Counting the number of 2-periodic O_K-points of a discrete dynamical system with applications from arithmetic statistics, V

2025/08/22 by Kintu, Brian
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.16393

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of 2-periodic points of a polynomial map φ_{d, c} defined by φ_{d, c}(z) = z^d + c for all c, z ∈ O_K and the coefficient c, where K is any number field of degree n≥ 2 and d>2 is an integer. As before, we again study here counting problems that are inspired by the exciting advances on 2-torsion point-counting in arithmetic statistics and on 2-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any given prime p≥ 3 and any integer ℓ ≥ 1, the average number of distinct 2-periodic integral points of any φ_{p^ℓ, c} modulo prime ideal pO_K is bounded (if ℓ ∈ ℤ⁺∖ \1,p\) or zero or unbounded (if ℓ ∈ \1,p\) as c tends to infinity. Motivated further by K-rational periodic point-counting work of Benedetto, along with a conjecture of Hutz on 2-periodic rational points of any φ_{(p-1)^ℓ, c} for any given prime p≥ 5 and any integer ℓ ≥ 1 in arithmetic dynamics, we then also prove that the average number of distinct 2-periodic integral points of any φ_{(p-1)^ℓ, c} modulo prime ideal pO_K is 1 or 2 or 0 as c tends to infinity. Finally, we then also apply here density and number field-counting results from arithmetic statistics, and as a result obtain again counting and statistical results on the irreducible monic integer polynomials and algebraic number fields arising naturally in our dynamical setting.

Citations

• Counting the number of 2-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV
• Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I
• Galois groups of random integer polynomials and van der Waerden's Conjecture
• Rational Periodic Points of x^d+c and Fermat-Catalan Equations
• Monogenic fields with odd class number Part II: even degree
• Upper bounds on number fields of given degree and bounded discriminant
• Squarefree values of polynomial discriminants I
• Sato-Tate equidistribution of certain families of Artin L-functions
• Families of L-functions and their Symmetry
• Determination of all rational preperiodic points for morphisms of PN
• On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps
• Preperiodic points of polynomials over global fields
• Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

Related

• Counting the number of 2-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV
• Counting the number of n-periodic ℤ_p-and \mathbbF_p[t]-points of a discrete dynamical system with applications from arithmetic statistics, VI
• Counting the number of O_K-fixed points of a discrete dynamical system with applications from arithmetic statistics, II
• Counting the number of ℤ_p-and \mathbbF_p[t]-fixed points of a discrete dynamical system with applications from arithmetic statistics, III
• Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I