# JMM 2016: Special Session on Arithmetic Dynamics

## Abstracts of Invited Talks (Alphabetical by Speaker)

 11-917 Robert Benedetto Non-archimedean connected Julia sets with branching Let $$K$$ be a complete and algebraically closed non-archimedean field, such as the $$p$$-adic field $${\mathbb C}_p$$. A rational function $$f(z)\in K(z)$$ acts on the Berkovich projective line $${\mathbb P}^1_{\text{Ber}}$$ over $$K$$. The Julia set of $$f$$ is a certain closed subset of $${\mathbb P}^1_{\text{Ber}}$$ that is invariant under application of $$f$$. Until recently, the only known examples of non-archimedean Julia sets were either disconnected, homeomorphic to an interval, or just a single point. In this talk, we describe some recent constructions of functions $$f$$ with non-archimedean connected Julia sets that are connected but much more complicated. As a result, these functions exhibit properties not previously seen both as regards their associated local canonical heights and their entropy. No prior knowledge of Berkovich spaces or entropy will be assumed for this talk. (PDF Abstract) 11-1804 T. Alden Gassert Discriminants of iterated quadratic extensions Let $$f(x)=x^2+c\in\mathbb{Z}[x]$$, and let $$K$$ be a number field generated by a root of $$f^n(x)$$ (assuming $$f^n(x)$$ is irreducible). The purpose of this talk is to determine the multiplicities of primes dividing the discriminant of $$K$$. As a consequence of our result, we identify a sufficient condition for $$K$$ to be monogenic. Namely, $$K$$ is monogenic if $$f(0),f^2(0),f^3(0),\ldots,f^n(0)$$ are all square-free. (PDF Abstract) 37-2433 William Gignac A nonarchimedean approach to local holomorphic dynamics in dimension two Let $$f$$ be a rational endomorphism of a complex algebraic surface $$X$$, and suppose that $$f$$ has a fixed point $$x$$. Analyzing the dynamics of $$f$$ near such a fixed point is often an essential step in understanding the global dynamical behavior of $$f$$ on $$X$$. In this talk, I will describe a nonarchimedean approach to analyzing the local dynamics in the case when $$f$$ is noninvertible near $$x$$. Instead of considering directly the dynamics of $$f$$ near $$x$$ in $$X$$, we will instead equip the field of complex numbers with the trivial absolute value and study the local dynamics of $$f$$ near $$x$$ in the corresponding Berkovich analytification of $$X$$. This will allow us to understand the dynamics of $$f$$ on certain birationally equivalent models of $$X$$, and in turn deduce concrete information about the original (archimedean) dynamical system. Our main application is that one can almost always find modifications of $$X$$ over $$x$$ on which $$f$$ exhibits a desirable "algebraic stability" property. (PDF Abstract) 11-593 Wade Hindes The average number of integral points in orbits Over a number field $$K$$, a celebrated result of Silverman's states that if $$\phi\in K(x)$$ is a rational function whose second iterate is not a polynomial, then the set of $$S$$-integral points in the orbit $$\mathcal{O}_\phi(b)=\bigl\{\phi^n(b)\bigr\}_{n\ge0}$$ is finite for all $$b\in\mathbb{P}^1(K)$$. In this talk, we show that if we vary $$\phi$$ and $$b$$ in a suitable family, the number of $$S$$-integral points of $$\mathcal{O}_\phi(b)$$ is absolutely bounded. In particular, if we fix $$\phi\in K(x)$$ and vary the base point $$b$$, we show that $$\#\bigl(\mathcal{O}_\phi(b)\cap\mathcal{O}_{K,S}\bigr)$$ is zero on average. Finally, we prove an analogous averaging result in general, assuming a standard conjecture in arithmetic geometry, and prove it unconditionally over global function fields. (PDF Abstract) 37-1810 Benjamin Hutz Automorphism Groups and Invariant Theory on $$\mathbb{P}^N$$ Let $$K$$ be a field and $$f:\mathbb{P}^N\to\mathbb{P}^N$$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $$\operatorname{PGL}_{N+1}$$. The group of automorphisms, or stabilizer group, of a given $$f$$ for this action is known to be a finite group. In this talk, we discuss a mainly computational problem concerning automorphism groups: Given a finite subgroup of $$\operatorname{PGL}_{N+1}$$ determine endomorphisms of $$\mathbb{P}^N$$ with that group as subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. (PDF Abstract) 11-1935 Patrick Ingram Canonical