Integral points of bounded degree on the projective line and in dynamical orbits

2016/07/27 by Gunther, Joseph, Hindes, Wade
#11D45 #11G50 #11R04 #14G05 #37P15 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.08272

Abstract

Let D be a non-empty effective divisor on ℙ¹. We show that when ordered by height, any set of (D,S)-integral points on ℙ¹ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let φ(z)∈ ℚ(z) be a rational function of degree at least two whose second iterate φ²(z) is not a polynomial. We show that as we vary over points P∈ℙ¹(ℚ) of bounded degree, the number of algebraic integers in the forward orbit of P is absolutely bounded and zero on average.

Related

• The average number of integral points in orbits
• S-Integral Points in Orbits on ℙ¹
• Integer Points in Backward Orbits
• Integral points in two-parameter orbits
• On non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem