Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions $\mathbb R\to\mathbb R$. In particular, he proved that there exists a continuous function $\mathbb R\to \mathbb R$ which in some sense “interpolates” all sequences $(x_n)_{n\in\mathbb Z}\in [0,1]^{\mathbb Z}$ “simultaneously.” In 2005, R. Naulin M. and C. Uzcátegui unifyed and generalized Benyamini's results. In this paper, the case of topological spaces $X$ and $Y$ with an abelian group acting on $X$ is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings $X\to Y$ is posed. Further generalizations of Naulin–Uncátegui theorems, in particular, multidimensional analogues of Benyamini's results are obtained.