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.
For a morphism $f:P^N to P^N$, the points whose forward orbit by $f$ is finite are called preperiodic points for $f$. This article presents an algorithm to effectively determine all the rational preperiodic points for $f$ defined over a given number
field $K$. This algorithm is implemented in the open-source software Sage for $Q$. Additionally, the notion of a dynatomic zero-cycle is generalized to preperiodic points. Along with examining their basic properties, these generalized dynatomic cycles are shown to be effective.
Given an endomorphism of a projective variety, by intersecting the graph and the diagonal varieties we can determine the set of periodic points. In an effort to determine the periodic points of a given minimal period, we follow a construction similar
to cyclotomic polynomials. The resulting zero-cycle is called a dynatomic cycle and the points in its support are called formal periodic points. This article gives a proof of the effectivity of dynatomic cycles for morphisms of projective varieties using methods from deformation theory.
