Periodic points and tail lengths of split polynomial maps modulo primes

Hutz, Benjamin; Patel, Teerth

doi:10.2140/involve.2022.15.185

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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

DOI: 10.2140/involve.2022.15.185

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 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.

Keywords

functional graph, periodic points, Chebyshev polynomial, power map

Mathematical Subject Classification 2010

Primary: 37P25, 37P35

Secondary: 11B50

Milestones

Received: 11 September 2018

Revised: 28 July 2021

Accepted: 27 August 2021

Published: 29 July 2022

Communicated by Kenneth S. Berenhaut

Authors

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

Vol. 15, No. 2, 2022