Effective divisors on the projective line having small diagonals and small heights and their application to adelic dynamics

Abstract

Abstract. We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small g-heights with respect to an adelic normalized weight g in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function f ∈ k(z) of degree > 1 and a rational function a ∈ k(z) of degree > 0 over a product formula field k of characteristic 0, applying this quantitative adelic equidistribution result to adelic dynamics of f .

A Mahler-type estimate of weighted Fekete sums on the Berkovich projective line

“…Remark 4.1. An estimate similar to (4.1) was obtained in [16,Lemma 3.2], where for archimedean K, the definition of [z] ǫ was slightly different (or, more precisely, [z] ǫ was defined to be more smooth for archimedean K).…”

“…Once (4.1) is at our disposal, (2.6) will follow by a computation similar to that in the proof of [16,Lemma 6.1].…”

“…For the foundation of the potential theory on P 1 for non-archimedean K, see Baker-Rumely [2], Favre-Rivera-Letelier [9], Thuillier [17], and also Jonsson [13]. In the following, we adopt the notation in [16]. Notation 1.1 (Potential theory on P 1 ).…”

Value distribution of derivatives in polynomial dynamics

“…For more details on canonical heights on P 1 , see [1,2,9,13]. For the treatment of effective divisors rather than Galois conjugacy classes, which are effective divisors represented by irreducible polynomials, see [21].…”

“…The proof is based on an adelic equidistribution result for effective divisors on P 1 (k) having small diagonals and small heights [21].…”