Annales de la Faculté des Sciences de Toulouse

Vincent Guedj

Joaquim Ortega-Cerdà

Pascal J. Thomas

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
VINCENT GUEDJ, JOAQUIM ORTEGA-CERDÀ, PASCAL J. THOMAS
KAWA Lecture Notes
Tome XXII, no 4 (2013), p. i-iv.
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© Université Paul Sabatier, Toulouse, 2013, tous droits réservés.
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org/legal/). Toute reproduction en tout ou partie de cet article sous quelque
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KAWA Lecture notes
This volume collects lecture notes of some of the mini-courses delivered
during the first four editions of KAWA,
Komplex Analysis Winterschool/Workshop and Applications.
Why KAWA?
Several Complex Variables in the 21st century. Complex analysis in
several variables (usually abbreviated as SCV) went through a golden age
during the second half of the 20th century. These last few years, what used
to be the core of SCV saw most activity gradually desert it, to move to
the interface with other subfields. Complex analysis methods thus proved
decisive in recent developments for (direct and) inverse problems in the
spectral theory of self-adjoint differential operators, in probability theory
(Schramm-Loewner evolution equation), in signal processing (Gabor anal-
ysis), in random matrix theory, in conformal field theory, in the statistical
analysis of critical two-dimensional networks.
Moving in a different direction, the ergodic theory of rational maps is a
very rich and vigorously expanding field. It is using many tools from, and
motivates many new questions in complex analysis and complex geometry.
One may mention, too, the recent and fruitful interaction between pluripo-
tential theory and differential and Kählerian geometry, as well as the – ever
stronger – interaction between complex analysis and analytic and algebraic
geometry.
In order to pass on the knowledge accumulated along the last few decades
by complex analysis, as well as to acquaint young and confirmed inves-
tigators with the methods of those domains where new interactions take
place, researchers from Toulouse and Barcelona with a well-established track
record of cooperation (within the Journées Complexes du Sud) have started
a yearly thematic event:
KAWA, Komplex Analysis Winterschool/Workshop. A yearly meet-
ing made up of a week of mini-courses followed by a workshop (short con-
– i –
Introduction
ference) with a stable location. The mini-courses being held in Toulouse,
Marseille or Barcelona, the workshop in the same place or in a neighboring
town. A short list of the main topics covered is:
• One and several variable complex dynamics
• Analytic, differential and almost complex geometry
• New developments in function and operator theory
• Pluripotential theory and applications
This project is meant to last. Once in a while, a larger conference, of
European scope, will be planned (the first of those will take place in 2014
at CIRM, in Marseille).
Organizing committee. The organizers of these first editions were
• Vincent GUEDJ, Université Paul Sabatier (FRANCE),
• Jordi MARZO1, Universitat de Barcelona (ESPAGNE),
• Joaquim ORTEGA-CERDÀ, Universitat de Barcelona (ESPAGNE),
• Pascal THOMAS, Université Paul Sabatier (FRANCE),
Scientific committee. The scientific committee of KAWA consisted of:
• Bo BERNDTSSON, University of Gothenburg (SWEDEN);
• Jean-Pierre DEMAILLY, Université J. Fourier (FRANCE);
• Julien DUVAL, Université Paris 11 (FRANCE);
• Franc FORSTNERIČ, Ljubljana University (SLOVENIA);
• Laszlo LEMPERT, Purdue University (USA).
Courses
This volume consists of sets of lecture notes of those courses given at
KAWA which were not already the object of a published article or survey.
Here follows a list of all the courses delivered (five hours each):
KAWA 1. [Toulouse & Albi, 25-29.01.2010]
