%0 Journal Article %A Mahmood, M. Kh. %A Anwar, L. %T The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations %J Iranian Journal of Mathematical Sciences and Informatics %V 17 %N 2 %U http://ijmsi.ir/article-1-1499-en.html %R 10.52547/ijmsi.17.2.307 %D 2022 %K Fixed points, Lambert map, Multiplicative order., %X Let $p$ be prime and $alpha:x mapsto xg^x$, the Discrete Lambert Map. For $kgeq 1,$ let $ V = {0, 1, 2, . . . , p^k-1}$. The iteration digraph is a directed graph with $V$ as the vertex set and there is a unique directed edge from $u$ to $alpha(u)$ for each $uin V.$ We denote this digraph by $G(g, p^{k}),$ where $gin (mathbb{Z}/p^kmathbb{Z})^*.$ In this piece of work, we investigate the structural properties and find new results modulo higher powers of primes. We show that if $g$ is of order $p^{d} ,1leq d leq k-1$ then $G(g, p^k)$ has $ p^{k-lceil frac{d}{2}rceil} $ loops. If $g = tp+1,~1leq t leq p^{k-1}-1$ then the digraph contains $frac{p^k+1}{2}$ cycles. Further, if g has order $p^{k -1}$ then $G(g, p^{k})$ has $p-1$ cycles of length $p^{k-1}$ and the digraph is cyclic. We also propose explicit formulas for the enumeration of components. %> http://ijmsi.ir/article-1-1499-en.pdf %P 307-314 %& 307 %! %9 Research paper %L A-10-4348-1 %+ Department of Mathematics, University of the Punjab, Lahore, Pakistan %G eng %@ 1735-4463 %[ 2022