The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications

TL;DR

This paper provides a complete structural description of the functional graph of Chebyshev polynomial iterations over finite fields and applies these results to estimate various dynamical properties such as cycle lengths and component counts.

Contribution

It offers a comprehensive analysis of the functional graph structure of Chebyshev polynomials over finite fields, which was previously not fully understood.

Findings

Structural description of the functional graph of Chebyshev polynomials

Estimates for average rho length and number of connected components

Expected values for period and preperiod of polynomial iterations

Abstract

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.