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• Publié par: Ucte
• Nombre de lectures: 19
• Langue: English

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Dynamics of holomorphic maps
Tien-Cuong Dinh and Nessim Sibony
March 01, 2007
Introductory Lectures (Master)
available at http://www.math.jussieu.fr/dinh23
Preface
This text is written for the students in the Master program at the University
of Paris 6. Only a knowledge in complex analysis in one variable and in mea-
sure theory is required. We begin with the theory of iteration of holomorphic
polynomials and of rational fractions, but our aim is to introduce the readers
to the current research in complex dynamics of several variables. We introduce
the main dynamical objects and their properties in a way so that they can be
easily extended to the case of higher dimension after introducing the necessary
tools in complex analysis of several variables. Since the number of lectures is
small, exercices are given; they contain important results which are used in the
text. This rst version may contain mistakes. Your remarks and comments are
welcome.4Contents
1 Fatou-Julia theory for rational fractions 7
1.1 Dynamics of polynomials . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Critical and periodic points of polynomials . . . . . . . . . 7
1.1.2 Dynamics near a xed point . . . . . . . . . . . . . . . . . 10
1.1.3 Fatou and Julia sets for polynomials . . . . . . . . . . . . 16
1.1.4 Periodic and wandering Fatou components . . . . . . . . . 17
1.1.5 Topological entropy and invariant measures . . . . . . . . 21
1.2 Dynamics of rational fractions . . . . . . . . . . . . . . . . . . . . 29
1.2.1 Riemann sphere and holomorphic maps . . . . . . . . . . . 29
1.2.2 Fatou and Julia sets for rational fractions. . . . . . . . . . 33
1.2.3 Topological entropy and invariant measures . . . . . . . . 35
2 Equilibrium measure and properties 37
2.1 Potential and quasi-potential of a measure . . . . . . . . . . . . . 37
2.1.1 Subharmonic and quasi-subharmonic functions . . . . . . . 37
2.1.2 Potential and quasi-potential of a measure . . . . . . . . . 43
2.1.3 Space of dsh functions and Sobolev space . . . . . . . . . . 46
2.2 Equilibrium measure . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 Constructions of the equilibrium measure . . . . . . . . . . 50
2.2.2 Equidistribution of preimages . . . . . . . . . . . . . . . . 56
2.2.3 Eqn of periodic points . . . . . . . . . . . . . 61
2.2.4 Mixing and rate of mixing . . . . . . . . . . . . . . . . . . 61
2.2.5 Lyapounov exponent and entropy . . . . . . . . . . . . . . 63
3 Dynamics in higher dimension 67
3.1 Pluripotential theory . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.1 Di erential forms and currents . . . . . . . . . . . . . . . . 67
3.1.2 Holomorphic maps and analytic sets . . . . . . . . . . . . 70
3.1.3 Positive forms and positive currents . . . . . . . . . . . . . 74
3.1.4 Plurisubharmonic functions . . . . . . . . . . . . . . . . . 79
3.1.5 Intersection of currents . . . . . . . . . . . . . . . . . . . . 81
3.1.6 Skoda’s extension theorem . . . . . . . . . . . . . . . . . . 83
3.1.7 Lelong number and Siu’s theorem . . . . . . . . . . . . . . 84
56 CONTENTS
3.1.8 Projective spaces . . . . . . . . . . . . . . . . . . . . . . . 85
k3.1.9 Quasi-psh functions and positive currents onP . . . . . . 89
3.1.10 Green quasi-potentials of positive closed currents . . . . . 91
3.1.11 Space of dsh functions . . . . . . . . . . . . . . . . . . . . 93
3.1.12 Complex Sobolev space . . . . . . . . . . . . . . . . . . . . 94
3.2 Equilibrium measure and Green currents . . . . . . . . . . . . . . 95
3.2.1 Construction of the equilibrium measure . . . . . . . . . . 95
3.2.2 Mixing and rate of mixing . . . . . . . . . . . . . . . . . . 97
3.2.3 Equidistribution of preimages and exceptional set . . . . . 98
3.2.4 Equidistn of periodic points . . . . . . . . . . . . . 102
3.2.5 Lyapounov exponents and entropy . . . . . . . . . . . . . 103
3.2.6 Green currents and equidistribution problems . . . . . . . 107
3.2.7 Some other open problems . . . . . . . . . . . . . . . . . . 108Chapter 1
Fatou-Julia theory for rational
fractions
1.1 Dynamics of polynomials
1.1.1 Critical and periodic points of polynomials
Letf beaholomorphicpolynomialofdegreed 2(wewillseethatthedynamics
ofpolynomialsofdegree1isnotinteresting). Thenf de nesacontinuoussurjec-
tive map fromC ontoC. For every z∈C the equation f(w) =z admits exactly
1d solutions counted with multiplicities. In other words, the bers f (z) of f
contain exactly d points counted with multiplicities. More precisely, f :C→C
is a rami ed covering of degree d.
