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John Milnor

Publications and source records attributed to John Milnor.

13 recordsLinked to original sources
arXiv2025

Cubic Polynomial Maps with Periodic Critical Orbit, Part III: Tessellations and Orbit Portraits

We study the parameter space ${\mathcal S}_p$ for cubic polynomial maps with a marked critical point of period $p$. We will outline a fairly complete theory as to how the dynamics of the map $F$ changes as we move around the parameter space ${\mathcal S}_p$. For every escape region ${\mathcal E}\subset {\mathcal S}_p$, every parameter ray in ${\mathcal E}$ with rational parameter angle lands at some uniquely defined point in the boundary $\partial{\mathcal E}$. This landing point is necessarily either a parabolic map or a Misiurewicz map. The relationship between parameter rays and dynamic rays is formalized by the period $q$ tessellation of ${\mathcal S}_p$, where maps in the same face of this tessellation always have the same period $q$ orbit portrait.

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arXiv2020

The W. Thurston Algorithm for Real Quadratic Rational Maps

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 application to topological entropy.

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arXiv2018

Group Actions, Divisors, and Plane Curves

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 transformations of ${\mathbb P}^1$; and then the moduli space of effective 1-cycles with finite stabilizer on ${\mathbb P}^2$ modulo the group ${\rm PGL}_3$ of projective transformations of ${\mathbb P}^2$.

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arXiv2015

Antipode Preserving Cubic Maps: the Fjord Theorem

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 rotation number.

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arXiv2012

Hyperbolic Components

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 components, and those hyperbolic components with compact closure (or equivalently those contained in the "connectedness locus") are called bounded hyperbolic components. It is shown that each bounded hyperbolic component is a topological cell containing a unique post-critically finite map called its center point. For each degree $d$, the bounded hyperbolic components can be separated into finitely many distinct types, each of which is characterized by a suitable reduced mapping scheme $\bar S_f$. Any two components with the same reduced mapping scheme are canonically biholomorphic to each other. There are similar statementsfor real polynomial maps, for polynomial maps with marked critical points, and for rational maps. Appendix A, by Alfredo Poirier, proves that every reduced mapping scheme can be represented by some classical hyperbolic component, made up of polynomial maps of $\C$. This paper is a revised version of [M2], which was circulated but not published in 1992.

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arXiv2009

Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions

The parameter space $\mathcal{S}_p$ 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 $\mathcal{S}_p$ 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 an essentially unique Puiseux series. This note will describe the topology of $\mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $\mathcal{S}_p$.

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arXiv2006

Schwarzian Derivatives and Cylinder Maps

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 $(\R/\Z)\times I$.

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arXiv2006

Elliptic Curves as Attractors in ${\mathbb P}^2$ Part 1: Dynamics

A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe ``blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, the elliptic curve (a topological torus) can never have a trapping neighborhood, yet it can have an attracting basin of large measure (perhaps even of full measure). We also describe examples where there appear to be Herman rings (that is topological cylinders mapped to themselves with irrational rotation number) with open attracting basin. In some cases we provide proofs, but in other cases the discussion is empirical, based on numerical computation.

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arXiv1999

On Entropy and Monotonicity for Real Cubic Maps

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 needs to be reformulated. For instance, one can say the entropy is monotone in a multiparameter family if the isentropes, or sets of maps with the same topological entropy, are connected. Here we reduce the problem of the connectivity of the isentropes in the real cubic families to a weak form of the Fatou conjecture on generic hyperbolicity, which was proved to hold true by C. Heckman. We also develop some tools which may prove to be useful in the study of other parameterized families, in particular a general monotonicity result for stunted sawtooth maps: the stunted sawtooth family of a given shape can be understood as a simple family which realizes all the possible combinatorial structures one can expect with a map of this shape on the basis of kneading theory. Roughly speaking, our main result about real cubic families is that they are as monotone as the stunted sawtooth families with the same shapes because of Heckman's result (there are two posible shapes for cubic maps, depending on the behavior at infinity).

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arXiv1992

Problems in holomorphic dynamics

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller 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 Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert Devaney, Open questions in non-rational complex dynamics; Alexandre Eremenko and Mikhail Lyubich, Wandering domains for holomorphic maps. 5. Newton's Method: Scott Sutherland, Bad polynomials for Newton's method

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