Rational map $ax+1/x$ on the projective line over $\mathbb{Q}\_{p}$

Abstract

The dynamical structure of the rational map ax + 1/x on the projective line P 1 (Qp) over the field Qp of p-adic numbers is described for p ≥ 3.

Rational map ax+1/x on the projective line over $\mathbb{Q}_2$

“…Except for one subcase (|a| p < 1, √ a ∈ Q p ), the global picture of the dynamical structure of φ on P 1 (Q p ) for p ≥ 3 was given in [14]. In this paper, we deal with the case p = 2 and we successfully show the global dynamical structure of φ for all cases.…”

“…By Lemma 3.7, we immediately have φ(X a ) ⊂ X a and X a ⊂ F φ . By the argument of the proof of Proposition 3 in [14], we obtain a minimal decomposition for the subsystem (X a , φ) as stated in Theorem 1.1. It remains to show that for each x ∈ F φ , there exists some positive integer N such that φ N (x) ∈ X a .…”

“…However, the dynamical behaviors of higher degree rational maps are far from clear. In [14], we have started an attempt with a family of rational maps of degree 2. That is…”