Transcendental Dynamics

Course Meeting Time and Location
Tuesday and Thursday
1:00 - 2:25 pm
131 Math Building (Building 15)

Course Instructor Contact Information and Office Hours
210-7 Math Building (Building 15)

Some resources used for the class:
Walter Bergweiler's survey on complex dynamics
Joel Schiff's text on normal families
Chris Bishop's notes on transcendental dynamics
Function Theory of one Complex Variable - Greene/Krantz
Complex dynamics - Carleson/Gamelin

 Date Topic 1/4/18 Review of some basic complex analysis: the difference between the behavior of an entire function with a removable singularity at infinity (i.e. a polynomial) and an entire function with an essential singularity at infinity (e.g. the exponential). Proof of the existence of a Laurent series expansion about an isolated singularity traced back to the Cauchy integral formula and Cauchy-Riemann equations. Mention of some general goals for the class. 1/9/18 The notion of a normal family of meromorphic functions (here convergence is with respect to the chordal metric on the Riemann sphere). A theorem on the possible limiting behavior of a family of normal functions on a subdomain of the Riemann sphere. Together with a criterion of Montel for normality (missing three points) this says that the only really interesting subdomains of the sphere on which to study complex dynamics are the sphere itself, the punctured sphere, and the twice punctured sphere (otherwise the family of iterates will always be normal and there will be no Julia set).
 1/11/18 A proof of Montel's criterion for normality (missing three points) using the elliptic modular function. (A discussion also on a proof using the fact that a subdomain of the sphere missing at least three points is covered by the unit disk). Notion of conjugacy between two dynamical systems. Definitions of the Julia and Fatou sets, invariance of the Julia/Fatou set, equality of Fatou set of f and Fatou set of f^n.
 1/16/18 The notion of an exceptional point: for rational maps there are at most two exceptional points, for transcendental maps there is at most one exceptional point, and none for dynamics on the twice punctured sphere. A proof of Picard's "great" theorem using Montel's criterion for normality. Some basic properties of the Julia set.
 1/18/18 Periodic points, cycles, multipliers. Repelling and rationally indifferent periodic points always lie in the Julia set. A proof that a rational map always has either a repelling or rationally indifferent periodic point, and hence the Julia set for a rational map is always infinite.
 1/23/18 Some basics facts about basins of attraction of attracting periodic points, a proof that the Julia set is always perfect (though in the transcendental setting we still haven't proved the Julia set is non-empty). Some examples: computations of Julia/Fatou sets for z->z^2+c for a couple values of c, a discussion of Newton's method associated to a polynomial.
 1/25/18 Nevanlinna theory: counting/proximity functions and the Nevanlinna characteristic. Jensen's formula and a derivation of the first fundamental theorem of Nevanlinna. Statement of second fundamental theorem (no proof) and some consequences: a transcendental map can have at most four perfectly branched values (sharp via the Weierstrass p-function).
 1/30/18 Proof that the Julia set of a transcendental entire function is infinite (via a lemma that the square of any such function has infinitely many fixpoints).
 2/1/18 Marty's theorem + Zalcman's Lemma. Deducing Montel's three point normality criterion from Zalcman+Little Picard, deducing Montel's bounded on compacta normality criterion from Zalcman+Louiville.
 2/6/18 Structure theorem for Julia set: the set of repelling periodic points is dense in the Julia set (transcendental+rational setting).
 2/8/18 Conjugacies near linearly attracting/repelling fixed points. Presence of a singular value in the immediate basin of attraction. Classification/definition of singular values (critical values in rational setting, critical values+asymptotic values in transcendental setting).
 2/13/18 Conjugacies near parabolic (rationally indifferent) fixed points.
 2/15/18 Continuation of dynamics near parabolic fixed points, presence of a singular value in the immediate basin of attraction for a parabolic fixed point. Superattracting fixed points and conjugacies near such fixed points (and relationship with singular values).
 2/19/18 Classification of periodic Fatou components: Herman ring/Siegel disc cases
 2/21/18 Classification of periodic Fatou components: (super)attracting basin case, baker domain/parabolic basin cases.
 2/27/18 no wandering domain theorem (rational+speiser class setting)
 03/01/18 approximation theory: Runge's theorem+examples
 03/06/18 Mergelyan's Theorem part I
 03/08/18 Mergelyan's Theorem part II