On the iterations of certain maps $x \mapsto k \cdot(x+x^{-1})$ over finite fields of odd characteristic

TL;DR

This paper investigates the behavior of specific rational maps over finite fields of odd characteristic, focusing on their iterative dynamics and how they evolve under repeated application.

Contribution

It provides a detailed analysis of the iteration patterns of maps of the form $k imes (x + x^{-1})$ over finite fields of odd characteristic, a topic not extensively explored before.

Findings

Characterization of the cycle structure of the maps

Identification of fixed points and periodic cycles

Insights into the distribution of orbits under iteration

Abstract

In this paper we describe the dynamics of certain rational maps of the form $k \cdot (x+x^{-1})$ over finite fields of odd characteristic.