Quasi-adelic measures and equidistribution on $\mathbb{P}^1$

TL;DR

This paper extends the concept of adelic measures to quasi-adelic measures on the projective line, proving equidistribution of small height points and exploring their properties in arithmetic dynamics.

Contribution

It introduces quasi-adelic measures, proves equidistribution results for them, and analyzes the rarity of adelic measures in dynamical systems on .

Findings

Equidistribution of small height points for quasi-adelic measures.

Canonical measures for certain dynamical pairs are rarely adelic.

Quasi-adelic measures arise naturally in families of rational functions.

Abstract

Baker-Rumely and Favre-Rivera-Letelier independently proved an important arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on $\mathbb{P}^1$. Around the same time, Chambert-Loir proved a more general version of this arithmetic equidistribution theorem in the setting of curves from a different approach. We generalize the notion of an adelic measure to that of a quasi-adelic measure on $\mathbb{P}^1$, and show that arithmetic equidistribution of points with small height holds for quasi-adelic measures as well. Moreover, we show that the canonical measure associated with a dynamical pair $(f,c)$ on $\mathbb{P}^1$ is rarely adelic. We prove that for certain examples of families of rational functions parameterized by $\mathbb{P}^1$, corresponding to the curve $\mathrm{Per}_1(\lambda)$ introduced by Milnor for a root of unity $\lambda$, the measure corresponding to a general starting point is quasi-adelic. Finally, we place our results in context by establishing their connection with two problems in arithmetic dynamics.