Douglas Lind (University of Washington):
Adelic Dynamics

Abstract: Algebraic actions of Z^d defined over the integers give rise to both complex and p-adic actions, which complement each other via various local-global ideas. I will describe how this works in several settings, and the role of complex and p-adic amoebas in answering natural dynamical questions such as expansiveness and entropy.

Manfred Einsiedler (Princeton University):
Measure rigidity and divisibility properties of Hamiltonian quaternions

Abstract: Recent progress on measure rigidity for higher-rank partially hyperbolic actions has lead to interesting applications to number theory. In this talk we will discuss measure rigidity and a particular such application to the divisibility properties of Hamiltonian quaternions: The non-zero integer quaternions modulo the center form a discrete group. The subgroup of elements having norm which is a power of a fixed prime is actually a free group with finitely many generators. However, the subgroup consisting of elements whose norm is a product of powers of two primes shows a surprisingly different structure. This is joint work with S. Mozes.

Thomas Ward (University of East Anglia):
Orbit growth for a p-adic extension: an exact result

Abstract: A natural family of p-adic extensions exhibit very complex growth in the number of orbits. For a simple example, the exact (real) asymptotics are related directly to sequences converging p-adically. This is joint work with Everest, Miles and Stevens.

Rafe Jones (Brown University):
The Density of the p-adic Hyperbolic Mandelbrot Set

Abstract: While the p-adic Mandelbrot Set is famously boring, the same is not true of its subset corresponding to maps where zero tends to an attracting cycle. I call this subset the hyperbolic Mandelbrot set. In this talk, I will explain a result showing that this set has density zero in a natural sense, which is in sharp contrast to its complex analogue. To get this result, ideas from several fields are used, including number theory, group theory, and probability.

Robert Rumely (University of Georgia):
Ih's integrality conjecture for preperiodic points

Abstract: By analogy with Siegel's theorem, and with input from Zhang and Silverman, Su-Ion Ih has conjectured that if K is a number field, E/K is an elliptic curve, and P is a non-torsion point in E(K), then there are only finitely many torsion points (over the algebraic closure of K) which are integral with respect to P. More generally he conjectures that for the dynamical system associated to a rational function f(x) in K(x) of degree at least 2, if P is a non-preperiodic point for f(x), then there are only finitely many preperiodic points (over the algebraic closure of K) which are integral with respect to P.

Baker, Ih and Rumely have proved the conjecture for elliptic curves and for the dynamical system attached to f(z) = z², using equidistribution of small points and lower bounds for linear forms in logarithms. This talk will outline the proof and discuss progress towards the general case.

Matthew Baker (Georgia Institute of Technology):
Rational dynamics on the Berkovich projective line

Abstract: We will define the Berkovich projective line and give a construction of a natural invariant measure attached to a rational map over C_p. As a consequence, we will obtain a purely ``analytic'' proof that the Berkovich Julia set of a rational map (defined via equicontinuity) is nonempty.

Jan Kiwi (Pontificia Universidad Católica de Chile):
Puiseux series dynamics: iteration of cubic polynomials

Abstract: We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of the dynamical and parameter space of cubic polynomials. In particular we characterize cubic polynomials with compact Julia sets. Also, we prove that any infraconnected connected component of a filled Julia set (of a cubic polynomial) is either a point or eventually periodic. Our techniques are based on the ideas introduced by Branner and Hubbard to study complex cubic polynomials. Thus, we show that fields such as L seem to be a natural dynamical space to explore the interplay between non-archimedean and complex dynamics.

Gabriela Fernández Lamilla (Pontificia Universidad Católica de Chile):
Wandering Fatou Components in p-adic Polynomial Dynamics

Abstract: Given a polynomial Q(z) in C_p[z] of degree at least p+1 satisfying a norm condition, we consider a certain family {Q_c} of polynomial perturbations of Q. We prove that for a dense set of parameters c, the polynomial Q_c has a wandering domain, generalizing a theorem of Benedetto. We also consider some extensions and consequences of our result.

Ghassan Sarkis (Pomona College):
Marking Formal Groups in Characteristic p

Abstract: A formal group over the p-adic integers is uniquely determined by any of its nontorsion endomorphisms. This is perhaps best seen by resorting to the logarithm, which encodes all the zeros of the noninvertible endomorphism and linearizes the formal group. While the setup does not reduce nicely to residue fields, there is some evidence suggesting that analogues do exist. In this talk, I will introduce dynamical systems arising out of formal groups over local rings and then discuss some of the problems encountered in reducing the whole big mess to characteristic p.

Franco Vivaldi (Queen Mary, University of London):
Maps over finite fields: integrability and reversibility

Abstract: In the theory of dynamical systems, integrability (existence of invariants of the motion) and reversibility (existence of conjugacy with inverse map) are important structural properties. We let two-dimensional algebraic mappings act on finite coordinate fields, and present experimental evidence for the existence of limit distributions of the length of the orbits for the integrable and reversible case. Such distributions feature considerable rigidity (independence from the mapping).