1. Sébastien BOUCKSOM (Université Paris 7, France): Transfinite
diameter and equilibrium measures on complex manifolds.
See Berman, R., Boucksom, S., Witt Nyström, D., Fekete points
and convergence towards equilibrium measures on complex manifolds,
Acta Math. 207 (2011), no. 1, 1–27.
(1) Starting with KAWA 3.
– ii –
Introduction
2. Misha LYUBICH (Stony Brook University, USA):
Lee-Yang zeros and 2D rational dynamics.
See http://www.math.sunysb.edu/∼mlyubich/papers/index.html
3. Alexandre SUKHOV (Université de Lille, France):
Complex methods in symplectic topology.
4. Dror VAROLIN (Stony Brook University, USA):
Bergman Kernels in Complex Analytic Geometry.
See http://arxiv.org/abs/math/0511225
KAWA 2. [CIRM, 31.01-05.02.2011]
1. Bo BERNDTSSON (CTH, Sweden):
Kähler Geometry and convexity.
2. Charles FAVRE (Ecole Polytechnique, France):
The Cremona group.
See : Favre, C., Le groupe de Cremona et ses sous-groupes de type
fini, Séminaire Bourbaki. Astérisque 332 (2010), Exp. 998, 11-43.
3. Kristian SEIP (NTNU, Trondheim, Norway): Hardy spaces of Dirich-
let series and function theory on polydiscs
See http://arxiv.org/abs/1206.2815
KAWA 3. [Barcelona, 30.01-04.02.2012]
1. Romain DUJARDIN (Ecole Polytechnique, France).
Parameter spaces of holomorphic dynamical systems.
See Dujardin, R. Bifurcation currents and equidistribution in param-
eter space. To appear in Frontiers in complex dynamics (celebrating
John Milnor’s 80th birthday).
2. Steve ZELDITCH (Northwestern University, USA).
Planck’s constant in stochastic complex geometry.
See http://euclides.imub.ub.es/kawa12/papers/Zelditch.pdf
3. Ahmed ZERIAHI (IMT, France). A viscosity approach to degen-
erate complex Monge-Ampère equations.
KAWA 4. [Toulouse & Albi, 21-27.01.2013]
1. Kari ASTALA (Helsinki University, Finland), Holomorphic defor-
mations in calculus of variations and geometry.
See Astala, K., Iwaniec, T., Prause, I., Saksman, E., Burkholder in-
tegrals, Morrey’s problem and quasiconformal mappings, J. Amer.
Math. Soc. 25 (2012), no. 2, 507-531.
– iii –
Introduction
2. Robert BERMAN (CTH, Sweden), Real Monge-Ampère equa-
tions and Kähler-Einstein metrics on toric varieties.
3. Franc FORSTNERIČ (Ljubljana University, Slovenia)
Oka manifolds.
Acknowledgements
First of all, the continued existence of the KAWA series wouldn’t have
been possible without the participation of dozens of people from all around
the world (from 60 to 100 for each edition), and we the organizers are
thankful for the trust they put in us. Also, thanks are due to the advisors
at various universities who encouraged their students to go and found them
funding. We also have to thank all those who gave talks, and all the others
who offered to do so, but couldn’t be included in our programme, severely
constrained by time; and the scientific committee who helped and guided
us through the difficult process of selecting which talks would be given.
We are thankful, too, for the financial support from various sources which
enabled us to subsidize, in particular, part of the expenses of most of the
younger participants in the various editions of KAWA:
Our institutions, the Institut de Mathématiques de Toulouse, the Univer-
sité Paul Sabatier, the Universitat de Barcelona; the Institut de Matemàtica
de la Universitat de Barcelona (IMUB), the Agence Nationale de la Recherche
(through its MACK programme), the Institut Universitaire de France, the
CNRS, the CIRM, the CTP (Communauté de Travail des Pyrénées), the
Région Midi-Pyrénées, the Département of Tarn.
Most of all, we wish to thank the Annales de la Faculté des Sciences de
Toulouse for offering us the opportunity to have this special issue devoted
to KAWA, the anonymous referees, and the authors of this volume who, in
addition to preparing the courses they gave, put in the extra work to write
up the papers presented here.
– iv –


