Combinatorics of cycle lengths on Wehler K3 Surfaces over finite fields

TL;DR

This paper investigates the cycle length distribution of dynamical maps derived from involutions on Wehler K3 surfaces over finite fields, extending previous combinatorial results to more general cases including degenerate fibers.

Contribution

It demonstrates that the dynamics of these K3 surfaces meet Roberts and Vivaldi's hypotheses, providing a detailed cycle distribution analysis, even with degenerate fibers included.

Findings

Cycle distribution matches Roberts and Vivaldi's combinatorial predictions.

Extension of involution analysis to cases with degenerate fibers.

Provides explicit descriptions of cycle lengths over finite fields.

Abstract

We study the dynamics of maps arising from the composition of two non-commuting involution on a K3 surface. These maps are a particular example of reversible maps, i.e., maps with a time reversing symmetry. The combinatorics of the cycle distribution of two non-commuting involutions on a finite phase space was studied by Roberts and Vivaldi. We show that the dynamical systems of these K3 surfaces satisfy the hypotheses of their results, providing a description of the cycle distribution of the rational points over finite fields. Furthermore, we extend the involutions to include the case where there are degenerate fibers and prove a description of the cycle distribution in this more general situation.