Michigan Lectures in Number Theory
April 19-21, 2010
Speaker: Matt Baker
(Georgia Tech)
April 19: Preperiodic points and unlikely intersections (3:00-5:00, room 4096)
We will discuss the proof of the following theorem: For any fixed
complex numbers a and b, and any integer d≥2, the set of
complex numbers c for which both a and b are preperiodic for zd+c is
infinite if and only if ad = bd. This provides an affirmative
answer to a question of Zannier. The main ingredients in the proof
are a complex-analytic study of certain generalized Mandelbrot sets
and an adelic equidistribution theorem for preperiodic points over
number fields and function fields. Somewhat surprisingly,
non-Archimedean Berkovich spaces play an essential role in the
arguments even though the theorem is purely about complex dynamics.
This is joint work with Laura DeMarco.
April 20: Potential theory on the Berkovich projective line (2:30-4:00, room 3096)
We will give an elementary introduction to Berkovich's theory of
non-Archimedean analytic spaces, focusing on the Berkovich projective
line (a non-Archimedean analog of the Riemann sphere). We will then
discuss the rudiments of potential theory on the Berkovich projective
line, including harmonic and subharmonic functions, capacities, and
the Laplacian operator. We will use this theory to construct a
canonical Green's function and a canonical probability measure
associated to the dynamics of a rational map of degree at least 2
defined over a non-Archimedean field.
April 21: Canonical heights, equidistribution, and dynamics (2:00-3:30, room 3096)
We will define global canonical heights associated to compact
Berkovich adelic sets and to the iteration of rational maps, and then
discuss some associated equidistribution theorems for Galois orbits of
points of small height. We will use these results to study the
relationship between preperiodic points and points of canonical height
zero in the dynamical function field setting. As an application, we
will complete our sketch of the proof of the main theorem from the April 19 talk.
If time permits, we will also discuss a local equidistribution
theorem for iterated preimages, due to Favre and Rivera-Letelier,
which is a powerful tool for studying Fatou-Julia theory on the
Berkovich projective line.
Background sessions (in room 4096, starting at 3pm and ending by consensus):
April 2: The Berkovich projective line. Session leaders: Alex Mueller and Julian Rosen
April 9: Canonical heights and Bilu's equidistribution theorem. Session leaders: Sijun Liu and Zach Scherr
April 16: Green's functions and the Mandelbrot set. Session leaders: Hunter Brooks and Ari Shnidman
Topics for background sessions:
- The Berkovich projective line
(see e.g. Matt's Arizona Winter School lecture notes,
especially Sections 1-3)
- Canonical heights for rational maps on P^1, canonical heights over
function fields, and Bilu's equidistribution theorem.
- The Green's function attached to a rational map,
(see Silverman's Arithmetic Dynamics book, or the recent
Baker-Rumely
book), and basics of the Mandelbrot set. Specifically:
(a) There is an explicit formula for the Riemann mapping from the
exterior of the Mandelbrot set to the unit disc (this is how Douady
and Hubbard proved that the Mandelbrot set is connected); the Green's
function for the Mandelbrot set is -log| | of this function.
(b) The values c for which 0 is preperiodic for z2+c come in two
flavors: the periodic ones (called hyperbolic centers) and the
strictly preperiodic ones (called Misiurewicz points). The former lie
in the interiors of the various bulbs, and the latter lie on the
boundary of the Mandelbrot set (and are in fact dense there).
Feedback to: numbertheory [at] umich.edu