Gaven J. Martineastwood/GDEpdfs/martin.pdf · Quasiregular mappings Rational Endomorphisms The...

Classical Conformal Dynamics on the Riemann SphereHigher Dimensions

Examples and Results

Curvature, Dynamics and Quasiregular Mappings

Gaven J. Martin

Australian National University, September 2011

G.J.Martin Curvature, Dynamics and Quasiregular Mappings

Introduction1 Classical Conformal Dynamics on the Riemann Sphere

Classical 2D TheoryQuestionsRational maps of surfaces

2 Higher DimensionsRigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem

3 Examples and ResultsNon-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings

Classical 2D Theory

Iteration of Rational Maps of C ≈ S2: Fatou-Julia Theoryconformal self mappings of the Riemann sphere are rational

f (z) =p(z)

p, q polynomials with no common roots.

Gaston Julia (right with Herglotz) and Pierre Fatou

Classical 2D Theory

Study the dynamics of the sequence

z → f (z)→ f (f (z))→ · · · → f (n)(z)→ · · ·

Fatou Set: the set where there is stable dynamics.

{z : ∃ nbd U of z and {f (n)∣∣∣U}∞n=1 forms a nomal family }

Julia Set: The complement of the Fatou set, where thedynamics is unstable.

Classical 2D Theory

z → f (z)→ f (f (z))→ · · · → f (n)(z)→ · · ·

Classical 2D Theory

z → f (z)→ f (f (z))→ · · · → f (n)(z)→ · · ·

Classical 2D Theory

Newtons method for solving z3 = 1. The iterative procedure isf : z 7→ z − g(z)/g ′(z) with g(z) = z3 − 1 giving the rational map3f (z) = 2z + z−2. The sequence z → f (z)→ · · · → f (n)(z) · · ·converges to the roots

Classical 2D Theory

Complex parameter spaces - for instance the Mandelbrot set

Parameterizes the dynamics of iterating z 7→ z2 + c .Black area is where the Julia set is connected.

Classical 2D Theory

Toy Models of more complex dynamical systems

Figure: A Julia set - dynamic plane

manageable examples with generic features

density of hyperbolicity/expanding dynamics

structural stability

Classical 2D Theory

classifications of components of the Fatou set

hyperbolic parabolic Siegel or rotational

Figure: A Siegel domain

Natural Questions

Conformal Dynamics on other surfaces ?

Conformal Dynamics in Higher Dimensions ?

Structure and Properties of Rational Maps ?- classification of dynamics of stable basins into attracting,super-attracting, parabolic, rotational and questions oflinearisation.- density of repellors- structure of branch sets

Relationship between Dynamics, Curvature and Topology

Natural Questions

Rational maps of Riemann sufaces

The Riemann-Hurwitz formula shows that analytic self maps of ahyperbolic surface are homeomorphisms. Let X and Y be compactRiemann surfaces with a non-constant analytic map f : X → Y .Then

2g(X )− 2 = deg(f )(2g(Y )− 2) +∑y

(by − 1)

where by are the branch points.

If X = Y , this leaves only the Riemann sphere and the torus andonly those on the sphere can be branched (not locally injective)

So nothing new !

Rational maps of Riemann sufaces

The Riemann-Hurwitz formula shows that analytic self maps of ahyperbolic surface are homeomorphisms. Let X and Y be compactRiemann surfaces with a non-constant analytic map f : X → Y .Then

2g(X )− 2 = deg(f )(2g(Y )− 2) +∑y

(by − 1)

where by are the branch points.

If X = Y , this leaves only the Riemann sphere and the torus andonly those on the sphere can be branched (not locally injective)

So nothing new !

RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem

Rigidity

Liouville Theorem:

Conformal maps in dimension n ≥ 3 are local restrictions ofMobius transformations

Look to more general notions of conformality

Definition If M and N are manifolds with conformal metrictensors G and H, then f : M → N is conformal iff∗〈·, ·〉G = λ(x)〈·, ·〉H for some scalar function λ. Equivalently

Df t(x)H(f (x))Df (x) = J(x , f )2/nG (x)

Rigidity

Liouville Theorem:

Conformal maps in dimension n ≥ 3 are local restrictions ofMobius transformations

Look to more general notions of conformality

Definition If M and N are manifolds with conformal metrictensors G and H, then f : M → N is conformal iff∗〈·, ·〉G = λ(x)〈·, ·〉H for some scalar function λ. Equivalently

Strong Rigidity

Liouville Theorem:

If G and H are close to continuous in BMO, then any f ∈W 1,nloc

satisfying the Beltrami system

is a local homeomorphism.

