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^{MATH 663: Complex Dynamics and the Mandelbrot set (Nataliya Goncharuk, spring 2025)}

Picture from Wikipedia, created by Wolfgang Beyer with the program Ultra Fractal 3, license CC BY-SA 3.0.

Final exam

Final exam.

Homeworks:

HW1, HW1 Solutions, HW2, HW2 Solutions, HW3, HW3 Solutions, HW4, HW4 Solutions, HW5, HW5 Solutions, HW6, HW6 Solutions, HW7, HW7 Solutions.

 

Textbooks:

[M] J. Milnor, ``One-dimensional complex dynamics'';

[CG] L. Carleson, T. Gamelin, ``Complex dynamics''.

Notes for the lectures:

Lec1, Lec2, Lec3, Lec4, Lec5, Lec6, Lec7, Lec8, Lec9, Lec10, Lec11, Lec12, Lec13, Lec14, Lec15, Lec16, Lec17, Lec18, Lec19, Lec20, Lec21, Lec22, Lec23, Lec24, Lec25, Lec26, Lec27.

 Schedule, reading, past assignments and solutions.

 

Week 1: Jan 14, 16. Introduction to dynamical systems: periodic points, dense orbits, chaotic behavior. Definition of Julia sets. [M] Sec.3-4, [CG] Sec. II.1 and III.1.

            On Tuesday, I introduced fixed, periodic, eventually periodic points; multipliers; attracting, superattracting, repelling,  parabolic and  irrationally indifferent fixed/periodic points; attracting basins. I discussed a special case when infinity is fixed or periodic. I introduced critical points and critical values, Finally, I discussed the dynamics z->z^2 and explain why it is chaotic on |z|=1.    On Thursday, I discussed uniform convergence on compacts (in spherical metric), gave definition of normal families and Fatou/Julia sets, definition of filled-in Julia sets for polynomials, explained why repelling and parabolic points belong to Julia sets. 

Week 2: Jan 21, 23. Normal families and Montel's theorem. Properties of Fatou and Julia sets. [M] Sec.3-4, [CG] Sec. III.1.

On Thursday, I proved that attracting basins belong to the Fatou set (example:Newton maps) and that Julia sets are forward and backward invariant. I used Montel-Caratheodori theorem to show that inverse images of any point z\in J(f) are dense in J(f), and that the boundary of any attracting basin coincides with the Julia set.

Week 3: Jan 28, 30. [M]Sec.7, [CG] II.1 Normal families and Montel's theorem. Lattes example.

On Tuesday, I explained Lattes example of a rational function whose Julia set is the whole sphere. On Thursday, I discusse classification of Riemann surfaces and almost proved Montel-Caratheodori theorem; I introduced Poincare metric and discussed Schwartz-Pick lemma that is needed for the last part of the proof.

Week 4: Feb 4, 6. Linearizing coordinates for attracting, super-attracting, and repelling periodic points. Quasiconformal maps, Ahlfors-Bers theorem.   [M] Sec. 8-9, [CG] Sec. II.2-4.

On Tuesday, I finished the proof of the Montel-Caratheodori theorem and proved existence of linearizing coordinate for attracting and repelling fixed points. On Thursday, I explained why any immediate attracting basin contains a critical point, and proved existence of uniformizing coordinates for superattracting points.

Week 5: Feb 11, 13. [M] Sec. 10 [CG] Sec. II.5  Parabolic points, Fatou flowers. Uniformization for parabolic points (using quasiconformal maps).

On Tuesday, I gave alternative proof of the existence of linearizing coordinates for attracting points (via the quotient space by f). For parabolic points, I almost proved that Fatou coordinates exist, and described Ecalle-Voronin moduli. On Thursday, I showed how parabolic points look in the c/z^n-chart that almost conjugates f to z->z+1. I showed that parabolic basins belong to the Fatou set, their boundaries belong to Julia sets, and each immediate parabolic basin contains a critical point. I showed some examples (fat rabbit, fat basilica) and introduced the parabolic checkerboard. [I took pictures from A.Cheritat's paper on Inou-Shishikura renormalization].

Week 6: Feb 18, 20. Quasiconformal maps. Cremer points, Siegel discs. [M] Sec. 10-11, [CG] Sec. I.5,I.7, II.5-6.

On Tuesday, I introduced quasiconformal maps, formulated Ahlfors-Bers theorem (a.k.a. measurable Riemann mapping thm), and used it to prove existence of Fatou coordinates. As other applications of this theorem, I sketched the proof of the classification of complex manifolds, and defined Teichmuller metric. On Thursday, I formulated Siegel-Brjuno theorem about linearization of irrationally neutral points and provided examples of Cremer points.

Week 7: Feb 25, 27. [M] Sec. 11, [CG] Sec. II.6.   Cremer points, Siegel discs.  Parabolic bifurcations in the Mandelbrot set. Real slice of the Mandelbrot set, period-doubling bifurcations. 

