A uniform Tits alternative for endomorphisms of the projective line

TL;DR

The paper addresses growth behavior in finitely generated semigroups of endomorphisms on projective varieties, focusing on polarized endomorphisms and their preperiodic points. Its main innovation is a height-based ping-pong method that yields a free semigroup of rank $2$ whenever two endomorphisms share the same polarization and have distinct preperiodic sets, enabling uniform statements about growth. It proves that for any semigroup $S$ of endomorphisms semi-polarized by a common line bundle, if two degree-$\ge 2$ elements have different $\mathrm{PrePer}$ sets, then the diameter of independence satisfies $\Delta(S)\le 2$, implying $\Sigma(S)\ge \log(2)/2$ and hence uniform exponential growth; in characteristic $0$ on $\mathbb{P}^{1}$, this yields a dichotomy: polynomial growth or finite $\Delta(S)$ and thus uniform exponential growth for exponential-growth semigroups. The approach blends height theory and dynamical systems (via canonical heights $h_{f}$ and their contraction maps $\alpha_{f}$ on $\mathcal{H}_{\mathcal{L}}$) with a refined ping-pong argument, and has concrete consequences for End$(\mathbb{P}^{1})$ through the linear-case Breuillard–Gelander results, contributing a uniform perspective on growth for dynamical semigroups.

Abstract

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions. Extending this method, we obtain a uniform version of the same alternative. In particular, we show that semigroups of $\mathrm{End}(\mathbb{P}^{1})$ of exponential growth are of uniform exponential growth.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Remark 2.2
• proof : Proof of Theorem \ref{['uni1']}
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['uni2']}