Counting fixed points and two-cycles of the singular map x ↦ x^(x^n)
The "self-power" map x↦x^x modulo m and its generalized form x↦x^(x^n) modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel's lemma, and lifting singular points modulo p, to count fixed points and two-cycles of equations related to these maps when m is a prime power.
Holden, Joshua; Richardson, Pamela A.; and Robinson, Margaret M., "Counting fixed points and two-cycles of the singular map x ↦ x^(x^n)" (2016). Mathematical Sciences Technical Reports (MSTR). 159.