Random complex dynamics and semigroups of holomorphic maps

0.88 MB·English

by  Hiroki Sumi

Random complex dynamics and
semigroups of holomorphic maps ∗
0
1
0 Hiroki Sumi
2
Department of Mathematics, Graduate School of Science, Osaka University

y
1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
a
M E-mail: [email protected]
http://www.math.sci.osaka-u.ac.jp/˜ sumi/welcomeou-e.html

7
1
May 15, 2010

]
S
D
. Abstract
h
t We investigate the random dynamics of rational maps on the Riemann sphere Cˆ and
ma thedynamics of semigroups of rational maps on Cˆ. We show that regarding random complex
dynamicsofpolynomials,inmostcases,thechaosoftheaveragedsystemdisappears,duetothe
[ cooperationofthegenerators. Weinvestigatetheiterationandspectralpropertiesoftransition

operators. We show that under certain conditions, in the limit stage, “singular functions
6
v on the complex plane” appear. In particular, we consider the functions T which represent
3 the probability of tending to infinity with respect to the random dynamics of polynomials.
8 Undercertain conditions thesefunctionsT arecomplex analogues of thedevil’s staircase and
4 Lebesgue’s singular functions. More precisely, we show that thesefunctions T are continuous
4 onCˆ andvaryonly ontheJulia setsofassociated semigroups. Furthermore,byusingergodic
2. theory and potential theory, we investigate the non-differentiability and regularity of these
1 functions. We find many phenomena which can hold in the random complex dynamics and
8 thedynamicsofsemigroupsofrationalmaps,butcannotholdintheusualiterationdynamics
0 of a single holomorphic map. We carry out a systematic study of these phenomena and their
: mechanisms.
v
i
X
1 Introduction
r
a
In this paper, we investigate the random dynamics of rational maps on the Riemann sphere Cˆ
andthedynamicsofrationalsemigroups(i.e.,semigroupsofnon-constantrationalmapswherethe
semigroup operation is functional composition) on Cˆ. We see that the both fields are related to
each other very deeply. In fact, we develop both theories simultaneously.
One motivation for research in complex dynamical systems is to describe some mathematical
models on ethology. For example, the behavior of the population of a certain species can be
described by the dynamical system associated with iteration of a polynomial f(z) = az(1−z)
such that f preserves the unit interval and the postcritical set in the plane is bounded (cf. [7]).
However, when there is a change in the natural environment, some species have several strategies
tosurviveinnature. Fromthis pointofview,itisverynaturalandimportantnotonlytoconsider
the dynamics of iteration, where the same survival strategy (i.e., function) is repeatedly applied,
but also to consider random dynamics, where a new strategy might be applied at each time step.
∗ToappearinProc. LondonMath. Soc. (3). 2000MathematicsSubjectClassification. 37F10,30D05. Keywords:
Random dynamical systems, random complex dynamics, random iteration, Markov process, rational semigroups,
polynomialsemigroups,Juliasets,fractalgeometry,cooperation principle,noise-inducedorder.
1
The firststudy of randomcomplex dynamics wasgivenby J. E.Fornaessand N. Sibony ([9]). For
researchonrandomcomplexdynamicsofquadraticpolynomials,see[2,3,4,5,6,10]. Forresearch
on randomdynamics of polynomials (of generaldegrees)with bounded planar postcriticalset, see
the author’s works [35, 34, 36, 37, 38, 39].
The first study of dynamics of rational semigroups was conducted by A. Hinkkanen and G.
J. Martin ([13]), who were interested in the role of the dynamics of polynomial semigroups (i.e.,
semigroups of non-constant polynomial maps) while studying various one-complex-dimensional
modulispacesfordiscretegroups,andbyF.Ren’sgroup([11]),whostudiedsuchsemigroupsfrom
the perspective of random dynamical systems. Since the Julia set J(G) of a finitely generated
rational semigroup G= hh1,...,hmi has “backward self-similarity,” i.e., J(G) = mj=1h−j1(J(G))
(see Lemma 4.1 and [26, Lemma 1.1.4]), the study of the dynamics of rational semigroups can be
S
regardedas the study of “backwarditeratedfunction systems,” and alsoas a generalizationofthe
study of self-similar sets in fractal geometry.