On Tuesday, I completed the proof of the Siegel-Brjuno theorem for Diophantine irrational numbers, following [CG]. Also, I discussed bifurcations that happen on the real slice of the Mandelbrot set: a parabolic bifurcation and a period-doubling bifurcation. On Thursday, I wrote out normal forms for parabolic and period-doubling bifurcations. I showed the bifurcation diagram of the real quadratic family and discussed its relation to Sharkovskii order.  Also, I discussed the cascade of period-doubling bifurcations, formulated the statement on Feigenbaum universality and gave a sketch of the proof (via renormalization operators).

Week 8: Mar 4, 6. [M] Sec 12-14, [CG] III 2-4. More about Julia sets. Repelling orbits are dense in the Julia set of a rational map. 

On Tuesday, I introduced indices of fixed points and proved that any rational function of degree at least 2 has either a repelling fixed point or a parabolic point with multiplier 1. Also, I provided an upper bound on the number of attracting and parabolic periodic cycles of any period for rational functions, via critical points.  On Thursday, I proved that repelling periodic orbits are dense in the Julia set. Also, I proved more elementary properties of the Julia sets: Julia set is nonempty, infinite, and has no isolated points. 

Spring break: Mar 10-14.

Week 9: Mar 18, 20. [M] Sec 11, 15; [CG] III.2, IV.2 Sullivan classification (statement). Counting neutral points; critical points accumulate on boundaries of Siegel discs and Herman rings. Examples of Herman rings: Blachke products.

On Tuesday, I proved an upper estimate on the number of neutral periodic cycles. I stated Sullivan classification of Fatou components for rational functions. I stated that critical orbits accumulate onto boundaries of Siegel discs and Herman rings, and extracted useful corollaries. On Thursday, I proved that critical orbits accumulate to Cremer points and boundaries of Siegel discs and Herman rings, introduced rotation number for circle maps and explained its relation to Herman rings.  

Week 10: Mar 25, 27. [M] Sec. 15, [CG] II.7 Rotation number and linearization of circle maps. Shishikura's surgery. 

On Tuesday, I proved properties of the rotation number for circle homeomorphisms and provided examples of Blaschke products that have Herman rings. Also, I started to explain the construction of the Shishikura's surgery. On Thursday, I discussed Shishikura's surgery. Also, I proved an upper estimate on the number of Herman rings (using quasiconformal deformations). 

Week 11:  Apr 1, 3. [M] Sec. 9,18, [CG] III.4, VI.1, VIII.5.  Cantor dusts. Bottcher coordinates for polynomial maps. Quadratic-like maps.  Caratheodori theorem and external rays. 

On Tuesday, I proved that the Julia set is either connected or has infinitely many connected components. For polynomials, I proved that the Julia set is a Cantor dust if all critical points escape to infinity, has infinitely many components if some critical points escape, and is connected if no critical points escape.  On Thursday, I explained why polynomial-like mappings are quasiconformally conjugate to polynomials (this is the reason to have baby quadratic Julia sets in e.g. cubic polynomials, and baby Mandelbrot sets in a large Mandelbrot set). Also, for connected Julia sets, I explained how to extend the Bottcher coordinate, defined external rays and laminations.  

Week 12:  Apr 8, 10. [M] Sec 18 [CG] VIII.3-5. External rays. Douady elephants and parabolic renormalization.

On Tuesday, I discussed external rays and their landing points. I formulated statements on landing of periodic rays and used the fat Douady rabbit to demonstrate applications. Also, I introduced the Green's function and external rays for the Mandelbrot set. I formulated the statement on the landing on rational external rays and explained the motivation for the Local Connectivity conjecture for M. On Thursday, I proved that M is connected. Also, I explained similarity of Douady elephants and seahorses of M using the Lavaurs map (a.k.a. the map through the eggbeater/through the parabolic point).

Week 13: Apr 15, 17. [M] Sec 5, 16; [CG] VIII.5,  IV.1-3 Denjoy-Wolf theorem. Sullivan classification of Fatou components (proof). 

On Tuesday, I explained which rational rays land together at parabolic points of the Mandelbrot set (without complete proof). Also, I started the proof of Sullivan classification of periodic Fatou components. On Friday, I completed the proof. 

Week 14.  Apr 22, 24.[M] Sec. 19, [CG] IV.1, V.2-4, VIII.6.  Absence of wandering Fatou components for rational maps.  Hyperbolicity/subhyperbolicity and local connectivity of Julia sets; Misiurewicz points and dendrites.

On Tuesday, I proved Sullivan's theorem on absence of wandering Fatou components (skipping details on Beltrami equations). Also, I introduced the notion of hyperbolic maps and proved a criterion of hyperbolicity. I mentioned Density of Hyperbolicity (DH) conjecture that would be implied by MLC (Mandelbrot  Local Connectivity conjecture). On Thursday, I proved local connectivity of Julia sets for hyperbolic polynomial maps. I sketched the proof of local connectivity for dendrite-like Julia set that appear for Misiurewicz points. Finally, I mentioned Tan Lei's theorem on similarities between Mandelbrot sets and dendrite Julia sets at Misiurewicz poins, and discussed external rays that land at Misiurewicz points of the Mandelbrot set.