Graphs associated with the map $x \mapsto x+x^{-1}$ in finite fields of characteristic two

TL;DR

This paper investigates the structure of graphs generated by iterating the map x ↦ x + x^{-1} over finite fields of characteristic two, revealing formulas for cycle lengths and tree depths based on algebraic properties.

Contribution

It provides new formulas for cycle lengths and tree depths in these graphs, connecting them to Koblitz curve groups and Kloosterman sum congruences.

Findings

Formulas for cycle lengths of the graphs

Formulas for tree depths in the graphs

Connection to Koblitz curve structures and Kloosterman sums

Abstract

In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic two. Formulas are given for the length of the cycles and the depth of the trees relying upon the structure of the group of the rational points of Koblitz curves and the congruences of Kloosterman sums modulo powers of 2.