Periodic points and tail lengths of split polynomial maps modulo primes

TL;DR

This paper derives explicit formulas for counting periodic points and maximum tail lengths of split polynomial maps over finite fields, analyzing their graph structures and applying results to classify maps via cycle statistics.

Contribution

It provides new explicit formulas and structural analysis for split polynomial maps over finite fields, including Chebyshev polynomials and powering maps, with applications to map classification.

Findings

Formulas for periodic points and tail lengths

Structural analysis of graph for Chebyshev polynomials

Algorithm for map classification based on cycle statistics

Abstract

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 for Chebyshev polynomials of non-prime degree in dimension 1 and the powering map in any dimension. The results are applied to an algorithm for determining the type of a given map through analysis of its cycle statistics modulo primes.