For recentworkonthe dynamicsof rationalsemigroups,see the author’spapers [26]–[39], [41],
and [25, 42, 43, 44, 45].
In order to consider the random dynamics of a family of polynomials on Cˆ, let T (z) be the
probability of tending to ∞ ∈ Cˆ starting with the initial value z ∈ Cˆ. In this paper,∞we see that
under certain conditions, the function T : Cˆ → [0,1] is continuous on Cˆ and has some singular
∞
properties (for instance, varies only on a thin fractal set, the so-called Julia set of a polynomial
semigroup), and this function is a complex analogue of the devil’s staircase (Cantor function) or
Lebesgue’s singular functions (see Example 6.2, Figures 2, 3, and 4). Before going into detail, let
us recallthe definitionofthe devil’sstaircase(Cantorfunction)andLebesgue’ssingularfunctions.
Note that the following definitions look a little bit different from those in [46], but it turns out
that they are equivalent to those in [46].
Definition 1.1 ([46]). Let ϕ : R → [0,1] be the unique bounded function which satisfies the
following functional equation:
1 1
ϕ(3x)+ ϕ(3x−2)≡ϕ(x), ϕ| ≡0, ϕ| ≡1. (1)
2 2 (−∞,0] [1,+∞)
The function ϕ| :[0,1]→[0,1] is called the devil’s staircase (or Cantor function).
[0,1]
Remark 1.2. The above ϕ : R → [0,1] is continuous on R and varies precisely on the Cantor
middle third set. Moreover,it is monotone (see Figure 1).
Definition 1.3 ([46]). Let 0 < a < 1 be a constant. We denote by ψa : R → [0,1] the unique
bounded function which satisfies the following functional equation:
aψ (2x)+(1−a)ψ (2x−1)≡ψ (x), ψ | ≡0, ψ | ≡1. (2)
a a a a ( ,0] a [1,+ )
−∞ ∞
For each a ∈ (0,1) with a 6= 1/2, the function L := ψ | : [0,1] → [0,1] is called Lebesgue’s
a a [0,1]
singular function with respect to the parameter a.
Remark 1.4. The function ψa : R → [0,1] is continuous on R, monotone on R, and strictly
monotone on [0,1]. Moreover, if a 6= 1/2, then for almost every x ∈ [0,1] with respect to the
one-dimensional Lebesgue measure, the derivative of ψ at x is equal to zero (see Figure 1). For
a
the details on the devil’s staircase and Lebesgue’s singular functions and their related topics, see
[46, 12].
These singular functions defined on[0,1]canbe redefinedby using randomdynamicalsystems
on R as follows. Let f1(x) := 3x,f2(x) := 3(x−1)+1 (x ∈ R) and we consider the random
dynamical system (random walk) on R such that at every step we choose f1 with probability 1/2
and f2 with probability 1/2. We set Rˆ := R∪{±∞}. We denote by T+ (x) the probability of
∞
2
Figure 1: (Fromleft to right)The graphsof the devil’s staircaseandLebesgue’s singularfunction.
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
tending to +∞ ∈ Rˆ starting with the initial value x ∈ R. Then, we can see that the function
T | :[0,1]→[0,1] is equal to the devil’s staircase.
+ [0,1]
∞
Similarly, let g1(x) := 2x,g2(x) := 2(x−1)+1 (x ∈ R) and let 0 < a < 1 be a constant.
We consider the random dynamical system on R such that at every step we choose the map
g with probability a and the map g with probability 1−a. Let T (x) be the probability
1 2 + ,a
∞
of tending to +∞ starting with the initial value x ∈ R. Then, we can see that the function
T | :[0,1]→[0,1]isequaltoLebesgue’ssingularfunctionL withrespecttotheparameter
+ ,a [0,1] a
∞
a.
Weremarkthatinmostoftheliterature,thetheoryofrandomdynamicalsystemshasnotbeen
useddirectlytoinvestigatethesesingularfunctionsontheinterval,althoughsomeresearchershave
used it implicitly.
One ofthe main purposes of this paper is to consider the complex analogueof the above story.
Inordertodothat,wehavetoinvestigatetheindependentandidentically-distributed(abbreviated
by i.i.d.) random dynamics of rational maps and the dynamics of semigroups of rational maps on
Cˆ simultaneously. We develop both the theory of random dynamics of rational maps and that of
the dynamics of semigroups of rational maps. The author thinks this is the best strategy since
when we want to investigate one of them, we need to investigate the other.
