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We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$, where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$. Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $, which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.
We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$, then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $, $\alpha $ and V) such that for each positive integer M,
For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue.
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 advance a new conjecture in the spirit of the dynamical Manin–Mumford conjecture. We show that our conjecture holds for all polarisable endomorphisms of abelian varieties and for all polarisable endomorphisms of $(\mathbb{P}^{1})^{N}$. Furthermore, we show various examples which highlight the restrictions one would need to consider in formulating any general conclusion in the dynamical Manin–Mumford conjecture.
Let $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.
We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.
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].
Fix
$d\geqslant 2$
and a field
$k$
such that
$\operatorname{char}k\nmid d$
. Assume that
$k$
contains the
$d$
th roots of
$1$
. Then the irreducible components of the curves over
$k$
parameterizing preperiodic points of polynomials of the form
$z^{d}+c$
are geometrically irreducible and have gonality tending to
$\infty$
. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of
$z^{d}+c$
. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.
Let $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$. The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$-sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.
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 $K(x)^{\ast }/K(x)^{\ast m}$.
We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.
We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms
$\unicode[STIX]{x1D6F7}$
of
$(\mathbb{P}^{1})^{n}$
. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.
We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.
In this paper we prove the following theorem. Let
$f$
be a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic
$0$
. If there is no nonconstant invariant rational function under
$f$
, then there exists a closed point in the plane whose orbit under
$f$
is Zariski dense. This result gives us a positive answer to a conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang for polynomial endomorphisms of the affine plane.
For a finite field of odd cardinality
$q$
, we show that the sequence of iterates of
$aX^{2}+c$
, starting at
$0$
, always recurs after
$O(q/\text{log}\log q)$
steps. For
$X^{2}+1$
, the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of
$X^{3}+c$
, when
$q$
is 2 mod 3.
We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are ‘optimal’ in the following sense: they minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher-degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sen’s theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of the iterative residue.
Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.