KAWA Lecture Notes

Nicola Ermotti

Cam Van Quach Hongler

Claude Weber

A proof of Tait’s Conjecture on prime alternating $-$achiral knots

M. Dickmann

A. Petrovich

Spectral Real Semigroups

Ohad Giladi

Assaf Naor

Gideon Schechtman

Bourgain’s discretization theorem

Guizhen Cui

Wenjuan Peng

Lei Tan

On a theorem of Rees-Shishikura

Gilles Carron

Inégalité de Sobolev et volume asymptotique

Nicolas Bousquet

Accelerated Monte Carlo estimation of exceedance probabilities under monotonicity constraints

Adam Epstein

Thomas Sharland

A classification of bicritical rational maps with a pair of period two superattracting cycles

John Hubbard

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
JOHN HUBBARD
Matings and the other side of the dictionary
Tome XXI, no S5 (2012), p. 1139-1147.
<http://afst.cedram.org/item?id=AFST_2012_6_21_S5_1139_0>
© Université Paul Sabatier, Toulouse, 2012, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci-
ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique
l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous quelque
forme que ce soit pour tout usage autre que l’utilisation à fin strictement
personnelle du copiste est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales de la Faculté des Sciences de Toulouse Vol. XXI, n◦ 5, 2012
pp. 1139–1147
Matings and the other side of the dictionary
John Hubbard
ABSTRACT. — In the theory of rational maps an important role is played
by matings. These are probably the best understood of all rational func-
tions, but they are bizarre, and involve gluing dendrites together to get
spheres carrying Peano curves.
In the theory of Kleinian groups, there is a parallel construction, the
construction of double limits, that is central to Thurston’s hyperbolization
theorem for 3-manifolds that fiber over the circle with pseudo-Anosov
monodromy. It also involves gluing dendrites and Peano curves.
Clearly these two constructions form one entry of the Sullivan dictionary.
This article attempts to spell out the similarities and differences.
RÉSUMÉ. — Les accouplements forment une classe essentielle d’applica-
tions rationelles, sans doute celle qui est la mieux comprise. Mais elle
fait intervenir des objets bizarres: recollements de dendrites, courbes de
Peano, etc.
La construction analogue pour les groupes Kleiniens est celle des limi-
tes doubles. Cette construction est essentielle pour l’hyperbolisation des
variétés de dimension trois fibrées sur le cercle.
Ces deux constructions se correspondent par le dictionnaire de Sullivan.
Cet article essaie de montrer les similitudes et les différences.
1. Introduction: the Sullivan dictionary
Sullivan’s dictionary pairs statements about 1-dimensional complex dy-
namics and statements about 3-dimensional hyperbolic geometry. For in-
stance
– 1139 –
John Hubbard
Dynamical systems
Julia set
non-attracting cycles
Yoccoz inequality
No wandering domains
Tuning
Topological characterization of
rational functions
Parabolic blow-ups
Hyperbolic geometry
Limit set
Non-elliptic fixed points
Bers compactness
Ahlfors finiteness
Klein-Maskit combination
Thurston’s hyperbolization theorem
for Haken manifolds
Geometric limits
In some cases, the constructions on the two sides look similar although
the proofs are different; for instance, the proof of the density of fixed points
in the limit set for Kleinian groups using the convex hull in hyperbolic 3-
space has no known analog in proving the density of repelling cycles in the
Julia set.
In other cases the actual details of the proofs are amazingly similar
even though the statements sound unrelated; this is particularly true of
Sullivan’s No-wandering Domains theorem, whose proof was modeled on
the Ahlfors finiteness theorem. It is also the case for the Yoccoz inequality
and the compactness of Bers slices: both depend on bounds on the modulus
of an annulus embedded in a torus. Further, Thurston’s theorem on the
Topological Characterization of Rational Functions was modeled after his