Tointroducethemainideaofthispaper,weletGbearationalsemigroupanddenotebyF(G)
the FatousetofG,whichisdefinedtobethemaximalopensubsetofCˆ whereGisequicontinuous
with respect to the spherical distance on Cˆ. We call J(G) := Cˆ \F(G) the Julia set of G. The
Julia set is backward invariant under each element h ∈ G, but might not be forward invariant.
This is a difficulty of the theory of rational semigroups. Nevertheless, we “utilize” this as follows.
The key to investigating random complex dynamics is to consider the following kernel Julia set
of G, which is defined by J (G)= g 1(J(G)). This is the largest forward invariant subset
ker g G −
of J(G) under the action ofG. Note tha∈t if G is a grouporif G is a commutativesemigroup,then
T
J (G) = J(G). However, for a general rational semigroup G generated by a family of rational
ker
maps h with deg(h)≥2, it may happen that ∅=J (G)6=J(G) (see subsection 3.5, section 6).
ker
LetRatbethespaceofallnon-constantrationalmapsontheRiemannsphereCˆ,endowedwith
the distance κ which is defined by κ(f,g) := sup d(f(z),g(z)), where d denotes the spherical
z Cˆ
distance on Cˆ. Let Rat+ be the space of all rationa∈l maps g with deg(g) ≥2. Let P be the space
of all polynomial maps g with deg(g) ≥ 2. Let τ be a Borel probability measure on Rat with
compactsupport. We considerthe i.i.d. randomdynamics onCˆ suchthatat everystepwe choose
a map h ∈ Rat according to τ. Thus this determines a time-discrete Markov process with time-
homogeneoustransitionprobabilitiesonthephasespaceCˆ suchthatforeachx∈Cˆ andeachBorel
measurable subset A of Cˆ, the transition probability p(x,A) of the Markov process is defined as
p(x,A) = τ({g ∈ Rat | g(x) ∈ A}). Let G be the rational semigroup generated by the support
τ
of τ. Let C(Cˆ) be the space of all complex-valued continuous functions on Cˆ endowed with the
supremum norm. Let Mτ be the operator on C(Cˆ) defined by Mτ(ϕ)(z) = ϕ(g(z))dτ(g). This
M is called the transition operator of the Markov process induced by τ. For a topological space
τ
X, let M (X) be the space of all Borel probability measures on X endoweRd with the topology
1
induced by the weak convergence(thus µ →µ in M (X) if and only if ϕdµ → ϕdµ for each
n 1 n
bounded continuous function ϕ:X →R). Note that if X is a compact metric space, then M1(X)
is compact and metrizable. For each τ ∈ M (X), we denote by suppτ tRhe topologRical support of
1
3
τ. LetM (X)be the spaceofallBorelprobabilitymeasuresτ onX suchthatsuppτ iscompact.
1,c
Let Mτ∗ : M1(Cˆ) → M1(Cˆ) be the dual of Mτ. This Mτ∗ can be regarded as the “averaged
map” on the extension M1(Cˆ) of Cˆ (see Remark 2.21). We define the “Julia set” Jmeas(τ) of the
dynamics of Mτ∗ as the set of all elements µ ∈ M1(Cˆ) satisfying that for each neighborhood B
of µ, {(Mτ∗)n|B : B → M1(Cˆ)}n N is not equicontinuous on B (see Definition 2.17). For each
sequence γ =(γ ,γ ,...)∈(Rat)N∈, we denote by J the set of non-equicontinuity of the sequence
1 2 γ
{γn◦···◦γ1}n N with respect to the spherical distance on Cˆ. This Jγ is called the Julia set of γ.
Let τ˜:=⊗ τ∈∈M ((Rat)N).
∞j=1 1
We prove the following theorem.
Theorem1.5(CooperationPrincipleI,seeTheorem3.14andProposition4.7). Letτ ∈M (Rat).
1,c
SupposethatJ (G )=∅.ThenJ (τ)=∅.Moreover, for τ˜-a.e. γ ∈(Rat)N,the2-dimensional
ker τ meas
Lebesgue measure of J is equal to zero.
