Combinations of Complex Dynamical Systems

121 Pages·2003·1.227 MB·English

by  Kevin M. Pilgrim (auth.)

1827
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Kevin M. Pilgrim
Combinations of Complex
Dynamical Systems
1 3
Author
KevinM.Pilgrim
DepartmentofMathematics
IndianaUniversity
Bloomington,IN47401,USA
e-mail:[email protected]
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MathematicsSubjectClassification(2000):37F20
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Preface
The goal of this research monograph is to develop a general combination, de-
composition, and structure theory for branched coverings of the two-sphere
to itself, regarded as the combinatorial and topological objects which arise
in the classification of certain holomorphic dynamical systems on the Rie-
mann sphere. It is intended for researchers interested in the classification of
those complex one-dimensional dynamical systems which are in some loose
sense tame, though precisely what this constitutes we leave open to interpre-
tation. The program is motivated in general by the dictionary between the
theoriesofiteratedrationalmapsandKleiniangroupsasholomorphicdynam-
ical systems, and in particular by the structure theory of compact irreducible
three-manifolds.
By and large this work involves only topological/combinatorial notions.
Apart from motivational discussions, the sole exceptions are (i) the construc-
tion of examples which is aided using complex dynamics in §9, and (ii) some
familiarity with the Douady-Hubbard proof of Thurston’s characterization of
rational functions in §§8.3.1 and §10.
The combination and decomposition theory is developed for maps which
are not necessarily postcritically finite. However, the proof of the main struc-
ture result, the Canonical Decomposition Theorem, depends on Thurston’s
characterization and is developed only for postcritically finite maps. A sur-
veyofknownresultsregardingcombinatoricsandcombinationproceduresfor
rational maps is included.
This research was partially supported by NSF grant No. DMS-9996070,
the University of Missouri at Rolla, and Indiana University. I thank Albert
Goodmanfortimelyadviceongroupactionswhichwereparticularlyhelpfulin
provingtheresultsin§7.IthankCurtMcMullenforencouragingmetothink
big. I am especially grateful to Mary Rees and to the referees for valuable
comments. Finally, I thank my family for their unwavering support.
Bloomington, Indiana, USA, Kevin M. Pilgrim
August, 2003
Contents
1 Introduction............................................... 1
1.1 Motivation from dynamics–a brief sketch ................... 1
1.2 Thurston’s Characterization and Rigidity Theorem. Standard
definitions .............................................. 2
1.3 Examples............................................... 6
1.3.1 A realizable mating................................ 6
1.3.2 An obstructed mating.............................. 6
1.3.3 An obstructed expanding Thurston map.............. 8
1.3.4 A subdivision rule ................................. 11
1.4 Summary of this work.................................... 12
1.5 Survey of previous results ................................ 14
1.5.1 Enumeration...................................... 14
1.5.2 Combinations and decompositions ................... 17
1.5.3 Parameter space .................................. 20
1.5.4 Combinations via quasiconformal surgery............. 22
1.5.5 From p.f. to geometrically finite and beyond .......... 23
1.6 Analogy with three-manifolds ............................. 24
1.7 Connections ............................................ 27
1.7.1 Geometric Galois theory ........................... 27
1.7.2 Gromov hyperbolic spaces and interesting groups...... 28
1.7.3 Cannon’s conjecture ............................... 29
1.8 Discussion of combinatorial subtleties ...................... 29
1.8.1 Overview of decomposition and combination .......... 30
1.8.2 Embellishments. Technically convenient assumption. ... 31
1.8.3 Invariant multicurves for embellished map of spheres.
Thurston linear map. .............................. 32
1.9 Tameness assumptions ................................... 33
2 Preliminaries .............................................. 37
2.1 Mapping trees .......................................... 39
2.2 Map of spheres over a mapping tree........................ 44
2.3 Map of annuli over a mapping tree......................... 46
VIII Contents
3 Combinations.............................................. 49
3.1 Topological gluing data .................................. 49
3.2 Critical gluing data...................................... 50
3.3 Construction of combination .............................. 52
3.4 Summary: statement of Combination Theorem .............. 53
3.5 Properties of combinations................................ 53
4 Uniqueness of combinations................................ 59
4.1 Structure data and amalgamating data. .................... 59
4.2 Combinatorial equivalence of sphere and annulus maps ....... 60
4.3 Statement of Uniqueness of Combinations Theorem.......... 61
