Preperiodic points and unlikely intersections

TL;DR

This paper investigates the conditions under which complex dynamical systems and rational functions share infinitely many preperiodic points, linking complex analysis, arithmetic geometry, and dynamical systems to answer longstanding questions.

Contribution

It proves that for fixed a, b, and degree d, the set of c making both a and b preperiodic is infinite iff a^d = b^d, and shows that sharing infinitely many preperiodic points implies identical Julia sets.

Findings

Infinite preperiodic points occur iff a^d = b^d.

Shared infinite preperiodic points imply identical Julia sets.

Uses adelic equidistribution and Berkovich spaces in proofs.

Abstract

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed complex numbers a and b, and any integer d at least 2, the set of complex numbers c for which both a and b are preperiodic for z^d+c is infinite if and only if a^d = b^d. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if two complex rational functions f and g have infinitely many preperiodic points in common, then they must have the same Julia set. This generalizes a theorem of Mimar, who established the same result assuming that f and g are defined over an algebraic extension of the rationals. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role.