Let q be a power of a prime p, let $\mathbb F_q$ be the finite field with q elements and, for each nonconstant polynomial $F\in \mathbb F_{q}[X]$ and each integer $n\ge 1$, let $s_F(n)$ be the degree of the splitting field (over $\mathbb F_q$) of the iterated polynomial $F^{(n)}(X)$. In 1999, Odoni proved that $s_A(n)$ grows linearly with respect to n if $A\in \mathbb F_q[X]$ is an additive polynomial not of the form $aX^{p^h}$; moreover, if q = p and $B(X)=X^p-X$, he obtained the formula $s_{B}(n)=p^{\lceil \log_p n\rceil}$. In this paper we note that $s_F(n)$ grows at least linearly unless $F\in \mathbb F_q[X]$ has an exceptional form and we obtain a stronger form of Odoni’s result, extending it to affine polynomials. In particular, we prove that if A is additive, then $s_A(n)$ resembles the step function $p^{\lceil \log_p n\rceil}$ and we indeed have the identity $s_A(n)=\alpha p^{\lceil \log_p \beta n\rceil}$ for some $\alpha, \beta\in \mathbb Q$, unless A presents a special irregularity of dynamical flavour. As applications of our main result, we obtain statistics for periodic points of linear maps over $\mathbb F_{q^i}$ as $i\to +\infty$ and for the factorization of iterates of affine polynomials over finite fields.

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

Given a number field $\mathbb {K} \subset \mathbb {C}$ that is not contained in $\mathbb {R}$, we prove the existence of a dense set (with respect to the topology of local uniform convergence) of entire maps $f \colon \mathbb {C} \rightarrow \mathbb {C}$ whose preperiodic points and multipliers all lie in $\mathbb {K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb {Q}$.

Given a self-morphism $\phi$ on a projective variety defined over a number field k, we prove two results which bound the largest iterate of $\phi$ whose evaluation at P is quasi-integral with respect to a divisor D, uniformly across P defined over a field of bounded degree over k. The first result applies when the pullback of D by some iterate of $\phi$ breaks up into enough irreducible components which are numerical multiples of each other. The proof uses Le’s algebraic-point version of a result of Ji–Yan–Yu, which is based on Schmidt subspace theorem. The second result applies more generally but relies on a deep conjecture by Vojta for algebraic points. The second result is an extension of a recent result of Matsuzawa, based on the theory of asymptotic multiplicity. Both results are generalisations of Hsia–Silverman, which treated the case of morphisms on ${\mathbb{P}}^1$.

In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda (x):=x^d+\lambda $ (parameterized by $\lambda \in \mathbb {C}$), given two starting points a and b in $\mathbb {C}$, if there exist infinitely many $\lambda \in \mathbb {C}$ such that both a and b are preperiodic under the action of $f_\lambda $, then $a^d=b^d$. In this paper, we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in ${\overline {\mathbb F}_p}$; (ii) d is a power of p; and (iii) not both a and b live in ${\overline {\mathbb F}_p}$, while d is not a power of p. Only in case (iii), one derives the same conclusion as in characteristic $0$ (i.e., that $a^d=b^d$). In case (i), one has that for each $\lambda \in {\overline {\mathbb F}_p}$, both a and b are preperiodic under the action of $f_\lambda $, while in case (ii), one obtains that also whenever $a-b\in {\overline {\mathbb F}_p}$, then for each parameter $\lambda $, we have that a is preperiodic under the action of $f_\lambda $ if and only if b is preperiodic under the action of $f_\lambda $.

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.

We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let $B_1,\dots ,B_r\in \mathrm {GL}_m(K)$ be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of $\mathrm {GL}_m(K)$ not passing through zero. Then there exist only finitely many elements of $\mathrm {GL}_m(K)$ of the form $B_1^{n_1}\cdots B_r^{n_r}$ (as we vary $n_1,\dots ,n_r$ in $\mathbb {Z}$) lying on the subvariety V.

In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$, where p is a prime number and where the orbit of $0$ is finite. For example, if $p=2$ and $0$ is periodic under $T^2+c$ with $c\in \mathbb {R}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in $\mathbb {Q}(y)[T]$.

We show that for every finite set of prime numbers $S$, there are at most finitely many singular moduli that are $S$-units. The key new ingredient is that for every prime number $p$, singular moduli are $p$-adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$-invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k} = \mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$, generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates $f^n$ grows geometrically fast with the degree $d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of $\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial $f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.

The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math. 1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J. 162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

Baker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

For any algebraically closed field K and any endomorphism f of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of f are the Möbius transformations that commute with f, and these form a finite subgroup of $\operatorname{PGL}_2(K)$ . In the moduli space of complex dynamical systems, the locus of maps with nontrivial automorphisms has been studied in detail and there are techniques for constructing maps with prescribed automorphism groups that date back to Klein. We study the corresponding questions when K is the algebraic closure $\bar{\mathbb{F}}_p$ of a finite field. We use the classification of finite subgroups of $\operatorname{PGL}_2(\bar{\mathbb{F}}_p)$ to show that every finite subgroup is realizable as an automorphism group. To construct examples, we use methods from modular invariant theory. Then, we calculate the locus of maps over $\bar{\mathbb{F}}_p$ of degree 2 with nontrivial automorphisms, showing how the geometry and possible automorphism groups depend on the prime p.

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic $0$ , endowed with a birational self-map $\phi $ of dynamical degree $1$ , we expect that either there exists a nonconstant rational function $f:X\dashrightarrow \mathbb {P} ^1$ such that $f\circ \phi =f$ , or there exists a proper subvariety $Y\subset X$ with the property that, for any invariant proper subvariety $Z\subset X$ , we have that $Z\subseteq Y$ . We prove our conjecture for automorphisms $\phi $ of dynamical degree $1$ of semiabelian varieties X. Moreover, we prove a related result for regular dominant self-maps $\phi $ of semiabelian varieties X: assuming that $\phi $ does not preserve a nonconstant rational function, we have that the dynamical degree of $\phi $ is larger than $1$ if and only if the union of all $\phi $ -invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation-theoretic questions about twisted homogeneous coordinate rings associated with abelian varieties.

Define the Collatz map ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ on the positive integers $\mathbb {N}+1 = \{1,2,3,\dots \}$ by setting ${\operatorname {Col}}(N)$ equal to $3N+1$ when N is odd and $N/2$ when N is even, and let ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ denote the minimal element of the Collatz orbit $N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $ . The infamous Collatz conjecture asserts that ${\operatorname {Col}}_{\min }(N)=1$ for all $N \in \mathbb {N}+1$ . Previously, it was shown by Korec that for any $\theta> \frac {\log 3}{\log 4} \approx 0.7924$ , one has ${\operatorname {Col}}_{\min }(N) \leq N^\theta $ for almost all $N \in \mathbb {N}+1$ (in the sense of natural density). In this paper, we show that for any function $f \colon \mathbb {N}+1 \to \mathbb {R}$ with $\lim _{N \to \infty } f(N)=+\infty $ , one has ${\operatorname {Col}}_{\min }(N) \leq f(N)$ for almost all $N \in \mathbb {N}+1$ (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a $3$ -adic cyclic group $\mathbb {Z}/3^n\mathbb {Z}$ at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.

We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

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.