MATH 81400 References
Lectures will be based on two book manuscripts that I've been writing in the past couple of years. As new topics are introduced, I give you access to a Dropbox folder where you can download copies of relevant chapters. Here is an incomplete list of some further reading material:
Background in complex analysis:
L. Ahlfors, Lectures on Quasiconformal Mappings, 2nd expanded edition,
AMS, 2006.
S. Krantz, Complex Analysis: The geometric viewpoint, 2nd edition, MAA, 2004.
W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1986.
Complex dynamics:
B. Branner, The Mandelbrot set, in Chaos and fractals pp. 75-105,
Proc. Sympos. Appl. Math., 39, Amer. Math. Soc., 1989.
L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, 1993.
M. Lyubich, The dynamics of rational transforms: the
topological picture, Russian Math Surveys 41:4 (1986) 43-117.
C. McMullen, Complex Dynamics and Renormalization, Annals of
Math Studies 135, Princeton University Press, 1996.
J. Milnor, Dynamics in One Complex Variable, 3rd edition, Princeton
University Press, 2006.
M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. Ecole Norm. Sup. 20 (1987) 1–29.
Siegel disks:
A. Douady, Disques de Siegel at aneaux de Herman, Seminar Bourbaki, Asterisque 152-153 (1987) 151-172.
C. Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Acta Math. 177 (1996) 163-224.
C. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Annals of Math. 159 (2004) 1-52.
S. Zakeri, Dynamics of cubic Siegel polynomials, Commun. Math. Phys. 206 (1999) 185-233.
S. Zakeri, On Siegel disks of a class of entire maps, Duke Math. Journal 152 (2010) 481-532.
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks, Inventiones Math.
185 (2011) 421-466.
Circle maps:
M. Herman, Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations, Pub. Math. IHES 49 (1979) 5-233.
C. Petersen, The Herman-Świątek theorems with applications, in "The Mandelbrot Set, Theme and Variations,"
edited by Tan Lei, London Mathematical Society Lecture Note Series No. 274, Cambridge University Press, 2000.
G. Świątek, Rational rotation numbers for maps of the circle , Commun. Math. Phys. 119
(1988) 109–128.
Rotation sets:
A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen, and D. Parris, Rotational subsets of the circle under $z^d$, Topology Appl. 153 (2006) 1540-1570.
S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc. 115 (1994) 451-481.
L. Goldberg, Fixed points of polynomial maps I: Rotation subsets of the circles, Ann. Sci. Ecole Norm. Sup. 25 (1992) 679-685.
L. Goldberg and J. Milnor, Fixed points of polynomial maps II: Fixed point portraits, Ann. Sci. Ecole Norm. Sup. 26 (1993) 51-98.
L. Goldberg and C. Tresser, Rotation orbits and the Farey tree, Ergodic Theory Dynam. Systems 16 (1996) 1011-1029.
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Math 81400