Niven’s theorem asserts that $\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap \mathbb {Q}=\{0,\pm 1,\pm 1/2\}.$ In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set $\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap K$, where K is an arbitrary number field.

We study the dynamics of the map $f:\mathbb {A}^N\to \mathbb {A}^N$ defined by

$$ \begin{align*} f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b}, \end{align*} $$

for $A\in \operatorname {SL}_N$ , $\mathbf {b}\in \mathbb {A}^N$ , and $d\geq 2$ , a class which specializes to the unicritical polynomials when $N=1$ . In the case $k=\mathbb {C}$ we obtain lower bounds on the sum of Lyapunov exponents of f, and a statement which generalizes the compactness of the Mandelbrot set. Over $\overline {\mathbb {Q}}$ we obtain estimates on the critical height of f, and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.

We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those that vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].

Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$. For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$-orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$, which measures the growth rate of the heights of the points $f^{n}(P)$. Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode[STIX]{x1D6FC}_{f}(P)$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $X=\mathbb{P}^{4}$.

Let $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$, we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$-th powers. For $m\geqslant 5$ we show that the only such functions are those of the form $cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$ with $\unicode[STIX]{x1D713}\in K(x)$, and for $m\leqslant 4$ we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.

With additional analysis, we show that the index set $\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$ is a union of finitely many arithmetic progressions, where $\unicode[STIX]{x1D719}^{n}$ denotes the $n$-th iterate of $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D706}\in K(x)$ is any map Möbius-conjugate over $K$ to $x^{m}$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^{m}=\unicode[STIX]{x1D719}^{n}(x)$. We describe all $\unicode[STIX]{x1D719}$ for which these curves have an irreducible component of genus at most 1, and show that such $\unicode[STIX]{x1D719}$ must have two distinct iterates that are equal in $K(x)^{\ast }/K(x)^{\ast m}$.

Given two monic polynomials $f$ and $g$ with coefficients in a number field $K$, and some $\alpha \,\in \,K$, we examine the action of the absolute Galois group $Gal\left( \bar{K}/K \right)$ on the directed graph of iterated preimages of $\alpha $ under the correspondence $g\left( y \right)\,=\,f\left( x \right)$, assuming that $\deg \left( f \right)\,>\,\deg \left( g \right)$ and that $\gcd \left( \deg \left( f \right),\deg \left( g \right) \right)\,=1$. If a prime of $K$ exists at which $f$ and $g$ have integral coefficients and at which $\alpha $ is not integral, we show that this directed graph of preimages consists of finitely many $Gal\left( \bar{K}/K \right)$-orbits. We obtain this result by establishing a $p$-adic uniformization of such correspondences, tenuously related to Böttcher’s uniformization of polynomial dynamical systems over $\mathbb{C}$, although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

The morphism $f\,:\,{{\mathbb{P}}^{N}}\,\to \,{{\mathbb{P}}^{N}}$ is called post-critically finite $\left( \text{PCF} \right)$ if the forward image of the critical locus, under iteration of $f$ , has algebraic support. In the case $N\,=\,1$ , a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattés maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of ${{\mathbb{P}}^{N}}$ for any $N$ .

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case.