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.
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