Power commuting and centralizing maps on the ring of strictly upper triangular matrices

Let $N_n(F)$ denote the ring of strictly upper triangular matrices with entries in a field $F$ of characteristic zero and center $Z(N_n(F))$. We characterize the $2$-power commuting maps over $N_n(F)$, maps satisfying the identity $[f(X),X^2]=0$ for all $X\in N_n(F)$. As a consequence, we also obtain a characterization of the maps centralizing maps over $N_n(F)$, maps satisfying $[f(X),X]\in Z(N_n(F))$ for all $X\in N_n(F)$.