So no branching and if M = N = Sn, then monodromy gives f ahomeomorphism.

Can there be any rational mappings ?

Strong Rigidity

Liouville Theorem:

If G and H are close to continuous in BMO, then any f ∈W 1,nloc

satisfying the Beltrami system

is a local homeomorphism.

So no branching and if M = N = Sn, then monodromy gives f ahomeomorphism.

Can there be any rational mappings ?

Quasiregular mappings

The theory of quasiregular mappings was developed in the 70’s byResetnyak, Martio-Rickman-Vaisala and Gehring to capture thegeometric aspects of analytic mappings in higher dimensions.

Basically a quasiregular mapping is a (possibly) branched mappingof bounded distortion. Typically one assumes the Sobolev regularityf ∈W 1,n

loc and the distortion inequality: there is K <∞ such that

‖Df (x)‖n ≤ K J(x , f ) almost everywhere

Easy examples of such mappings would be piecewise linearbranched coverings between spaces with fat triangulations.Another example is the winding map of Rn in spherical coordinates

(r , θ, ψ) 7→ (r , nθ, ψ), r ≥ 0, θ ∈ [0, 2π]

However they can be much more complicated.

Quasiregular mappings are discrete (point inverses are discrete)and open. By definition, for any H there is G such that f satisfiesthe Beltrami system

Df t(x)H(f (x))Df (x) = J(x , f )2/nG (x), almost everywhere

but G and H are only measurable ! Thus we consider measurableconformal structures

There are many interesting aspects of the theory of quasiregularmappings in higher dimensions, but a real highlight is Rickman’sversion of the value distribution theory, and with it the approriateversions of Montel’s theorem and so forth.

Rational Maps of Manifolds

Definition. Let Mn be a closed manifold. We say thatf : Mn → Mn is rational if there is a bounded measurableconformal structure G on Mn preserved by f : that is

Df t(x)G (f (x))Df (x) = J(x , f )2/nG (x), almost everywhere

If f is a rational map, then so is f (n) = f ◦ f ◦ · · · ◦ f - the space ofsolutions to this equation is closed under composition. This impliesthe semigroup {f (n) : n ∈ Z} consists of quasiregular mappings allwith a uniform distortion bound - f is uniformly quasiregular. NowRickman’s theorem on normal families applies and there is a verynice Fatou-Julia theory.

But : are there any examples ?

Rational Maps of Manifolds

Definition. Let Mn be a closed manifold. We say thatf : Mn → Mn is rational if there is a bounded measurableconformal structure G on Mn preserved by f : that is

Df t(x)G (f (x))Df (x) = J(x , f )2/nG (x), almost everywhere

If f is a rational map, then so is f (n) = f ◦ f ◦ · · · ◦ f - the space ofsolutions to this equation is closed under composition. This impliesthe semigroup {f (n) : n ∈ Z} consists of quasiregular mappings allwith a uniform distortion bound - f is uniformly quasiregular. NowRickman’s theorem on normal families applies and there is a verynice Fatou-Julia theory.

But : are there any examples ?

The Lichnerowicz Problem

Lichnerowicz Theorem (Ferrand 70’s)

If Mn is closed and admits a non-compact conformal automorphismgroup, then Mn is (in the appropriate category) the n-sphere Sn

This motivates us to formulate the following

Rational Lichnerowicz Problem

Classify those closed n-manifolds which admit a non-injectiverational map. Such a map will generate a non-compact semigroupof rational transformations.

The Lichnerowicz Problem

Lichnerowicz Theorem (Ferrand 70’s)

If Mn is closed and admits a non-compact conformal automorphismgroup, then Mn is (in the appropriate category) the n-sphere Sn

This motivates us to formulate the following

Rational Lichnerowicz Problem

Classify those closed n-manifolds which admit a non-injectiverational map. Such a map will generate a non-compact semigroupof rational transformations.

Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings

Existence: Non-negative Curvature

Positive Curvature Theorem

Every n-sphere and Lens space admits a non-injective (indeed notlocally injective) rational endomorphism.

