Instructor: 
Nils Bruin
nbruin@sfu.ca
SC K 10507
(778) 782 3794

Webpage: 
http://www.cecm.sfu.ca/~nbruin/math845ADS 
Textbook:

Silverman, Joseph H. The arithmetic of dynamical systems.
Graduate Texts in Mathematics, 241. SpringerVerlag, New York, 2007.
x+511 pp. ISBN: 9780387699035.
Questions and errata

Course materials: 
Assignments and handouts are available on this
page. 
Prerequisites:

 Algebraic number theory (MATH 842). Can be taken together with this course.
 Algebraic geometry is useful but not required.
 A reasonable knowledge of algebra (in particular galois theory)
will be very useful.


Meetings: 
Scheduled timeslot is Monday, Wednesday 9:30  11:20 in K9509. However, if that is not suitable for a participant,
we can try to change to a slot that DOES work for everybody. See
this table.
First meeting will be to determine our further schedule and will be Wednesday January 6.
Lecture schedule
Jan 11. Nils
Chapter 1: Introduction to classical discrete dynamical systems
Critical points, fixed points, multipliers, relations on multipliers.

Jan 13. Fatou and Julia sets: Pictures.
Start on padics.

Jan 18. Kevin
Chapter 2: padic dynamical systems with good reduction (part 1)

Jan 20. Local rings and their completions, projective limits. Resultants as linear algebra.

Jan 25. Karen
Chapter 2: padic dynamical systems with good reduction (part 2)

Jan 27. Introduction to heights and geometry (roughly sections 3.1, 3.2) 
Feb 1. Kate
Chapter 3: Canonical heights

Feb 3. Heights and morphisms 
Feb 8. Kate
Chapter 3: Dynamical Diophantine Approximation

Feb 10. Iain
Newton Polygons

Feb 15  26: Olympic break 
Mar 1 Alex Molnar
Chapter 5: Nonarchimedean Julia and Fatou sets; the dynamics of (z^2z)/p

Mar 3 Nils Bruin
Proof of Proposition 5.24 (step to symbolic dynamical characterisation of the Julia set of (z^2z)/p, Theorem 5.26 (towards a nonarchimedean Montel Theorem).

Mar 8 Alex Molnar
Julia sets and the closure of the periodic points.

Mar 10. Exercises 
Mar 13  17: Arizona Winter School 
Mar 22
Sander Dahmen
Dynatomic polynomials (Section 4.1)

Mar 24. Exercises 
Mar 29
Sander Dahmen
Dynatomic Curves
Nils Bruin
Moduli spaces of rational function; Multiplier spectra

Mar 31
Nils Bruin
Dynamics from multiplicative groups; Twists

Apr 5: Easter 
Apr 7
Nils Bruin
Chebyshev Polynomials

Apr 12
Nils Bruin
Elliptic curves; Lattès maps

Apr 14
Nils Bruin
Multipliers for flexible Lattès maps


Description: 
Arithmetic Dynamics is a relatively new area of research. It may be
viewed as the transposition of classical results in the theory of
Diophantine equations to the setting of discrete dynamical systems,
such as iterated polynomial maps.
The interplay between arithmetic and dynamics yields a particularly
rich structure, which resembles the much older and established field
of arithmetic algebraic geometry. The course text gives a particularly
accessible introduction to the field.
We will follow the book quite closely. The chapters are:
 Classical dynamics
 Dynamics over local fields: good reduction
 Dynamics over global fields
 Families of dynamical systems
 Dynamics over local fields: bad reduction
 Dynamics associated to algebraic groups
 Dynamics in dimension greater than one

Format: 
Given the advanced nature of the material, the course will be run in a
mixed seminar/workgroup style. Every week, there will be 2 hours of
seminar and an additional 2 hours of workgroup, where the lecturer and
the students can look at problems and further delve into the theory.

Grading: 
 Participating students are expected to prepare at least one seminar
contribution, which will be assessed and count towards the grade.
 Students will regularly hand in assignments, which will be marked.
 Students will be graded on participation in the problem sessions.

Lecture schedule: 
To be announced 
Field trip: 
The Arizona Winter School 2010, held March 1317, 2010 at the University of Arizona in Tucson, AZ,
will be on Number Theory and Dynamics. It is warmly recommended that you attend this event. Application deadline for students seeking funding is
January 11, 2010. 