John Milnor's research while affiliated with Stony Brook University and other places
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Publications (144)
This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.
A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one exceptional badly behaved case, and provide a new description of the appropriate moduli spaces. There is also an ap...
This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.
After a general discussion of group actions,
orbifolds, and ``weak orbifolds''
this note will provide elementary introductions to two basic moduli spaces
over the real or complex numbers: First the moduli space of effective
divisors with finite stabilizer on the projective space $\bP^1$,
modulo the group of projective transformations of $\bP^1$...
After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space ${\mathbb P}^1$ modulo the group ${\rm PGL}_2$ of projective transformat...
An expository description of smooth cubic curves in the real or
complex projective plane.
An expository description of smooth cubic curves in the real or complex projective plane.
This note will study a family of cubic rational maps which carry antipodal
points of the Riemann sphere to antipodal points. We focus particularly on the
fjords, which are part of the central hyperbolic component but stretch out to
infinity. These serve to decompose the parameter plane into subsets, each of
which is characterized by a corresponding...
This note will provide a lightning tour through the centuries, concentrating on the study of manifolds of dimension 2, 3, and 4. Further comments and more technical details about many of the sections may be found in the Appendix.
Edited by Araceli Bonifant: University of Rhode Island, Kingston, RI
This volume is the seventh in the series “Collected Papers of John Milnor.” Together with the preceding Volume VI, it contains all of Milnor's papers in dynamics, through the year 2012. Most of the papers are in holomorphic dynamics; however, there are two in real dynamics and on...
This chapter studies complex polynomials with only one critical point, relating arithmetic properties of the coefficients to those of periodic orbits and their multipliers and external rays. It first defines the complex polynomial maps of degree n ≥ 2, and draws an alternate normal form for studying periodic orbits. The chapter also discusses the n...
This note is a brief description of my life, both personal and mathematical.
Edited by Araceli Bonifant: University of Rhode Island, Kingston, RI
This book, the sixth in the series “Collected Papers of John Milnor”, contains all of Milnor's work on Real and Complex Dynamics from 1953 to 1999, plus one paper from 2000.
These papers provide important and fundamental material in real and complex dynamical systems. Many of t...
Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally
polynomial maps from a finite union of copies of $\C$ to itself. In the space
of suitably normalized maps of this type, the hyperbolic maps form an open set
called the hyperbolic locus. The various connected components of this
hyperbolic locus are called hyperbolic component...
This note will study complex polynomial maps of degree $n\ge 2$ with only one
critical point.
I n the 1965 Hedrick Lectures, 1 I described the state of differential topology, a field that was then young but growing very rapidly. During the intervening years, many problems in differential and geometric topology that had seemed totally impossible have been solved, often using drastically new tools. The following is a brief survey, describing...
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed...
In this note we fill in some essential details which were missing from our paper. In the case of an escape region E h \mathcal {E}_h with non-trivial kneading sequence, we prove that the canonical parameter t t can be expressed as a holomorphic function of the local parameter η = a − 1 / μ \eta =a^{-1/\mu } (where a a is the periodic critical point...
The parameter space S_p for monic centered cubic polynomial maps with a marked critical point of period p is a smooth affine algebraic curve whose genus increases rapidly with p . Each S_p consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by...
This is a discussion of the dynamic plane and the parameter space for complex cubic maps which have a superattracting periodic orbit. It makes essential use of Hubbard trees to describe associated Julia sets.
We study rational maps of the real or complex projective plane of degree two or more, concentrating on those that map a genusone curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of interesting dynamical behaviors that are perhaps familiar to the applied dynamics community but not to spec...
We study rational maps of the real or complex projective plane of degree two or more, concentrating on those that map a genus-one curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of interesting dynamical behaviors that are perhaps familiar to the applied dynamics community but not to spe...
We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval $I$ affects the dynamics of an associated many-to-one skew product map of the cylinder $(\R/\Z)\times I$.
The 1950's and 1960's were exciting times to study the topology of manifolds. This lecture will try to describe some of the more interesting developments. The first two sections describe work in dimension 3, and in dimensions n ≥ 5, while 3 discusses why it is often easier to work in higher dimensions. The last section is a response to questions fr...
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed...
The operation of "mating'' two suitable complex polynomial maps {\small $f_1$} and {\small $f_2$} constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map {\small $f_1 \mat f_2$} from this sphere to itself. This construction i...
nine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3-dimensional manifolds. The final paragraph provides a brief description of th...
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points.
The Mandelbrot set is a fractal shape that classifies the dynamics of quadratic polynomials. It has a remarkably rich geometric and combinatorial structure. This volume provides a systematic exposition of current knowledge about the Mandelbrot set and presents the latest research in complex dynamics. Topics discussed include the universality and th...
Carathéodory’s theory of “prime ends” is the basic tool for relating an open set of complex numbers to its complementary closed set. Let U be a simply connected subset of \(\widehat {\Bbb C}\) such that the complement \(\widehat {\Bbb C}\)\ U is infinite. The Riemann Mapping Theorem asserts that there is a conformal isomorphism, which we write as $...
