Preperiodic points, finiteness, and structures of semigroups of algebraic morphisms

2025/08/12 by Jason P. Bell, Bell, Jason P., Thomas J. Tucker +1
Computer Science · Mathematics · #20D15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT) #Primary: 20M05 #Secondary: 14H37 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2508.09114

openalex publication_date 2025/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we explore a variety of finiteness questions for preperiodic points of morphisms. We begin by treating a group action analog of the Burnside problem for torsion groups using the p-adic arc method. We then prove some results connecting commonality of preperiodic points for elements of an automorphism group with structural properties of the group; these results are related to well-known results of Tits and Borel. We finish by proving some Northcott-type results for finite morphisms.

Citations

• A uniform Tits alternative for endomorphisms of the projective line
• Uniform Bounds for Periods of Endomorphisms of Varieties
• Applications of p-adic analysis for bounding periods of subvarieties\n under etale maps
• Periods of rational maps modulo primes
• The dynamical Mordell-Lang problem for etale maps
• A generalised Skolem-Mahler-Lech theorem for affine varieties
• Boundedness results for periodic points on algebraic varieties
• The canonical arithmetic height of subvarieties of an abelian variety over a finitely generated field
• Arithmetic height functions over finitely generated fields

Related

• Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height
• Determination of all rational preperiodic points for morphisms of PN
• On algebraic integers of bounded house and preperiodicity in polynomial\n semigroup dynamics
• S-integral preperiodic points for monomial semigroups over number fields
• Rigidity and unlikely intersections for formal groups