examples of monomial maps f, where this condition is not satisfied and where the degree sequences deg_κ(fⁿ) do not satisfy any linear recurrence. It follows that such an f is not κ-stable on any toric model with at worst quotient singularities.