Flat case I. Strong Rigidity Theorem

Let Mn admit a locally injective (but not globally injective) rationalmap f . Then up to change of coordinates by a homeomorphism ofbounded distortion (quasiconformal) Mn is a Bieberbach manifoldand f is multiplication on the universal cover Rn.

Flat case II. Topological Rigidity Theorem

A Bieberbach manifold Mn admits no continuous open branchedself maps

Julia sets and dynamics

Linearisation: a local quasiconformal change of coordinates to astandard conformal model such as rotation or dilation. We haveexamples of rational maps of the n-sphere Sn with

Julia set a Cantor set with attracting or super-attracting basin.

Julia set = Sn - chaotic Lattes examples

Julia set a codimension one sphere withsuper-attracting basins, no model for linearisation !attracting basins, can linearize by quasiconformal to x 7→ 1

2xparabolic basin. topological obstruction to linearisation - wildtranslation arcs parabolic examples with linearisation and alsowithout linearisation.

Siegel basins ? No known examples with rotational dynamics,however if they exist they can be “linearised”

Repelling fixed points can be linearised to x 7→ 2x

Julia set a codimension one sphere withsuper-attracting basins, no model for linearisation !

attracting basins, can linearize by quasiconformal to x 7→ 12x

parabolic basin. topological obstruction to linearisation - wildtranslation arcs parabolic examples with linearisation and alsowithout linearisation.

Other examples:

Figure: A knotted Julia set

Three dimensions:

Julia sets can be

A circle or any (p, q)-torus knot. Must be a parabolic basin.

A square [0, 1]× [0, 1]× {0} ⊂ S3

Can’t be any other knot or link.

Can’t be any closed surface other than the sphere.

For example:

Figure: A Julia set in S3 with infinitely many Fatou components;period two cycle 3rd generation

Stoilow Theorem:

A discrete open mapping of the Riemann sphere is a conformalmapping after a topological change of coordinates.

We have the following version of this theorem.

Stoilow Factorisation (rational maps are generic topologically)

Every quasiregular mapping g : Sn → Sn is of the form g = h ◦ fwhere h is a quasiconformal homeomorphism of Sn and f isrational self mapping of Sn

Stoilow Theorem:

A discrete open mapping of the Riemann sphere is a conformalmapping after a topological change of coordinates.

We have the following version of this theorem.

Stoilow Factorisation (rational maps are generic topologically)

Every quasiregular mapping g : Sn → Sn is of the form g = h ◦ fwhere h is a quasiconformal homeomorphism of Sn and f isrational self mapping of Sn

Negative Curvature

We have

Negative Curvature Theorem

A closed negatively curved manifold cannot admit a non-injectiverational map.

The proof is based around a dynamical renormalisation argument toshow qr-ellipticity (existence of a non-constant map h : Rn → Mn

of bounded distortion). Then Gromov, Varopoulos, Saloff-Coste,Coulhon show this implies that Mn has amenable fundamentalgroup. Negative curvature implies non-amenable though

What about topological rigidity ? Can a negatively curvedmanifold admit a branched open self mapping ?

Negative Curvature

We have

Negative Curvature

We have

Negative Curvature

Of course there must be some restriction on the type of mapping.

Wilsons solution of Whyburn Conjecture

Every topological n-manifold, n ≥ 3, admits a non-injective openself mapping with point inverses totally disconnected - finite or aCantor set (called light open mappings)

Topological Rigidity I

Let Mn be closed and negatively curved. Then every discrete openself mapping is a homeomorphism

Topological Rigidity II

Let Mn be closed and negatively curved (n 6= 4). Then every openself mapping is homotopic to a homeomorphism

Negative Curvature

Open mappings and curvature

Walsh Smale Theorem

Let f : Mn → Nn be a proper open surjection between topologicalmanifolds. Then |f∗(π1(Mn)) : π1(Nn)| <∞

Next we have the following refinement of Sela’s work on the Hopfproperty for torsion free Gromov hyperbolic groups.

Virtual Hopf Property for Hyperbolic Groups

Let Γ be a torsion free Gromov hyperbolic group and ϕ : Γ→ Γ ahomomorphism with image of finite index |ϕ(Γ) : Γ| <∞. Then ϕis an isomorphism.

In the case of negative curvature our open mapping is evidently ahomotopy equivalence so we may apply the beautiful results ofFarrell and Jones to deduce homotopic to a homeomorphism.

Walsh Smale Theorem

Coauthors

Figure: A ragtag bunch

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