Section numbers refer to Iteration of Rational Functions, by A.F. Beardon.

Classes meet at 2:45pm on Mondays and Wednesdays, in Beckman 134.
We will also meet on some Thursdays at 4:15pm.

Monday Wednesday Thursday
1
week of
Mar 5
Introduction. Key questions, rational maps,
iteration, fixed points, the Mandelbrot set,
the Julia and Fatou sets
Example: R(z)=z2, backwards iterates,
invariant sets, periodic points,
properties of the Julia set
no meeting
week of
Mar 12
spring break
2
week of
Mar 19
Course goals, character of fixed points,
multipliers, iteration near a fixed point,
character of periodic points
Basin of attraction of a fixed point or cycle,
conjugacy (via Mobius transformations),
conjugation preserves dynamics, example
Short presentations
1.2, 1.4, 1.7, 1.8
3
week of
Mar 26
The Mandelbrot set M:
iteration of quadratic polynomials,
attracting cycles, period-doubling cascade
Mandelbrot set (continued),
Green's function for an external domain
no meeting
4
week of
Apr 2
Conjugation of R near a super-attracting
fixed point (Boettcher's theorem),
dynamics in several complex variables
(paper by Malgorzata Stawiska)
Critical points, begin proving equivalence
of two definitions of M, when is the Julia set
of a polynomial connected?
no meeting
5
week of
Apr 9
finish proof of last theorem,
sufficient condition for Julia set of
polynomial to be totally disconnected
finish proof of last theorem,
equicontinuity, chordal and spherical metrics,
rigorous definitions of Fatou and Julia sets
no meeting
6
week of
Apr 16
Normal families, Arzela-Ascoli theorem
Cam McLeman:
The Complex Dynamics of Quadratic Polynomials
(Robert L. Devaney)
Adam Bliss:
Complex Dynamics and Entire Functions
(Robert L. Devaney)
week of
Apr 23
Presentation Days
7
week of
Apr 30
conditions for normality,
if deg(R)>2 then the Julia set lies in
the closure of the set of periodic points
John Cloutier and Steve Haas:
Puzzles and Para-Puzzles of
Quadratic and Cubic Polynomials
(Bodil Branner)
Mike Schubmehl and David Uminsky:
The Dynamics of Newton's Method
(Paul Blanchard)


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