We find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely, pure power maps $\phi (z) = z^d$ and Chebyshev polynomials.
"Periodic points in towers of finite fields for polynomials associated to algebraic groups." Rocky Mountain J. Math. 49 (1) 171 - 197, 2019. https://doi.org/10.1216/RMJ-2019-49-1-171