Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height

TL;DR

This paper investigates the distribution and boundedness of preperiodic points in affine space under semigroup morphisms, extending previous results to higher dimensions and cyclotomic closures.

Contribution

It generalizes prior work by proving non-density of preperiodic points on cyclotomic closures and extends boundedness results for house and height in higher-dimensional semigroup dynamics.

Findings

Preperiodic points on cyclotomic closures are not dense under certain conditions.

Boundedness of house and height is established for higher-dimensional preperiodic sets.

Results extend previous work to more general affine space morphisms.

Abstract

Under special conditions, we prove that the set of preperiodic points for semigroups of self-morphisms of affine spaces falling on cyclotomic closures is not dense. generalising results of Ostafe and Young (2020). We also extend previous results about boundness of house and height on certain preperiodicity sets of higher dimension in semigroup dynamics.