Bound for preperiodic points for maps with good reduction

2016/08/20 by Troncoso, Sebastian · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1608.05849

Abstract

Let K be a number field and let ϕ in K(z) be a rational function of degree d≥ 2. Let S be the places of bad reduction for ϕ (including the archimedan places). Let Per(ϕ,K), PrePer(ϕ, K), and Tail(ϕ,K) be the set of K-rational periodic, preperiodic, and purely preperiodic points of ϕ, respectively. The present paper presents two main results. The first result gives a bound for |PrePer(ϕ,K)| in terms of the number of places of bad reduction |S| and the degree d of the rational function ϕ. This bound significantly improves a previous bound given by J. Canci and L. Paladino 2014. For the second result, assuming that |Per(ϕ,K)| ≥ 4 (resp. |Tail(ϕ,K)| ≥ 3), we prove bounds for |Tail(ϕ,K)| (resp. |Per(ϕ,K)|) that depend only on the number of places of bad reduction |S| (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when |Per(ϕ,K)| < 4 (resp. |Tail(ϕ,K)| < 3).

Cited by

• On the Number of Small Points for Rational Maps

Related

• Preperiodic points for rational functions defined over a rational function field of characteristic zero
• Preperiodic points for rational functions defined over a global field in\n terms of good reductions
• Scarcity of finite orbits for rational functions over a number field
• On preperiodic points of rational functions defined over \mathbbF_p(t)
• Distribution of preperiodic points in one-parameter families of rational maps