γ
This theorem means that if all the maps in the support of τ cooperate, the set of sensitive
initialvalues ofthe averagedsystemdisappears. Note thatforanyh∈Rat , J (δ )6=∅.Thus
+ meas h
the above result deals with a phenomenon which can hold in the random complex dynamics but
cannot hold in the usual iteration dynamics of a single rational map h with deg(h)≥2.
Fromtheaboveresultandsomefurtherdetailedarguments,weprovethefollowingtheorem. To
state the theorem,for aτ ∈M (Rat), wedenote by U the spaceofallfinite linearcombinations
1,c τ
of unitary eigenvectors of Mτ : C(Cˆ) → C(Cˆ), where an eigenvector is said to be unitary if the
absolute value of the corresponding eigenvalue is equal to one. Moreover, we set B := {ϕ ∈
0,τ
C(Cˆ)|Mn(ϕ)→0}. Under the above notations, we have the following.
τ
Theorem 1.6 (Cooperation Principle II: Disappearance of Chaos, see Theorem 3.15).
Let τ ∈ M (Rat). Suppose that J (G ) = ∅ and J(G ) 6= ∅. Then we have all of the following
1,c ker τ τ
statements.
(1) There exists a direct decomposition C(Cˆ)=Uτ⊕B0,τ. Moreover, dimCUτ <∞and B0,τ is a
closed subspace of C(Cˆ). Moreover, there exists a non-empty Mτ∗-invariant compact subset A
of M1(Cˆ) with finite topological dimension such that for each µ∈M1(Cˆ), d((Mτ∗)n(µ),A)→
0 in M1(Cˆ) as n → ∞. Furthermore, each element of Uτ is locally constant on F(Gτ).
Therefore each element of Uτ is a continuous function on Cˆ which varies only on the Julia
set J(G ).
τ
(2) For each z ∈ Cˆ, there exists a Borel subset Az of (Rat)N with τ˜(Az) = 1 with the following
property.
– For each γ = (γ ,γ ,...) ∈ A , there exists a number δ = δ(z,γ) > 0 such that
1 2 z
diam(γ ···γ (B(z,δ))) → 0 as n → ∞, where diam denotes the diameter with respect
n 1
to the spherical distance on Cˆ, and B(z,δ) denotes the ball with center z and radius δ.
(3) There exists at least one and at most finitely many minimal sets for (Gτ,Cˆ), where we say
that a non-empty compact subset L of Cˆ is a minimal set for (Gτ,Cˆ) if L is minimal in
{C ⊂Cˆ |∅6=C is compact,∀g ∈Gτ,g(C)⊂C} with respect to inclusion.
(4) LetSτ betheunionofminimalsetsfor (Gτ,Cˆ). Then for eachz ∈Cˆ thereexistsaBorelsub-
set C of (Rat)N with τ˜(C )=1 suchthat for each γ =(γ ,γ ,...)∈C , d(γ ···γ (z),S )→
z z 1 2 z n 1 τ
0 as n→∞.
This theorem means that if all the maps in the support of τ cooperate, the chaos of the
averaged system disappears. Theorem 1.6 describes new phenomena which can hold in random
complex dynamics but cannot hold in the usual iteration dynamics of a single h ∈ Rat . For
+
example, for any h ∈ Rat , if we take a point z ∈ J(h), where J(h) denotes the Julia set of the
+
4
semigroup generated by h, then for any ball B with B∩J(h)6=∅, hn(B) expands as n→∞, and
we have infinitely many minimal sets (periodic cycles) of h.
InTheorem3.15,wecompletely investigatethe structureofU andthe setofunitaryeigenval-
τ
uesofMτ (Theorem3.15). Usingtheaboveresult,weshowthatifdimCUτ >1andint(J(Gτ))=∅
where int(·) denotes the set of interior points, then F(G ) has infinitely many connected compo-
τ
nents (Theorem 3.15-20). Thus the random complex dynamics can be applied to the theory of
dynamics ofrationalsemigroups. The key to provingTheorem1.6 (Theorem3.15) is to show that
for almost every γ = (γ ,γ ,...) ∈ (Rat)N with respect to τ˜ := ⊗ τ and for each compact set
1 2 ∞j=1
Q contained in a connected component U of F(G ), diamγ ◦···◦γ (Q)→ 0 as n →∞. This is
τ n 1
shownbyusingcarefulargumentsonthehyperbolicmetricofeachconnectedcomponentofF(G ).
