On some issues concerning polynomial cycles

summary:We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain $R$ of positive characteristic (for $N\ge 1$) or for any Dedekind domain $R$ of positive characteristic (but only for $N\ge 2$), we give a closed formula for a set ${\cal CYCL}(R,N)$ of all possible cycle-lengths for polynomial mappings in $R^N$. Then we give a new property of sets ${\cal CYCL}(R,1)$, which refutes a kind of conjecture posed by W. Narkiewicz