First recall the following. Let f: \(\widehat {\Bbb C} \to \widehat {\Bbb C}\) be a rational map of degree d ≥ 0
Consider real cubic maps of the interval onto itself, either with
positive or with negative leading coefficient. This paper completes the
proof of the ``monotonicity conjecture'', which asserts that each locus
of constant topological entropy in parameter space is a connected set.
The proof makes essential use of the thesis of Christopher Heckman, a...
The next two sections will be surveys only, with no proofs for several major statements. This section will describe a close relative of the Siegel disk.
The next four sections will study the dynamics of a holomorphic map in some small neighborhood of a fixed point. This local theory is a fundamental tool in understanding more global dynamics. It has been studied for well over a hundred years by mathematicians such as Schröder, Koenigs, Leau, Böttcher, Fatou, Cremer, Siegel, Ecale, Voronin, Cherry,...
The first three sections will present an overview of some background material.
The local study of iterated holomorphic mappings, in a neighborhood of a fixed point, was quite well developed in the late 19th century. (Compare §§8–10, and see Alexander.) However, except for one very simple case studied by Schröder and Cayley (see Problem 7-a), nothing was known about the global behavior of iterated holomorphic maps until 1906,...
This article discusses the dynamics of iterated cubic maps
An exposition of results of Yoccoz, Branner, Hubbard and Douady concerning polynomial Julia sets. The contents are as follows: 1. Local connectivity of quadratic Julia sets (following Yoccoz) . 2 2. Polynomials for which all but one of the critical orbits escape (following Branner and Hubbard) . . . . . . . . . . . . 21 3. An innitely renormalizabl...
There are fifteen prime numbers p for which the normalizer of the congruence subgroup-0 (p) in SL(2; R) has the "genus-zero property"; that is, the compactification of the upper half-plane modulo this normalizer is a Riemann surface of genus zero, so that the field of modular functions in-variant under this discrete group is generated by only one f...
ere. However, McMullen [Mc2] produced very simple examples for the more general question in which we replace the rational map by a transcendental function, such as the map z 7! sin(z) of Figure 1. (A dierent and more complicated example was given in [EL].) Figure 1. The Julia set for z 7! sin(z), shown in black, has positive area but no interior po...
with his father at age seventeen. His thesis, at age twenty-one, presented clear and elementary mathematical ideas that inaugurated a slow revolution in fields as diverse as economics, political science, and evolutionary biology. During the following nine years, in an amazing surge of mathematical activity, he sought out and often solved the toughe...
It has been known for some time that the topological entropy is a nondecreasing function of the parameter in the real quadratic family, which corresponds to the intuitive idea that more nonlinearity induces more complex dynamical behavior. Polynomial families of higher degree depend on several parameters, so that the very question of monotonicity n...
The Novikov Conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds. These two volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September 1993, on the subject of 'Novikov Conjectures, Index Theorems and Rigidity'. They...
John E Nash created an impressive array of exciting mathematics during the 10 years of his mathematical activity. To some, the brief paper written at age 21, for which he has won a Nobel prize I in economics, may seem like the least of his achievements. Nevertheless, I applaud the wisdom of the selection committee in making this award. It is notori...
This article is an expository description of quadratic rational maps from the Riemann sphere to itself.
This article is an expository description of quadratic rational maps from the Riemann sphere to itself.
Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichm\"uller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of J...
This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.
This note will show, as an immediate consequence of a theorem of Fried, that many Hénon maps are not expansive.
Consider a fixed lattice L in n -dimensional euclidean space, and a finite set K of symbols. A correspondence a which assigns a symbol a(x) ∈K to each lattice point x ∈ L will be called a configuration. An n -dimensional cellular automaton can be described as a map which assigns to each such configuration a some new configuration a′ = f (a) by a fo...
This note proposes a definition for the concept of attractor, based on the probable asymptotic behavior of orbits. The definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor.
It will be convenient to use the term Kepler orbit for any curve x = x(t) in 3-space which arises as a solution to the Newtonian two body problem. Hamilton showed that the velocity vector v = dx/dt, associated with any nondegenerate Kepler orbit, moves along a circle. Following Györgyi, Moser, Osipov and Belbruno, this velocity circle can be interp...
This article outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a Riemannian metric invariant under left translation.
Nature is the international weekly journal of science: a magazine style journal that publishes full-length research papers in all disciplines of science, as well as News and Views, reviews, news, features, commentaries, web focuses and more, covering all branches of science and how science impacts upon all aspects of society and life.
In this concluding chapter we briefly describe some examples of bilinear forms which arise naturally in topology, in differential geometry, and in number theory. The three sections of this chapter are completely independent.
This chapter will define the concept of an inner product space over a commutative ring R, and describe basic constructions which are independent of the ring R. In particular it introduces the Witt ring W(R), which will play a central role in later chapters. Roughly speaking, W(R) is the collection of all symmetric inner product spaces X over R modu...