Hyperbolization Theorem for Haken Manifolds.
There is a glaring gap in the dictionary: there is no known analog on
the dynamical side of hyperbolic space H3. Thurston claims that this is the
greatest weakness of the theory.
In this article I will look at the entries
Dynamical systems
Matings
Hyperbolic geometry
Double limit theorem
In this case, the constructions are clearly parallel, but the known proofs
are quite different. Maybe they don’t need to be; a familiarity with one side
of the dictionary may inspire new and better proofs on the other.
There is nothing really new here, but there is a lack of communication
between the two fields and many practitioners of one are unaware of the
other.
– 1140 –
Matings and the other side of the dictionary
2. The two settings
2.1. The mating of polynomials
Let p1, p2 be two polynomials of the same degree d with connected filled-
in Julia sets K1,K2. Choose Böttcher coordinates
φ1 : C−K1 → C− D and φ2 : C−K2 → C− D.
When d = 2, the Böttcher coordinate is unique, but when d > 2 there
are d− 1 choices, and they matter: the mating may exist for some and not
for others.
Let C̃1, C̃2 be two copies of the complex plane compactified by adding
a circle at infinity, adding a point t∞,i ∈ R/Z at limr→∞ φ−1i
(
re2πit
)
. Both
p1 and p2 extend to maps
p̃1 : C̃1 → C̃1 and p̃2 : C̃2 → C̃2
that map t∞,i to dt∞,i.
Construct a topological 2-sphere
Sp1,p2 =
(
C̃1  C̃2
)
/ ∼
where ∼ identifies t∞,1 to −t∞,2.
This sphere carries no natural complex structure, but it does have a map
to itself which is p̃1 on C̃1 and p̃2 on C̃2; these maps are compatible with
the gluing and induce a ramified covering map
f̃p1,p2 : Sp1,p2 → Sp1,p2
of degree d. The space Sp1,p2 together with the map f̃p1,p2 is called the
formal mating of p1, p2; it actually depends also on the choices of Böttcher
coordinates.
The map f̃p1,p2 is compatible with the equivalence relation ≈p1,p2 on
Sp1,p2 defined as follows: it is the smallest equivalence relation with closed
equivalence classes, such that for all t ∈ R/Z, all points of φ−11 (re2πit) are
equivalent to all points of φ−12 (re
−2πit).
As such, f̃p1,p2 induces a map
fp1,p2 : Sp1,p2 → Sp1,p2 ,
where Sp1,p2 = Sp1,p2/ ≈p1,p2 .
– 1141 –
John Hubbard
There is another way of thinking about Sp1,p2 , that is far easier to visua-
lize. This way applies only to the case whenK1 andK2 are locally connected.
There are then Caratheodory loops γi : R/Z → ∂Ki, and the space Sp1,p2
is the quotient of K1 K2 obtained by identifying γ1(t) with γ2(−t). The
resulting space and map is often called the topological mating of p1 and p2.
This space Sp1,p2 may appear very bizarre and rather scary, especially
when K1 and K2 are dendrites: Sp1,p2 is obtained by gluing these dendrites
to each other.
There is a reassuring theorem of R. L. Moore that asserts that if the
equivalence classes do not separate the sphere, are not the whole sphere,
and if the quotient space is Hausdorff, then the quotient space is home-
omorphic to the sphere, and the quotient map can be approximated by
homeomorphisms. The hypotheses of Moore’s theorem are often satisfied
by ≈p1,p2 , and the mating construction yields beautiful examples of Moore
quotients. But they are still pretty bizarre: each of the dendrites maps sur-
jectively to the quotient sphere, as does the equator of Sp1,p2 , which is then
a Peano curve in Sp1,p2 .
We say that p1 and p2 are mateable if Sp1,p2 is homeomorphic to a sphere,
and if this sphere carries a complex structure such that fp1,p2 : Sp1,p2 →
Sp1,p2 is analytic; it is then automatically conjugate to a rational function
of degree d.
Note that in this case where p1 and p2 are mateable, we have a surjec-
tive continuous map γ : R/Z → P1 (a Peano curve) giving a commutative
diagram
R/Z γ−→ P1t → dt
fp1,p2
R/Z γ−→ P1
So γ semiconjugates the complicated map fp1,p2 to t → dt; it somehow
unwinds the sphere like a ball of yarn, turning it into a circle, and the
