Abstract :
Let 𝑝 be prime and : x-- 𝑥g𝑥, the Discrete Lambert Map. For 𝑘≥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 (u) for each u 2 V. We denote this digraph by G(g, 𝑝^𝑘), where g 2 (Z/pkZ) . 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 𝑝𝑑, 1 d k − 1 then 𝐺(𝑔, 𝑝^𝑘) has 𝑝^𝑘−𝑑 𝑑 2 e loops. If g = tp + 1, 1 t 𝑝^𝑘−1 − 1 then the digraph contains 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.
