### A local-global criterion for dynamics on ℙ¹

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We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages...

Let $\varphi :X\to X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${\mathcal{O}}_{\varphi}\left(P\right)=\{{\varphi}^{n}\left(P\right):n\ge 0\}$ be the orbit of a point $P\in X\left(K\right)$. We describe a local-global principle for the intersection $V\cap {\mathcal{O}}_{\varphi}\left(P\right)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V\left(K\right)$ are Brauer–Manin unobstructed for power maps on ${\mathbb{P}}^{2}$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical...

We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials x²+c whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only integer value with this property is c=3, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational...

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...