complicated rational function fp1,p2 unwinds to the simple mapping t → dt.
In dynamical systems, the main question is: to what extent does the
statement that Sp1,p2 is homeomorphic to a sphere imply that p1 and p2
are mateable, and to what extent is the resulting mating unique.
There are many results in this direction: the present volume is largely
devoted to this question, and I will not pursue it further here.
– 1142 –
Matings and the other side of the dictionary
Figure 1. — On the left, we see a sphere C̃1 ∪ C̃2, with sketches of Julia sets and
external rays drawn in. On the right, we see a sphere H ∪H∗ with the horizontal
trajectories of a Γ-invariant quadratic differential sketched in H and the vertical
trajectories of q∗ sketched in H∗. In both figures, we can consider a family of Beltrami
forms µK with eccentricity K and big axes of the infinitesimal ellipses aligned with the
external rays on the left, and with the horizontal trajectories of q and −q∗ on the right.
The effect of conjugating by a solution of the Beltrami equation ∂f = µK∂f is to shrink
these curves.
2.2. The setting for hyperbolic manifolds
We set H to be the upper half-plane, and H∗ to be the lower half-plane.
Let Γ be a torsion-free Fuchsian group, such thatX := H/Γ is a Riemann
surface of finite type; then H∗/Γ is the conjugate surface X∗.
Choose q ∈ Q(H) a Γ-invariant quadratic differential, coming from an in-
tegrable quadratic differential on X; then q∗(z) = q(z) defines a Γ-invariant
quadratic differential on H∗, the reflection of q in the real axis.
In analogy with matings of polynomials, the sphere, with the measured
foliation F1 := |Im√q| on H and the measured foliation F2 := |Im
√−q∗| on
H∗, could be called the formal mating of F1 and F2 (as far as I know, this
terminology is not used).
Now take s, t ∈ [0, 1) and consider the Beltrami form on C given by
µq,s,t =
{
s q|q| in H;
t−q
∗
|q| in H
∗
This Beltrami form corresponds to a field of ellipses on C
– 1143 –
John Hubbard
• of eccentricity (1 + s)/(1 − s) in H, with major axes aligned along the
leaves of F1, i.e., along the horizontal trajectories of q,
• of eccentricity (1 + t)/(1 − t) on H∗, with major axes aligned along the
leaves of F2, i.e., along the vertical trajectories of q
∗.
The object of taking the horizontal and vertical trajectories of the same
quadratic differential was to make the two foliations transverse on X; and
conversely, if two measured foliations are transverse, they correspond to the
horizontal and vertical trajectories of a quadratic differential.
Let fs,t : C→ C be a quasi-conformal map such that
∂̄fs,t = µq,s,t∂fs,t.
Roughly speaking, the limit of the fs,t as s, t→ 1 consists of collapsing
the trajectories of F1 in H and the trajectories of F2 in H∗ to points. In fact,
one can define a “topological mating of measured foliations”, analogous to
the topological mating of manifolds. Consider the leaf spaces of F1 and F2;
they are R-trees with an action of Γ, and there is a natural equivalence rela-
tion on their disjoint union; the quotient space might be called to topological
mating of F1 and F2 (although again I don’t believe that the terminology is
used). Since these R-trees are always locally connected (unlike the case of
polynomials, where Kp might not be), it is quite reasonable to view them
as intermediate objects in the process of constructing double limits. Mahan
[7] does this.
Here the question is whether, up to conjugacy, the group representations
ρs,t : Γ→ PSL2C given by ρs,t : (γ) = fs,t ◦ γ ◦ f−1s,t
have non-empty compact accumulation sets when s, t both tend to 1, and
whether this set consists of a single point. If the accumulation set is compact
and non-empty, we say that the measured foliations F1 and F2 are mateable.
If ρ∞ is an accumulation point of the representations ρs,t, there is a
map γ : RP1 → CP1 that maps the circle RP1 surjectively to the limit set
of ρ∞(Γ) ⊂ AutP1, that semiconjugates the action of the Fuchsian group
Γ on RP1 to the action of the Kleinian group ρ∞(Γ) on its limit set. Most