heights and preperiodic points for a special class of polynomials To each (non-linear) polynomial defined of a number field is a canonical height function, which vanishes precisely at points with finite orbit under that map. This function has a smallest positive value, and it is natural to ask for some lower bound on this quantity in terms of some data about the polynomial. We present some results in this direction for certain families generalizing or related to the unicritical family $$z^d + c$$. (PDF Abstract) 37-1171 Kenneth Jacobs Lyapunov Exponents in non-Archimedean Dynamics The Lyapunov exponent of a rational map $$\phi$$ measures the rate of growth of a point in a generic orbit. It is related to the orbits of the critical points of $$\phi$$, and when $$\phi$$ is defined over $$\mathbb C$$, a sharp lower bound is $$\frac12\log d$$, where $$d$$ is the degree of the map. Much less is known about Lyapunov exponents for maps defined over non-Archimedean fields. In this talk, we will give an explicit lower bound similar to the one over $$\mathbb C$$ which is sharp for maps of good reduction. We will also give a formula relating Lyapunov exponents to Silverman's critical height. (PDF Abstract) 11-1181 Rafe Jones Eventually stable rational functions Let $$K$$ be a field, $$f$$ a rational function with coefficients in $$K$$, and $$\alpha\in K$$. We say that the pair $$(f,\alpha)$$ is eventually stable over $$K$$ if the the number of irreducible factors of the numerator of $$f^n(x)-\alpha$$ is bounded as $$n$$ grows, where $$f^n(x)$$ denotes the $$n$$'th iterate of $$f$$. This is a natural finiteness condition that should hold in great generality: we conjecture that a given pair $$(f,\alpha)$$ is eventually stable unless $$\alpha$$ is periodic under $$f$$. We summarize what little is known in the direction of this conjecture, and in the process give several equivalent conditions for eventual stability. We also give a new result showing that pairs $$(f,\alpha)$$ satisfying a weak version of the Eisenstein criterion are eventually stable. (PDF Abstract) 30-475 Sarah Koch Postcritical sets in moduli space Consider the moduli space $$\mathcal{M}_{0,n}$$ of curves of genus 0 with $$n$$ marked points. Call a point $$x\in\mathcal{M}_{0,n}$$ postcritically special if there is a postcritically finite rational map $$F:\mathbb{P}^1\to\mathbb{P}^1$$ whose postcritical set $$P$$ is a representative of the point $$x$$ in moduli space. In an email conversation, L. DeMarco posed the following question: in $$\mathcal{M}_{0,n}$$, what does the locus of postcritically special points look like? We prove that this locus is dense in $$\mathcal{M}_{0,n}$$, with respect to the complex-analytic topology. (PDF Abstract) 37-747 Holly Krieger The dynamical Andre-Oort conjecture I will discuss recent progress on the dynamical Andre-Oort conjecture concerning the geometry of post-critically finite maps in algebraic families of rational maps. (PDF Abstract) 37-835 ChongGyu (Joey) Lee Generalized dynamical systems, from monoid actions to homomorphisms We can define a dynamical system with a self-map $$f$$ on a set $$S$$. We may consider this dynamical system as a monoid action on a set $$S$$. In such point of view, we can generalize the concept of dynamical systems. For example, when we consider an algebraic group, we consider $$n$$-multiplication map on a group and consider dynamical systems defined by $$n$$-multiplication. Then the torson group is the set of preperiodic points. We may consider not only iterations of $$n$$-multiplication, but all multiple maps and get the same result. So we can generalize the dynamical system as an action by a set of self maps with some specific conditions. Such points of view will give us one more step, we may have a set of homomorphisms whose codomain is not the same with the domain, like isogenies of two elliptic curves. In this talk, we examine sime examples of such generalization and find some conditions under which dynamical system between projective spaces can have small preperiodic points. (PDF Abstract) 11-754 Alon Levy Questions in higher-dimensional non-archimedean dynamics We develop some machinery to study self-morphisms of $$\mathbb{P}^n$$ and other higher-dimension varieties. At a fixed point, there's an action