This chapter will describe some of the highlights of the theory of the Witt ring W(F), where F is an arbitrary field. We are particularly careful to give proofs which are valid also in characteristic 2. The classical theory for number fields, as described for example in [O’Meara], is largely ignored.
The first section of this chapter defines the two residue class form homomorphisms associated with a discrete valuation of a field. Section 2 uses the second residue class form homomorphisms to compute the Witt ring W(Q) of the rational numbers, and to give a new proof that W(Z)≅Z.
This chapter will discuss the classification problem for inner product spaces over the ring Z of rational integers. All inner products are to be symmetric. Our presentation is based on the classical theorem of Minkowski concerning lattice points in a convex symmetric subset of R
n
. This theorem is first used to classify inner product spaces of ran...
Thesis--University of Tsukuba, D.Sc.(B), no. 188, 1984. 3. 22
By an inner product space V will be meant a finite dimensional vector space V over a field F together with a non-degenerate, bilinear, F-valued inner product (u, v) which is either symmetric or skew symmetric. The group of isometries t of V (that is F-linear automorphisms satisfying the identity (tu, t v)=(u, v)) will be denoted by O(V) in the symm...
Citations
... Of particular interest is the case of VU maps with finite turning point orbits as they will correspond to post-critically finite polynomials. We will follow the approach in [BMS21] to describe the combinatorics of these maps. We will say that a piecewise monotone map f : Thus, for each j ∈ {0, 1, . . . ...
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... One key open question from both the complex dynamical and number theoretic perspectives is the following variant of a conjecture of Milnor from [21]. ...
... We recall classical results that are either explicitly in [12] or can easily be deduced from the material there. ...
... The proof of Theorem 4.1(2) is very complicated. Firstly, we will use the polynomiallike mapping theory introduced by Douady and Hubbard in[14]and the Branner– Hubbard–Yoccoz puzzle technique[5]; see also[18]. Recall that a polynomial-like mapping of degree d is a triple (U, V, g), where U and V are simply connected planar domains with V ⊂ U and g : V → U is a holomorphic proper mapping of degree d. The filled Julia set K g of the polynomial-like mapping g is defined as K g = {z ∈ V : g k (z) ∈ U for all k ≥ 0}.In the following, we introduce the Branner–Hubbard–Yoccoz puzzle technique. ...
Reference: On the dynamics of generalized McMullen maps
... Carathéodory's prime end theory received developments on the plane R 2 (see [3], [4]) and in the space R n for n > 2 (see [5], [6]), in studying Dirichlet problems for elliptic equations [7], and in the theory of dynamical systems (see [8], [9]). For more detailed surveys of the available results and literature, see [10]- [13]. ...
... Proof. Recall that K 2 (Z) = Z/2 (see, e.g., [M,Corollary 10.2]). However, the map K 2 (C l , Z) → K 2 (C l+1 , Z) is surjective by [St78,Corollary 3.2], and lim l K 2 (C l , Z) = K 2 Sp(Z) = Z, see, e.g., [Sch,Theorem 2.1]. ...
... 1/.d 2/, one of the analytic singularities in this topological type is the hypersurface .X d ; 0/ in C 3 , given by x d C y d C z d D 0. For each d 5, the results of Baykur, Monden and Van Horn-Morris [7] produce arbitrarily long positive factorizations of the corresponding open book monodromy, which in turn yields infinitely many Stein fillings for the link .Y d ; d /; in particular, there are Stein fillings with arbitrarily large b 2 . One might hope that most of these Stein fillings are unexpected: indeed, a hypersurface singularity has a unique Milnor fiber, and its topology is well understood; see Milnor [39] and Tyurina [64]. However, the question is more subtle: because .X d ; 0/ is not (pseudo)taut (see Laufer [32]), there are infinitely many singularities with the same link .Y d ; d /. ...
... In this paper we determine the topology of the Riemann surfaces L m . To state our main result, we recall the notion of a 2-dimensional orbifold [7], [27]. ...
... In other words, the basin of an attractor is the set of initial points whose trajectories tend toward the attractor. In summary, the theory of attractor in dynamical systems simply provides a way of describing the asymptotic behaviour of typical trajectories in a dynamical system (Auslander et al., 1964;Buescu, 1991;Dénes and Makey, 2011;Milnor, 2010;Ruelle, 1981). ...
... For each Möbius transformation T ∈ PSL 2 (C) and each rational map φ ∈ Rat d , it holds that φ T := T •φ•T −1 ∈ Rat d , and the induced map φ → φ T is a holomorphic automorphism of Rat d . The quotient space M d = Rat d /PSL 2 (C), the moduli space of complex rational maps of degree d, has the structure of a connected complex orbifold of dimension 2(d − 1) [17]. If φ ∈ Rat d and Aut(φ) is its PSL 2 (C)-stabilizer (called its group of holomorphic automorphisms), then [φ] ∈ M d is a cone point of the orbifold structure of M d if and only if Aut(φ) is non-trivial. ...
Reference: On Real and Pseudo-Real Rational Maps
