Schedule for: 17w5081 - Workshop on Arithmetic and Complex Dynamics

We prove that the dynatomic curves associated with iteration of $z^d+c$ in any fixed characteristic (not dividing d) have gonalities tending to infinity. This implies a uniform upper bound on the number of L-rational preperiodic points of $z^d+c$ as L varies over extensions of bounded degree over a fixed function field K and c varies over nonconstant elements of L. It also reduces the strong uniform boundedness conjecture for preperiodic points over number fields to the conjecture for periodic points. This is joint work with John R. Doyle.

Consider a birational transformation $f$ of a projective variety $M$. Say that $f$ is undistorted if for every finitely generated subgroup $H$ of Bir($M$) that contains $f$ and every finite subset $S$ that generates $H$, the word length of $f^n$ with respect to the set of generators $S$ grows linearly with $n$; on the opposite, $f$ is distorted in $H$ if there is a sequence of iterates $f^{n_i}$ which can be written as a word of length $o(n_i)$ in the elements of $S$. The problem is to decide which $f$ in Bir($M$) are distorted : I will look at this problem when $M$ is a projective space, with a special emphasis on dimension 2.

This talk concerns joint work with Xander Faber. Let $K$ be a nonarchimedean local field of characteristic 0 and residue characteristic $p > 0.$ Let $q = p^f$ be the order of its residue field, and let $\mathbf{C}_K$ be the completion of an algebraic closure of $K$. The group of continuous automorphisms $Gal_c(\mathbf{C}_K/K)$ acts on the Berkovich Projective Line $\bf{P}^1_{\mathbf{C}_K}$. We show that the Galois invariant locus in $\bf{P}^1_{\mathbf{C}_K}$ is a densely branched tree which properly contains the path-closure of $\mathbf{P}^1(K)$, and is contained in a tube of path-distance radius $1/(p-1)*[1 + 1/(p-1)]$ around the path-closure. The radius can probably be improved to $ 1/(p-1)$. The Galois invariant locus has $q+1$ branches at each type II point in the locus corresponding to a disc $D(a,p^b) $, with $b$ rational, and no other branches. We construct a conjecturally dense subset of the Galois invariant locus. We also establish a conjecture of Benedetto, that each Galois invariant point is defined over a totally ramified extension of $K$.

Let $A$ be a non-isotrivial family of abelian varieties over a smooth irreducible curve $S$. Suppose the generic fiber of $A$ is simple and call $R$ its endomorphism ring. We consider an irreducible curve $C$ in the $n$-fold fibered power of $A$ and suppose that everything is defined over a number field $k$. Then $C$ defines $n$ points $P_1, ... P_n$ points on $A(k(C))$. Then, there are at most finitely many points $c$ on the curve such that the specialized $P_1(c), ... P_n(c)$ are dependent over $R$, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.

Let $A$ be a non-isotrivial family of abelian varieties over a smooth irreducible curve $S$. Suppose the generic fiber of $A$ is simple and call $R$ its endomorphism ring. We consider an irreducible curve $C$ in the $n$-fold fibered power of $A$ and suppose that everything is defined over a number field $k$. Then $C$ defines $n$ points $P_1, ... P_n$ points on $A(k(C))$. Then, there are at most finitely many points $c$ on the curve such that the specialized $P_1(c), ... P_n(c)$ are dependent over $R$, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.

We will discuss entropy of real and complex dynamics for automorphisms of rational surface that are birationally equivalent to quadratic birational maps of the plane and give examples where the entropy of the full (complex) automorphism is the same as its real restriction. This is a joint work with J. Diller.

Pick any family of degree $d>1$ polynomials $P_t$ parametrized by a complex irreducible affine curve $C$ and assume there exists two regular functions $a,b\in\mathbb{C}[C]$ such that $a(t)$ and $b(t)$ are both preperiodic under iteration of $P_t$ for infinitely many parameters $t\in C$. Baker and DeMarco conjectured that this implies the orbit of $a$ and $b$ are dynamically related in the whole family $P_t$ (and proved it when $C=\mathbb{A}^1$ is the affine line). In a joint work with Charles Favre, we prove a more precise version of this conjecture in the case of curves defined over a number field. The outline of the proof follows closely the one established by Baker and DeMarco in the case of family parametrized by the affine line. The main new ingredients are a rigidity phenomenon for marked points with a smooth bifurcation locus and the fact that a marked point always induces a continuous adelic metrization of an ample line bundle on $C$.

