MATH 845, Arithmetic of Dynamical Systems (Selected topics in Number Theory)

Instructor: Nils Bruin
SC K 10507
(778) 782 3794
Textbook: Silverman, Joseph H. The arithmetic of dynamical systems.
Graduate Texts in Mathematics, 241. Springer-Verlag, New York, 2007.
x+511 pp. ISBN: 978-0-387-69903-5.

Questions and errata

Course materials: Assignments and handouts are available on this page.
  • 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 p-adics.
Jan 18. Kevin

Chapter 2: p-adic 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: p-adic 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^2-z)/p

Mar 3 Nils Bruin

Proof of Proposition 5.24 (step to symbolic dynamical characterisation of the Julia set of (z^2-z)/p, Theorem 5.26 (towards a non-archimedean 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:

  1. Classical dynamics
  2. Dynamics over local fields: good reduction
  3. Dynamics over global fields
  4. Families of dynamical systems
  5. Dynamics over local fields: bad reduction
  6. Dynamics associated to algebraic groups
  7. 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.
  • 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 13-17, 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.