often this limit set is the entire sphere; this will happen for instance when
each of F1 and F2 has a dense leaf on H/Γ. This is generic in the space of
quadratic differentials on H/Γ.
In this case, we see that again γ is a Peano curve: again we can unwind
the sphere like a ball of yarn and get the much simpler action of a Fuchsian
group on RP1.
– 1144 –
Matings and the other side of the dictionary
3. Parallel answers to the questions
For matings of polynomials, two of the most striking results are the
following.
Theorem 3.1. — (Rees-Tan Lei [14]) Two post-critically finite quadratic
polynomials are mateable if and only if they do not belong to conjugate limbs
of the Mandelbrot set.
Theorem 3.2. — (Yampolsky-Zakeri [16]) Let θ1, θ2 ∈ [0, 1) be irra-
tionals of bounded type. Then the polynomials e2πiθ1z + z2 and e2πiθ2z + z2
are mateable if and if θ1 = 1− θ2.
It should be true that any pair of quadratic polynomials from non-
conjugate limbs of the Mandelbrot set are mateable: computer pictures cer-
tainly suggest it. Theorems 3.2 and 3.1 would both be special cases. Very
recent work of Dzmitry Dudko, following earlier work of Jiaqi Luo, gives
hope that something along these lines can be proved; in particular, Dudko
gives a definition of a mating when the Julia sets are not locally connected
The “parallel statement” for Kleinian groups is the double limit theorem:
Theorem 3.3. — (Thurston) Two measured foliations F1, F2 are mate-
able if any third measured foliation can be made transverse to one of F1 or
F2.
This is not quite the original statement of the double limit theorem [13]
which involves sequences of pairs of points in Teichmüller space approaching
the Thurston Boundary.
I do not know if in this generality the accumulation set of the ρs,t above
is always reduced to a single point; I believe this follows from the Ending
Lamination Conjecture (ELC, now proved [1],[6], which asserts that a limit
is determined by the topological type (in this case, that of a surface) and
the end-data. The end-data is rather difficult to define in general, but when
the foliations F1 and F2 have dense leaves, it is the measured laminations
associated to F1 and F2, forgetting about the associated measures (which
may not be unique). So the accumulation set should be a single point in
this case. I don’t yet understand the statement or the proof of the ending
lamination conjecture, so I will say nothing more about it here. Neither
will I speak of the work of Mahan Mj [7], which describes the geometry of
limit sets of limits of Kleinian groups in a way very similar to the quotient
Sp1,p2 = Sp1,p2/ ≈p1,p2 : I don’t understand the work well enough.
But in some cases the existence and uniqueness of the limit is true (and
I understand the proof), most crucially when F1 and F2 are the stable and
– 1145 –
John Hubbard
unstable foliations of a pseudo-Anosov homeomorphism of a surface. The
existence of the limit
lim
s,t→1
ρs,t
in this case is fairly easily shown to be equivalent to the existence of a
hyperbolic structure on 3-manifolds that fiber over the circle with pseudo-
Anosov holonomy.
Neither do I know if the condition stated is necessary as well as sufficient.
Note that it implies, by taking F3 = F1 that F1 and F2 can be made
transverse, and that is necessary, essentially for the same reason that the
condition in theorem 3.1 is necessary: in both cases, there are essential closed
curves on the formal mating that would collapse to points in the true mating
(or on the topological mating).
4. The proofs
There are many similarities in the proofs in the two fields, particularly
when the two sides of the dictionary are specialized to postcritically finite
polynomials on the dynamical systems side, and 3-d manifolds that fiber
over the circle on the Kleinian groups side.
In that case, on both sides either the rational function or Kleinian group
exist, or there is a multicurve that is an obstruction.
