Vol. 15, No. 2, 2022

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Periodic points and tail lengths of split polynomial maps modulo primes

Benjamin Hutz and Teerth Patel

Vol. 15 (2022), No. 2, 185–206

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.

functional graph, periodic points, Chebyshev polynomial, power map
Mathematical Subject Classification 2010
Primary: 37P25, 37P35
Secondary: 11B50
Received: 11 September 2018
Revised: 28 July 2021
Accepted: 27 August 2021
Published: 29 July 2022

Communicated by Kenneth S. Berenhaut
Benjamin Hutz
Department of Mathematics and Statistics
Saint Louis University
St. Louis, MO
United States
Teerth Patel
Saint Louis University
St. Louis, MO
United States