Schedule for Math 189A, Complex Dynamics --- Spring 2001

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 1week 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 2week 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 3week 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 4week 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 5week 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 6week 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 7week 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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