4.4 Proof of Uniqueness of Combinations Theorem .............. 62
4.4.1 Missing disk maps irrelevant ........................ 62
4.4.2 Reduction to fixed boundary values.................. 62
4.4.3 Reduction to simple form........................... 63
4.4.4 Conclusion of proof of Uniqueness Theorem........... 67
5 Decomposition............................................. 69
5.1 Statement of Decomposition Theorem...................... 69
5.2 Standard form with respect to a multicurve................. 71
5.3 Maps in standard forms are amalgams ..................... 71
5.4 Proof of Decomposition Theorem.......................... 76
6 Uniqueness of decompositions ............................. 79
6.1 Statement of Uniqueness of Decompositions Theorem ........ 79
6.2 Proof of Uniqueness of Decomposition Theorem ............. 79
7 Counting classes of annulus maps ......................... 83
7.1 Statement of Number of Classes of Annulus Maps Theorem... 83
7.2 Proof of Number of Classes of Annulus Maps Theorem ....... 84
7.2.1 Homeomorphism of annuli. Index. ................... 84
7.2.2 Characterization of combinatorial equivalence by
group action. ..................................... 85
7.2.3 Reduction to abelian groups ........................ 86
7.2.4 Computations and conclusion of proof ............... 86
8 Applications to mapping class groups...................... 89
8.1 The Twist Theorem ..................................... 89
8.2 Proof of Twist Theorem.................................. 90
8.2.1 Combinatorial automorphisms of annulus maps ....... 90
8.2.2 Conclusion of proof of Twist Theorem ............... 91
8.3 When Thurston obstructions intersect...................... 92
8.3.1 Statement of Intersecting Obstructions Theorem ...... 92
8.3.2 Maps with intersecting obstructions have large
mapping class groups .............................. 93
Contents IX
9 Examples.................................................. 95
9.1 Background from complex dynamics ....................... 95
9.2 Matings ................................................ 96
9.3 Generalized matings ..................................... 98
9.4 Integral Latt`es examples .................................101
10 Canonical Decomposition Theorem ........................105
10.1 Cycles of a map of spheres, and their orbifolds ..............105
10.2 Statement of Canonical Decomposition Theorem ............107
10.3 Proof of Canonical Decomposition Theorem.................108
10.3.1 Characterization of rational cycles with hyperbolic
orbifold ..........................................108
10.3.2 Conclusion of proof................................109
References.....................................................111
Index..........................................................117
1
Introduction
1.1 Motivation from dynamics–a brief sketch
Thisworkisaboutthecombinatorialaspectsofrigidityphenomenaincomplex
dynamics. It is motivated by discoveries of Douady-Hubbard [DH1], Milnor-
Thurston[MT],andSullivanmadeduringtheearly1980’s(seetheprefaceby
Hubbard in [Tan4] for a firsthand account).
In the real quadratic family f (x)=(x2+a)/2,a∈R, it was proven [MT]
a
that the entropy of f as a function of a is continuous, monotone, and in-
a
creasing as the real parameter varies from a=5 to a=8. A key ingredient of
theirproofisacompletecombinatorialcharacterizationandrigidityresultfor
critically periodicmapsf ,i.e.thoseforwhichtheuniquecriticalpointatthe
a
originisperiodic.Toanymapf inthefamilyoneassociatesacombinatorial
a
invariant, called its kneading invariant. Such an invariant must be admissible
in order to arise from a map f . It was shown that every admissible kneading
a
invariant actually arises from such a map f , and that if two critically peri-
a
odic maps have the same kneading invariant, then they are affine conjugate.
In a process called microimplantation the dynamics of one map f could be
a
“glued”intothatofanothermapf wheref iscriticallyperiodictoobtain
a0 a0
a new map fa∗a0 in this family. More precisely: a topological model for the
new map is constructed, and its kneading invariant, which depends only on
thetopologicaldata,iscomputed.Theresultturnsouttobeadmissible,hence
by the characterization theorem defines uniquely a new map fa0∗a. This con-
struction interprets the cascade of period-doublings as the limit limn→∞fan
where an+1 = an ∗a0 and a0 is chosen so that the critical point is periodic
of period two. As an application, it is shown that there exists an uncountable
family of maps with distinct kneading invariants but with the same entropy.
Similar combinatorial rigidity phenomena were also observed for maps
f (z) = z2 +c, c ∈ C in the complex setting. For “critically periodic” pa-
c
rameters c for which the critical point at the origin is periodic, the dynamics
restricted to the filled-in Julia set K ={z|f◦n(z)(cid:3)→∞} looks roughly like a
c
map from a tree to itself (here f◦n is the n-fold iterate of f). The dynamics
K.M.Pilgrim:LNM1827,pp.1–35,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003