De nition 1.1.1.1. A point c is critical of multiplicity m if it is a solution of
0multiplicity m of the equation f (z) = 0. The set of all critical points is the
critical set of f. If c is a critical point, f(c) is a critical value of f.
0Since degf =d 1, f admits exactly d 1 critical points inC, counted with
nmultiplicities. The polynomial f := f f is the iterate of order n of f; it
nis of degree d .
De nition 1.1.1.2. A point p is xed of multiplicity m if p is a solution of
multiplicity m of the equation f(z) =z. A point p is periodic of period n and of
nmultiplicity m if p is a xed point of multiplicity m of f , that is, p is a solution
nof multiplicitym of the equationf (z) =z. We say thatp is pre-periodic if there
Nis N 0 such that f (p) is a periodic point.
There are exactly d xed points in C counted with multiplicities (we will see
nthat the in nity can be considered as a xed point of multiplicity 1). Since f is
n na polynomial of degree d , f admits exactly d periodic points of period n inC.
If p is periodic of period n then it is also periodic of period kn for every k 1.
76
8 CHAPTER 1. FATOU-JULIA THEORY FOR RATIONAL FRACTIONS
0 0 nNote that f is the identity, that is, f (z) = z. One should distinguish f (z)
nfrom the power [f(z)] .
+ 2De nition 1.1.1.3. The sequence of points O (p) :={p,f(p),f (p),...} is the
orbit of the point p. Any sequence {p ,...,p ,p ,p } such that p = p andn 2 1 0 0
f(p ) =f(p ) for 0in 1, is an inverse branch of order n of p.i 1 i
The orbit of a periodic point is called a (periodic) cycle. In general, a point
may have several inverse branches. Pre-periodic points are the points whose
orbits take only a nite number of values.
Exercise 1.1.1.4. Describe the bers, the critical set, the set of critical values,
dthe periodic and pre-periodic points of the polynomial f(z) =z . Study the orbits
a general point p. Find the number of inverse branches of order n of p. Same
questions for a polynomial f of degree d = 1.
nExercise 1.1.1.5. Describe the critical set and the set of critical values of f in
term of the critical set of f. Let d be the number (counted with multiplicities)n
of periodic points whose minimal periods are equal to n. Show that
nlim d d = 1.n
n→∞
De nition 1.1.1.6. Let p be a periodic point of period n of f. We say that p is
n 01a. repelling if |(f )(p)|> 1.
n 01b. attracting if |(f )(p)|< 1.
n 02. super-attracting if (f )(p) = 0.
n 03. rationally indi erent if (f )(p) is a root of unity.
n 0 n 04. irrationally indi erent if |(f )(p)| = 1 and (f )(p) is not a root of unity.
The orbit ofp is called respectively a repelling cycle, an attracting cycle, a super-
attracting cycle, a rationally indi erent cycle or an irrationally indi erent cycle .
De nition 1.1.1.7. A subset K of C is invariant if f(K) = K and totally
1invariant if f (K) =K.
kn 0Exercise 1.1.1.8. Let p be a periodic point of period n of f. Compute (f )(p)
n 0for k 1. If (f )(p) satis es one of the properties in De nition 1.1.1.6, show
kn 0that (f )(p) satis es the same property.
Exercise 1.1.1.9. Determine the type of the periodic points of the polynomial
d d nz . Determine all the nite totally invariant sets of z . Hint: f (z) contains
nd distinct points if z = 0.6
1.1. DYNAMICS OF POLYNOMIALS 9
Exercise 1.1.1.10. Show that completely invariant sets are invariant. Show that
the complement of a totally invariant set is totally invariant. Describe the mimi-
maltotallyinvariantsetcontainingagivenpointp. Determineallthepolynomials
f admitting a totally invariant point.
1Exercise 1.1.1.11. Let h be a polynomial of degree 1. De ne g := hf h .
n n 1We say that f and g are analytically conjugate. Show that g = hf h .
Determine the relations between the bers of f and the bers of g. Same questions
for the critical sets, the sets of critical values, the periodic and pre-periodic points,
the type of periodic points, the orbits and inverse branches, the invariant and
totally invariant sets. Show that any polynomial of degree d 2 is conjugate to
d d 2a polynomial of the form z +a z ++a .2 d
Exercise 1.1.1.12. Let f be a polynomial of degree d 2. Let
be the set of∞
npoints z such that |f (z)|→ +∞. Show that
is a connected open neighbour-∞
hood of in nity which is totally invariant. Show also that the complement and
the boundary of
are totally invariant. We call
the basin of ∞. We will∞ ∞
see that the boundary J of
is the Julia set of f and its complement F is the∞
Fatou set. The compact setC\
is called the lled Julia set. Show that C\
∞ ∞
is the largest compact set invariant by f.
Exercise 1.1.1.13. Let p ,...,p be a repelling periodic cycle of period m of0 m 1
f with p =f(p ) and p =p . Let D denote the open disc of center p and ofi