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For a prime p, positive integers r , n , and a polynomial f with coefficients in p r , let W p , r , n (f) = f n p r \ f n + 1 p r . As n varies, the W p , r , n (f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on p r . In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of p r lying in a given W p , r , n (f). Moreover, when we generalize our definition of W p , r , n (f) , we obtain both upper and lower bounds for the resulting averages. [ABSTRACT FROM AUTHOR]

In discrete dynamical systems, the ultimate goal is to understand the asymptotic behavior of all points under iterated compositions of a certain transformation (self-map) of a certain space. In arithmetic dynamics, the asymptotic behavior of points with coordinates of arithmetic interest (algebraic numbers or p-adic numbers) is examined. In connection with arithmetic dynamics, some problems are reduced to the determination of rational points on curves. This article introduces the issues in arithmetic dynamics related to these problems. [ABSTRACT FROM AUTHOR]

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit Orb f (t) = { t , f (t) , f (f (t)) , ... } , where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer k ≥ 2 , we prove that it is impossible to cover Orb f (t) using k such arithmetic progressions unless Orb f (t) is contained in one of these progressions. In fact, we show that the relative density of terms covered by k such arithmetic progressions in Orb f (t) is uniformly bounded from above by a bound that depends solely on k. In addition, the latter relative density can be made as close as desired to 1 by an appropriate choice of k arithmetic progressions containing t if k is allowed to be large enough. [ABSTRACT FROM AUTHOR]

Working over a base number field K , we study the attractive question of Zariski non-density for (D , S) -integral points in O f (x) the forward f -orbit of a rational point x ∈ X (K). Here, f : X → X is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f -quasi-polarizable Cartier divisor D on X and defined over K , our main result gives a sufficient condition, that is formulated in terms of the f -dynamics of D , for non-Zariski density of certain dynamically defined subsets of O f (x). For the case of (D , S) -integral points, this result gives a sufficient condition for non-Zariski density of integral points in O f (x). Our approach expands on that of Yasufuku, [13] , building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13] ; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6]. [ABSTRACT FROM AUTHOR]

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring D [ x , σ , δ ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D [ x ]. In particular, we show that if a ∈ D and f ∈ D [ x , σ , δ ] satisfy f (a) = a , then f ∘ n (a) = a for every formal power of f. More generally, we give a sufficient condition for a point a to be r -periodic with respect to a polynomial f. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy. [ABSTRACT FROM AUTHOR]

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors. [ABSTRACT FROM AUTHOR]

Let be an odd prime, and let be the Shortened map, defined by if is even and if is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed -adic analysis, the study of functions from the -adics to the -adics, where and are distinct primes. In this, the first paper, working with the maps as a toy model for the more general theory, for each odd prime , we construct a function (the Numen of ) and prove the Correspondence Principle (CP): is a periodic point of if and only there is a so that . Additionally, if makes , then the iterates of under tend to or . [ABSTRACT FROM AUTHOR]

We examine the Galois groups of the extensions K ((f ′ ∘ f n) − 1 (0)) / K where K is a number field for polynomials f (x) ∈ K [ x ]. We use our understanding of this group to study the proportion of primes for which f has a p -adic attracting periodic point for a "typical" f and apply the result to the split case of the Dynamical Mordell-Lang Conjecture. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the dynamics of the unicritical polynomial family f d , c (z) = z d + c ∈ ℂ [ z ]. The c -values for which f d , c has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when d is a prime power and c is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of f d , c . We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases. [ABSTRACT FROM AUTHOR]

We construct many irreducible polynomials within semigroups generated by sets of the form S = { x 2 + c 1 , ... , x 2 + c s } under composition. [ABSTRACT FROM AUTHOR]

Let f be a polynomial, a , b be two algebraic numbers, and (a n) n ≥ 1 be a sequence satisfying f (a 1) = a and f (a n) = a n − 1 for all n ≥ 2. We generalize the concept of primitive prime divisors to the shifted sequence (a n − b) n ≥ 1 and study the generalized Zsigmondy set, that is, the set of n such that all prime ideal divisors of a n − b divide at least one prior term. The finiteness of Zsigmondy sets are determined in various cases. [ABSTRACT FROM AUTHOR]

The numerical hailstone sequences, or orbits, generated by the Collatz map have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n , the Collatz map f reads $f(n) = n/2$ ( $f(n) = 3 n + 1$ ) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended fluctuation, auto-correlation and entropy measure. Besides confirming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our findings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical system which could also be regarded as displaying key characteristics of an arithmetic statistical physics system, explaining its dynamical richness.

