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John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal , the Wolf Prize , the Abel Prize and ...
John Milnor was a key figure in this revival, and his fascination with holomorphic dynamics helped to make it so prominent. Milnor's book Dynamics in One Complex Variable [M8], his volumes of collected papers [M10, M11], and the surveys [L1, L5] are exemplary introductions into the richness and variety of Milnor's work in dynamics.
- Kneading Theory
- Milnor’s Attractors
- Milnor–Palis Conjecture
- Self-Similarity and Hairiness of The Mandelbrot Set
- Beyond The Quadratic Family
- Two-Dimensional Dynamics
- Art Gallery
Let us consider the class 풮 of continuous piecewise monotone interval maps f:I→I. The Milnor-Thurston Kneading TheoryFootnote 1provides us with a combinatorial classification of the corresponding dynamical systems. To such a map one can associate a sequence called the kneading invariant that determines a symbolic model for the map. This model is ni...
The notion of a “strange” attractor played an inspiring role in the 1970–1980’s. Any invariant set with somewhat complicated topology that attracts “many” points was regarded to be a strange attractor. Examples included the Smale solenoid, Lorenz, Hénon, and Feigenbaum attractors. The notion itself was coined by Ruelle and Takens who proposed it as...
For a typical smooth dynamical system f:M→M, the global attractor Af is decomposed into finitely many minimal attractors Ai. Moreover, for almost every point x∈M, the ω-limit set ω(x) is equal to one of the Ai. In fact, Palis put forward a stronger conjecture asserting that typically each minimal attractor supports a unique SRB measureFootnote 2μ t...
2.3.1 Pinched Model for Julia Sets
Let us now provide some brief background in holomorphic dynamics. Consider a polynomial \(f: \mathbb {C}\to \mathbb {C}\) of degree d≥2 normalized so that its leading coefficient is equal to 1. The basin of infinity ℬf(∞) is the set of points z whose orbits \(\{f^{n} z\}_{n=0}^{\infty}\) escape to infinity. The complementary set of non-escaping points is called the filled Julia setK(f). The Julia setJ(f) is the boundary of K(f) (and of ℬf(∞) as well). The (filled) Julia set is connected if an...
2.3.2 Hyperbolic and Superattracting Polynomials
Let α be a periodic point of period p, and let \(\boldsymbol {\alpha }= \{ f^{k} \alpha\}_{k=0}^{p-1}\) be the corresponding cycle. The derivative λ=(fp)′(α) is called the multiplier of α (and its cycle). The periodic point and its cycle are called attracting if |λ|<1. They are called superattracting if λ=0 (note that such a cycle contains a critical point). The basin ℬ(α)=ℬf(α) of an attracting cycle is the set of points whose orbits converge to α. The immediate basin ℬ∗(fkα) is the componen...
2.3.3 Little Mandelbrot Copies and Tuning
Let us now consider the quadratic family fc:z↦z2+c. The Mandelbrot setM is the set of parameters c for which the Julia set J(fc) is connected. One of the prominent features of the Mandelbrot set is that it contains many copies of itself that look identical to the main set (Fig. 1). Douady and Hubbard showed that these copies can be produced by a surgery called tuning. Let us take some superattracting quadratic polynomial fs:z↦z2+s with period p>1. For this map, the critical point 0 is periodi...
At the time when the world was rotating around the quadratic family fc:z↦z2+c, Milnor kept reminding people that there are plenty of very interesting research areas beyond it, concerning polynomial dynamics of higher degree and rational dynamics. As the quadratic family was eventually getting exhausted, with few outstanding very difficult problems ...
Milnor’s work sparked an intense interest to yet another direction of research: Higher Dimensional Complex Dynamics. An article by Friedland and Milnor was probably the first paper since Fatou’s time dedicated to global complex dynamics in several variables. It particularly emphasized the role of (generalized) complex Hénon maps (where p is a poly...
Let us conclude with an exhibition of a few dynamics pictures from various Milnor’s papers: see Fig. 7–10. We stop here. Of course, our brief survey is far from being complete. The reader is encouraged to go directly to Milnor’s papers in dynamics (whose collection up to 2000 has just appeared as a separate volume ) and to learn more about this bea...
- Mikhail Lyubich
- mlyubich@math.sunysb.edu
- 2014
Aug 18, 2014 · At the International Congress of Mathematicians (ICM) 2014 we had the great privilege to meet a true legend of mathematics: John Milnor. Milnor is famous not only for his work in mathematics, but also for his writing which has inspired generations of mathematicians.
John Willard Milnor. From the Abel Prize: "John Milnor’s profound ideas and fundamental discoveries have largely shaped the mathematical landscape of the second half of the 20th century…All of Milnor’s work display features of great research: profound insights, vivid imagination, striking surprises and supreme beauty ...
02:33 Beginnings, Aptitude, "socially maladjusted"03:40 Putnam, Math. as problem-solving04:10 First paper (at 18 yo)06:10 John Nash, Princeton07:45 games: Kr...
- 58 min
- 13.7K
- The Abel Prize
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Aug 26, 2014 · One of the highlights of the International Congress of Mathematicians (ICM) was the opportunity to meet a true mathematical legend. His name is John Milnor, and he's won several high-profile prizes over the years, including the Fields Medal in 1962 and the Abel Prize in 2011.