Let f:S^2 —> S^2 be a postcritically finite branched covering from the 2-sphere to itself with postcritical set P. Thurston studied the dynamics of f using an induced holomorphic self-map T(f) of the Teichmuller space of complex structures on (S^2, P). Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems: 1. T(f) always descends to a multivalued self map H(f) of the moduli space M_{0,P} of markings of the Riemann sphere by the finite set P 2. When P contains a point x at which f is fully ramified, under certain combinatorial conditions on f, the inverse of T(f) descends to a rational self-map M(f) of projective space CP^n. When, in addition, x is a fixed point of f, i.e. f is a `topological polynomial’, the induced self-map M(f) is regular. The dynamics of H(f) and M(f) may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties. I will introduce the dynamical systems T(f), H(f) and M(f), and dynamical degrees. I will then discuss why it is useful to study H(f) (resp. M(f)) simultaneously on several compactifications of M_{0,P}. We find that the dynamical degrees of H(f) (resp. M(f)) are algebraic integers whose properties are constrained by the dynamics of f on the finite set P. In particular, when M(f) exists, then the more f resembles a topological polynomial, the more M(f) : CP^n - - -> CP^n behaves like a regular map.

Rational functions of degree d > 1 in one variable are parametrized by a quasi-projective variety. From the point of view of arithmetic geometry, it is natural to study points on this variety through the machinery of Weil heights, while the dynamical interpretation suggests other measures of complexity, such as the "critical height" introduced by Silverman. This talk will present some recent work relating the critical height to Weil heights on moduli space, and then go on to suggest some further directions involving variants of the critical height.

We present several results about solenoidal manifolds motivated by results by Dennis Sullivan in [1] with commentaries developed in [2]. Solenoidal manifolds of dimension $n$ are topological spaces which are locally homeomorphic to the product of a Cantor set with an open subset of ${\mathbb R}^n$. We will give some applications to 3-dimensional manifolds. References: [1] D. Sullivan, Solenoidal manifolds, J. Singul. 9 (2014), 203–205. [2] A. Verjovsky, Commentaries on the paper “Solenoidal manifolds” by Dennis Sullivan. J. Singul. 9 (2014), 245–251.

Consider a rational map $R$ of degree $ d>1$ with coefficients over a non-archimedean field $\mathbb C_p$, with $p\geq 2$ a fixed prime number. If $R$ has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier (2000) that a new rational map $Q$ can be constructed from $R$, in such a way that $Q$ will exhibit a cycle of $m$-Herman rings. In this talk, we address the case of rational maps with bad reduction and provide an extension of Rivera-Letelier's result for this case. This is a joint work with Victor Nopal-Coello (CIMAT).

I shall explain how the study of positivity properties of numerical cycles allows one to understand the asymptotic behaviour of sequences of degrees of iterates of rational maps over any normal projective variety.

In this talk, firstly I would like to survey some arithmetic properties of Henon maps. Then I would like to talk about joint work in progress with Liang-Chung Hsia on some arithmetic properties of a certain family of Henon maps.

I will present a recent joint work with S. Vishkautsan where we provide an explicit bound on the number of periodic points of a rational function of degree at least 2 defined over a number field. The bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. We show that under stronger assumptions (but not so strong, in terms of ramification) the dependence on the degree of the map in the bounds can be removed.

A description of the equidistribution properties of Hecke correspondances acting on the moduli space of p-adic elliptic curves. The most difficult case, of elliptic curves with supersingular reduction, is analyzed using Lubin-Katz theory of the canonical subgroup, and the Gross-Hopkins period map coming from Lubin-Tate and Serre's theory of deformations of Abelian varieties. A key ingredient is a p-adic version Linnik's equidistribution theorem. This is a joint work with Sebastian Herrero and Ricardo Menares