on the tangent space. There are $$n$$ eigenvalues, and we can decompose them into attracting, indifferent, and repelling directions. This allows local linearization in some cases, and allows generalizing some of the basic features of non-archimedean dynamics in one variable. (PDF Abstract) 11-1349 Michelle Manes Characterizing cyclic quartic extensions by automorphism polynomials Following Morton's work with cyclic cubic extensions, we use the dynamics of a family of cubic polynomials to characterize the cyclic quartic extensions of a field $$K$$, provided that $$K$$ has characteristic different from 2 and 3. (PDF Abstract) 37-929 Clayton Petsche On the distribution of orbits in affine varieties. Given an affine variety $$X$$, a morphism $$\phi: X \to X$$, a point $$\alpha\in X$$, and a Zariski closed subset $$V$$ of $$X$$, we show that the forward $$\phi$$-orbit of $$\alpha$$ meets $$V$$ in at most finitely many in finite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces. A more general result has been independently obtained, using different methods, by Bell-Ghioca-Tucker and Gignac. (PDF Abstract) 37-695 Paul Reschke Complex dynamics of birational surface maps defined over number fields For a birational self-map with non-trivial first dynamical degree on a complex surface, Bedford and Diller defined an energy condition which when satisfied guarantees nice dynamical properties for the map (with regard, in particular, to a naturally defined invariant measure). However, Buff showed that the energy condition can fail and that in fact maps without the nice dynamical properties do exist. We show that the energy condition is always satisfied when the birational self-map is defined over a number field. Our proof relies in part on a construction of a natural dynamical height function for the map, which expands upon work by Silverman and Kawaguchi. (PDF Abstract) 37-753 Robert Rumely Arithmetic Coordinates on Dynamical Moduli Space For the moduli space of quadratic rational functions, work of Doyle, Jacobs, and Rumely shows that the usual coordinates $$\sigma_1,\sigma_2$$ distinguish the different reduction types of non-Archimedean quadratic rational functions. This talk will present similar results for cubic polynomials, and work in progress for cubic rational functions. (PDF Abstract) 11-941 Katherine Stange The dynamics of Apollonian circle packings Reduction to the root quadruple of an Apollonian circle packing can be viewed as a piece of Asmus Schmidt's continued fraction algorithm for Gaussian integers. We study the dynamics of this process, in analogy to the dynamical system $$T:[0,1]\to[0,1]$$ given by $$x\mapsto\{1/x\}$$ related to classical continued fractions. (PDF Abstract) 11-476 Bianca Thompson A very elementary proof of a conjecture of B. and M. Shapiro for cubic rational functions Using essentially only algebra, we give a proof that a cubic rational function over $$\mathbb{C}$$ with only real critical points is equivalent to a real rational function. We also determine all fields $$\mathbb{Q}_p$$ over which a reasonable generalization holds. (PDF Abstract) 11-1392 Thomas Tucker Uniform boundedness for positive dimensional varieties Let $$X$$ be a variety over a number field $$K$$ and let $$f:X\to X$$ be a morphism. Morton, Silverman, Zieve, Pezda, Hutz and others have proved that there are bounds on the number of $$f$$-periodic points in $$X(K)$$ that depend only on $$X$$, $$K$$, and the size of the residue field for a prime of good reduction for $$f$$. If one looks more generally at periodic subvarieties of arbitrary dimension, the situation becomes quite different. We present some partial results, joint with Bell and Ghioca, and present some simple questions that we cannot yet answer. (PDF Abstract) 37-418 Hexi Ye Some applications of the Arithmetic equidistribution theorem in Complex Dynamics Arithmetic equidistribution theorems play an important role in the study of Algebraic Dynamics. In recent years, Arithmetic questions arose and were studied in Complex Dynamics. One of the more important tools is the equidistribution theorem. In this talk, I would like to discuss some applications of the equidistribution theorem in Complex Dynamics. (PDF Abstract)

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