Characterization of n-valued dynamics with polynomial growth

Characterize all n-valued dynamics T: Y → Sym^n(Y) such that, for every y ∈ Y, the growth function ξ_y(r) = |Set(T^{r}(y))| is polynomial in r.

Background

The paper defines n-valued dynamics T on a space Y and the associated growth function ξ_y(r)=|Set(T^r(y))|, and relates dynamics induced by elements of n-valued groups to growth of the corresponding Cayley graphs. Using results on coset groups, the authors give partial answers and bounds for specific constructions.

They quote a question (originating from Buchstaber) asking for a full characterization of n-valued dynamics with polynomial growth of ξ_y(r) for all y, which remains unresolved in general beyond the specific cases treated in the paper.