Moreover, the existence in both cases comes down to finding a fixed point
of a mapping in Teichmüller space (or a product of compactified Teichmüller
spaces).
In the more general case, the only method that I know of to approach
existence of matings is puzzle techniques (that is what Yampolsky and Za-
keri use to prove Theorem 3.2; it is also what Dudko and earlier Luo used).
These have no parallel in the Kleinian world. In the Kleinian world, the
proof uses Beltrami forms aligned with the horizontal and vertical foliations
that are allowed to degenerate, i.e., whose norms are allowed to tend to 1.
This is very similar to slow mating ; the beautiful movies of slow mating
have not yet led to proofs, but may well do so in the future. For examples
of slow matings, go to Arnaud Chéritat’s web page
http://www.math.univ-toulouse.fr/ cheritat/Gall/galery.html,
and find the film “two mating polynomial Julia sets”; there are several
others equally as wonderful.
– 1146 –
Matings and the other side of the dictionary
And on the other side of the dictionary, maybe puzzles do have analogs.
For instance, Masaaki Wada has written the marvelous program Opti , which
explores 2-generator Kleinian groups with parabolic commutator. The space
of conjugacy classes of such discrete groups is a product of two copies of the
Teichmüller space for the punctured torus. The program draws the param-
eter space of such groups; it isn’t obvious how to do this. In the process
it “paves” the space by pieces corresponding to the different geometries
of fundamental domains: not so different from paving parameter space for
polynomials according to the corresponding dynamical puzzle. Maybe such
pieces could be used as some sort of parapuzzle pieces.
Bibliography
[1] Brock (J.), Canary (R.), Minsky (Y.). — The Classification of Kleinian Surface
Groups, II: the Ending Lamination Conjecture, arXiv:math/0412006.
[2] Hubbard (J. H.). — Teichmüller theory, vol. I, Matrix Editions (2006).
[3] Hubbard (J. H.). — Teichmüller theory, vol. II, Matrix Editions, to appear.
[4] McMullen (C.). — Iteration on Teichmüller space, Invent.math. 99, p. 425-454
(1990).
[5] McMullen (C.). — Rational maps and Kleinian groups, In Proceedings of the
International Congress of Mathematicians (Kyoto, 1990), p. 889-900. Springer-
Verlag (1991).
[6] Minsky (Y.). — The Classification of Kleinian Surface Groups, I. Models and
Bounds, Ann. Math., 171(1), p. 1-107 (2010).
[7] Mahan (Mj). — Cannon-Thurston Maps for Kleinian Groups, arXiv:math
arxiv:1002.0996.
[8] Otal (J.-P.). — The hyperbolization theorem for fibered 3-manifolds, Translated
from the 1996 French original by Leslie D. Kay. SMF/AMS Texts and Mono-
graphs, 7. American Mathematical Society, Providence, RI; Société Mathématique
de France, Paris (2001).
[9] Rees (M.). — A partial description of parameter space of rational maps of degree
two: Part I, Acta Math. 168, p. 11-87 (1992).
[10] Sullivan (D.). — Quasiconformal Homeomorphisms and Dynamics, I. Solution of
the Fatou-Julia problem on wandering domains, Annals of math 122, p. 401-418
(1985).
[11] Thurston (W.). — On the geometry and Dynamics of iterated Rational Maps,
in Complex Dynamics: Families and Friends, Schleicher ed., AK Peters, p. 3-130
(2009).
[12] Thurston (W.). — Hyperbolic structures on 3-manifolds, I: Deformation of
acylindrical manifolds, Annals of Math 124, p. 203-246 (1986).
[13] Thurston (W.). — Hyperbolic structures on 3-manifolds, II: Surface Groups and
3-Manifolds that fiber over the Circle, arXiv:math/9801045.
[14] Lei (T.). — Matings of quadratic polynomials, Ergod. th. and dyn. syst., vo. 12,
p. 589-620 (1992).
[15] Wada (M.). — Opti, http:\\delta.math.sci.osaka-u.ac.jp\OPTi
[16] Wada (M.). — Mating Siegel Quadratic Polynomials, JAMS 14, p. 25-78 (2000).
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Matings and the other side of the dictionary

Zoé Chatzidakis

Ehud Hrushovski

An invariant for difference field extensions

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