Let S be the collection of quadratic polynomial maps, and degree 2-rational maps whose automorphism groups are isomorphic to C 2 defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in S , we give a complete description of triples (f 1 , f 2 , p) such that p is a rational periodic point for both f i ∈ S , i = 1 , 2. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers a , b are periodic points of the map ϕ t 1 , t 2 (z) = t 1 z + t 2 / z for infinitely many nonzero rational pairs (t 1 , t 2) if and only if a 2 = b 2. [ABSTRACT FROM AUTHOR]

Let L d be the Lattès map associated to the multiplication-by- d endomorphism of an elliptic curve E defined over a finite field F q. We determine the density δ (L d , q) of periodic points for L d in P 1 (F q). We show that the periodic point densities δ (L d , q n) converge as n → ∞ along certain arithmetic progressions, and compute simple explicit formulas for δ (L ℓ , q) when ℓ is a prime and E belongs to a special family of supersingular elliptic curves. [ABSTRACT FROM AUTHOR]

The arboreal Galois group of a polynomial f over a field K encodes the action of Galois on the iterated preimages of a root point x 0 ∈ K , analogous to the action of Galois on the ℓ -power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial f (z) = z 2 − 1 when the field K and root point x 0 satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of f. For K = Q , our condition holds for infinitely many choices of x 0. [ABSTRACT FROM AUTHOR]

Fix an odd prime p. If r is a positive integer and f is a polynomial with coefficients in \mathbb {F}_{p^r}, let P_{p,r}(f) be the proportion of \mathbb {P}^1\left (\mathbb {F}_{p^r}\right) that is periodic with respect to f. We show that as r increases, the expected value of P_{p,r}(f), as f ranges over quadratic polynomials, is less than 22/\left (\log {\log {p^r}}\right). This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, we study rational periodic points of polynomial f d , c (x) = x d + c over the field of rational numbers, where d is an integer greater than two. For period two, we describe periodic points for degrees d = 4 , 6. We also demonstrate the nonexistence of rational periodic points of exact period two for d = 2 k such that 3 | 2 k − 1 and k has a prime factor greater than three. Moreover, assuming the a b c -conjecture, we prove that f d , c has no rational periodic point of exact period greater than one for sufficiently large integer d and c ≠ − 1. [ABSTRACT FROM AUTHOR]

Let F and G be integer polynomials where F has degree at least 2. Define the sequence (an) by an = F(an-1) for all n ≥ 1and a0 = 0. Let BF,G,k be the set of all positive integers nsuch that k | gcd(G(n), an) and if p | gcd(G(n), an) for some p, then p | k. Let AF,G,k be the subset of BF,G,k such that AF,G,k = {n ≥ 1 : gcd(G(n), an) =k}. In this article, we prove that the asymptotic density of AF,G,k and BF,G,k exists for a class of (F,G) and also compute the explicit density of AF,G,k and BF,G,k for G(x) = x. [ABSTRACT FROM AUTHOR]

We study intersections of semigroup orbits in polynomial dynamics with multiplicative subgroups and its approximations, extending results of Ostafe and Shparlinski (Orbits of algebraic dynamical systems in subgroups and subfields. Number theory: diophantine problems, Uniform Distribution and Applications. pp 347–368, 2010). [ABSTRACT FROM AUTHOR]

Given a field K and n > 1 , we say that a polynomial f ∈ K [ x ] has newly reducible n th iterate over K if f n − 1 is irreducible over K , but f n is not (here f i denotes the i th iterate of f). We pose the problem of characterizing, for given d , n > 1 , fields K such that there exists f ∈ K [ x ] of degree d with newly reducible n th iterate, and the similar problem for fields admitting infinitely many such f. We give results in the cases (d , n) ∈ { (2 , 3) , (3 , 2) , (4 , 2) } as well as for (d , 2) when d ≡ 2 mod 4. In particular, we show that for all these (d , n) pairs, there are infinitely many monic f ∈ ℤ [ x ] of degree d with newly reducible n th iterate over ℚ. Curiously, the minimal polynomial x 2 − x − 1 of the golden ratio is one example of f ∈ ℤ [ x ] with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions. [ABSTRACT FROM AUTHOR]

Let K be the function field of a smooth irreducible curve defined over Q. Let ƒ ∈ K[x] be of the form ƒ(x) = xq + c, where q = pr, r ≥ 1, is a power of the prime number p, and let β ∈ K. For all n ∈ N ∪∞, the Galois groups Gn(β) = Gal(K(ƒ−n(β))/K(β)) embed into [Cq]n, the n-fold wreath product of the cyclic group Cq. We show that if ƒ is not isotrivial, then [[Cq]∞ : G∞ (β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if ƒ1(x) = xq + c1 and ƒ2(x) = xq + c2 are two such distinct polynomials, then the fields ∪n=1∞ K(ƒ1−n(β)) and ∪n=1∞ K(ƒ2−n(β)) are disjoint over a finite extension of K. [ABSTRACT FROM AUTHOR]

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]. [ABSTRACT FROM AUTHOR]

Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree 4 0 2 0 , while for the 5 2   3 7 7 period-20 orbits the degree rises already to 1   0 4 7   5 4 0. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map. [ABSTRACT FROM AUTHOR]

