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
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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)
q(z)
p, q polynomials with no common roots.
Gaston Julia (right with Herglotz) and Pierre Fatou
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
Classical 2D Theory
classifications of components of the Fatou set
hyperbolic parabolic Siegel or rotational
Figure: A Siegel domain
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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 !
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Classical 2D TheoryQuestionsRational maps of surfaces
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 !
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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)
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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)
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
Strong Rigidity
Liouville Theorem:
If G and H are close to continuous in BMO, then any f ∈W 1,nloc
satisfying the Beltrami system
Df t(x)H(f (x))Df (x) = J(x , f )2/nG (x)
is a local homeomorphism.
So no branching and if M = N = Sn, then monodromy gives f ahomeomorphism.
Can there be any rational mappings ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
Strong Rigidity
Liouville Theorem:
If G and H are close to continuous in BMO, then any f ∈W 1,nloc
satisfying the Beltrami system
Df t(x)H(f (x))Df (x) = J(x , f )2/nG (x)
is a local homeomorphism.
So no branching and if M = N = Sn, then monodromy gives f ahomeomorphism.
Can there be any rational mappings ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
Quasiregular mappings
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
Quasiregular mappings
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
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 ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
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 ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
RigidityLocal RigidityQuasiregular mappingsRational EndomorphismsThe Lichnerowicz Problem
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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→ 12x
parabolic 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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
2x
parabolic 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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
Julia sets and dynamics
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
Julia sets and dynamics
For example:
Figure: A Julia set in S3 with infinitely many Fatou components;period two cycle 3rd generation
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
Julia sets and dynamics
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
Julia sets and dynamics
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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 ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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 ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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 ?
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
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.
G.J.Martin Curvature, Dynamics and Quasiregular Mappings
Classical Conformal Dynamics on the Riemann SphereHigher Dimensions
Examples and Results
Non-negative CurvatureJulia SetsNegative CurvatureVirtual Hopf Property and Open Mappings
Coauthors
Figure: A ragtag bunch
G.J.Martin Curvature, Dynamics and Quasiregular Mappings