# Combinatorics of Cycle Lengths on Wehler K3 Surfaces Over Finite Fields

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 45, (2015), **Pages** 19-31

**João Alberto de Faria**

Department of Mathematical Sciences

Florida Institute of Technology

150 W. University Blvd

Melbourne, FL 32901

mailto:jdefaria2010@my.fit.edu

**Benjamin Hutz **

Department of Mathematical Sciences

Florida Institute of Technology

150 W. University Blvd

Melbourne, FL 32901

**Abstract** We study the dynamics of maps arising from the composition of two non-commuting involutions 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.

**Keywords** Wehler K3 surface, reversible maps, dynamical systems, cycle lengths.

**Classification** (MSC2000) 37P55, 37E30, 37C80, 11T99.