Let φ be a quadratic, monic polynomial with coefficients in OF,D[t], where OF,D is a localization of a number ring OF. In this paper, we first prove that if φ is non-square and non-isotrivial, then there exists an absolute, effective constant Nφ with the following property: for all primes p ⊆ OF,D such that the reduced polynomial φp ∈ (OF,D/p)[t][x] is non-square and non-isotrivial, the squarefree Zsigmondy set of φp is bounded by Nφ. Using this result, we prove that if φ is non-isotrivial and geometrically stable, then outside a finite, effective set of primes of OF,D the geometric part of the arboreal representation of φp is isomorphic to that of φ. As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial x2 + t. [ABSTRACT FROM AUTHOR]

In this paper, algebraic relations were established that determined the invariance of a transformed number after several transformations. The restrictions that determine the group structure of these relationships were analyzed, as was the case of the Klein group. Parametric Kr functions associated with the existence of cycles were presented, as well as the role of the number of their links in the grouping of numbers in higher-order equivalence classes. For this, we developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.

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 $X=\mathbb{P}^{4}$. [ABSTRACT FROM AUTHOR]

Given a field K , a polynomial f ∈ K [ x ] of degree d , and a suitable element t ∈ K , the set of preimages of t under the iterates f ∘ n carries a natural structure of a d -ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d ≥ 20 is even and K = Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K = F (t) and f ∈ F [ x ] in which the corresponding Galois groups are the monodromy groups of the ramified covers f ∘ n : P F 1 → P F 1. [ABSTRACT FROM AUTHOR]

We give a short proof of an idempotent relation formula for counting periodic points of endomorphisms defined over finite fields. The original proof of this result, due to Walton, uses formal manipulation of arithmetic zeta functions, whereas we deduce the result directly from a related theorem of Kani and Rosen. [ABSTRACT FROM AUTHOR]

Let D be a residually finite Dedekind domain, a ∈ D be a nonzero element and n be a nonzero ideal of D. In this paper we describe the dynamics of the map x ↦ a x over the quotient ring D / n. We further present some applications of our main result. [ABSTRACT FROM AUTHOR]

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. [ABSTRACT FROM AUTHOR]

Let S = { x 2 + c 1 , x 2 + c 2 , ... , x 2 + c s } be a set of quadratic polynomials with rational coefficients, and let P be a rational basepoint. We classify the pairs (S , P) for which P has finite orbit for S , assuming that the maximum period length for each individual polynomial is at most three (conjectured by Poonen). In particular, under these hypotheses we prove that if s ≥ 4 , then there are no points P with finite orbit for S. Moreover, we use this perspective to formulate an analog of the Morton–Silverman Conjecture for sets of rational functions. [ABSTRACT FROM AUTHOR]

Let ρ be a Drinfeld 𝔽 q [ T ] -module defined over a global function field K. Let α ∈ K be a non-torsion point of ρ with infinite ρ T -orbit. For each n ≥ 1 , write the ideal (ρ T n (α)) = 𝔄 n 𝔅 n − 1 as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence { 𝔄 n } n ≥ 1 , i.e. for all but finitely many n ≥ 1 , there exists a prime ideal 𝔭 n such that 𝔭 n | 𝔄 n and 𝔭 n ∤ 𝔄 i for all i < n. [ABSTRACT FROM AUTHOR]

We give quantitative bounds for the number of quasi-integral points in orbits of semigroups of rational maps under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits gen- erated by the iterations of one rational map. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let φ ∊ K(z) have degree d ≥ 2. We characterize maps for which the minimal resultant of an iterate φn is given by a simple formula in terms of d, n, and the minimal resultant of φ. Three characterizations of such maps are given, one measure-theoretic and two in terms of the indeterminacy locus I(d) in the parameter space P2d+1 of (possibly degenerate) rational maps. As an application, we are able to give a new explicit formula involving the Arakelov-Green's function attached to '. We end by illustrating our results with some explicit examples. [ABSTRACT FROM AUTHOR]

Let ( F n ) n ≥ 0 be the Fibonacci sequence. The order of appearance z ( n ) of an integer n ≥ 1 is defined by z ( n ) = min { k ≥ 1 : n ∣ F k } . Marques, and Somer and Křížek proved that all fixed points of the function z ( n ) have the form n = 5 k or 12 · 5 k . In this paper, we shall prove that z ( n ) does not have any k-periodic points, for k ≥ 2 .

In this paper we study the geometric location of periodic points of power series defined over fields of positive characteristic p . We find a lower bound for the norm of all nonzero periodic points in the open unit disk of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of p -power iterates of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series. [ABSTRACT FROM AUTHOR]

Let K be a global function field and let φ(x) ∈ K[x]. For all wandering basepoints b ∈ K, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set Z(φ, b) that depends only on φ, the poles of the b, and K. Moreover, when we order b ∈ OK,S by height, we show that Z(φ, b) is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of φ(x) = xd + f is a finite index subgroup of an iterated wreath product of cyclic groups. In particular, since our methods translate to rational function fields in characteristic zero, we establish the inverse Galois problem for these groups via specialization. [ABSTRACT FROM AUTHOR]