Periodic points and tail lengths of split polynomial maps modulo primes

Explicit formulas are obtained for the number of periodic points and maximum tail length of split polynomial maps over finite fields for affine and projective space. This work includes a detailed analysis of the structure of the directed graph over finite fields for Chebyshev polynomials of nonprime degree in dimension 1 and the powering map in any dimension. The results are applied to provide an algorithm for determining the type of a given map defined over the rational numbers through analysis of its cycle statistics modulo primes.