τ
Combining this withthe decompositiontheoremon“almostperiodic operators”onBanachspaces
from [18], we prove Theorem 1.6 (Theorem 3.15).
Considering these results, we have the following natural question: “When is the kernel Julia
setempty?” SincethekernelJuliasetofGisforwardinvariantunderG,Montel’stheoremimplies
thatifτ isaBorelprobabilitymeasureonP withcompactsupport,andifthesupportofτ contains
an admissible subset of P (see Definition 3.54), then J (G )=∅ (Lemma 3.56). In particular, if
ker τ
the support of τ contains an interior point with respect to the topology of P, then J (G ) = ∅
ker τ
(Lemma 3.52). From this result, it follows that for any Borel probability measure τ on P with
compact support, there exists a Borel probability measure ρ with finite support, such that ρ is
arbitrarily close to τ, such that the support of ρ is arbitrarily close to the support of τ , and such
thatJ (G )=∅(Proposition3.57). Theaboveresultsmeanthatinacertainsense,J (G )=∅
ker ρ ker τ
for mostBorelprobabilitymeasuresτ onP. Summarizingthese resultswe canstatethe following.
Theorem 1.7 (CooperationPrinciple III, see Lemmas 3.52, 3.56, Proposition3.57). Let M (P)
1,c
be endowed with the topology O such that τ → τ in (M (P),O) if and only if (a) ϕdτ →
n 1,c n
ϕdτ for each bounded continuous function ϕ on P, and (b) suppτ →suppτ with respect to the
n
Hausdorff metric. WesetA:={τ ∈M (P)|J (G )=∅}andB :={τ ∈M (P)|JR (G )=
1,c ker τ 1,c ker τ
R
∅,♯suppτ <∞}. Then we have all of the following.
(1) A and B are dense in (M (P),O).
1,c
(2) If the interior of the support of τ is not empty with respect to the topology of P, then τ ∈A.
(3) For each τ ∈A, the chaos of the averaged system of the Markov process induced by τ disap-
pears (more precisely, all the statements in Theorems 1.5, 1.6 hold).
In the subsequent paper [40], we investigate more detail on the above result (some results of
[40] are announced in [41]).
Weremarkthatin1983,bynumericalexperiments,K.MatsumotoandI.Tsuda([20])observed
that if we add some uniform noise to the dynamical system associated with iteration of a chaotic
map onthe unit interval[0,1],thenunder certainconditions, the quantities whichrepresentchaos
(e.g., entropy, Lyapunov exponent, etc.) decrease. More precisely, they observed that the entropy
decreasesandtheLyapunovexponentturnsnegative. Theycalledthisphenomenon“noise-induced
order”,andmanyphysicistshaveinvestigateditbynumericalexperiments,althoughtherehasbeen
only a few mathematical supports for it.
Moreover, in this paper, we introduce “mean stable” rational semigroups in subsection 3.6. If
G is meanstable,then J (G)=∅ anda smallperturbationH ofG is stillmeanstable. We show
ker
that if Γ is a compact subset of Rat and if the semigroup G generated by Γ is semi-hyperbolic
+
(see Definition 2.12) and J (G) = ∅, then there exists a neighborhood V of Γ in the space of
ker
non-empty compact subset of Rat such that for each Γ ∈V, the semigroup G generated by Γ is
′ ′ ′
mean stable, and J (G)=∅.
ker ′
By using the above results, we investigate the random dynamics of polynomials. Let τ be
a Borel probability measure on P with compact support. Suppose that J (G ) = ∅ and the
ker τ
smallest filled-in Julia set Kˆ(G ) (see Definition 3.19) of G is not empty. Then we show that the
τ τ
5
functionT ,τ ofprobabilityoftendingto∞∈Cˆ belongstoUτ andisnotconstant(Theorem3.22).
Thus T ,τ∞is non-constantandcontinuous on Cˆ and varies only onJ(Gτ). Moreover,the function
T ,τ is∞characterizedas the unique Borel measurable bounded function ϕ:Cˆ →R which satisfies
∞
M (ϕ) = ϕ, ϕ| ≡ 1, and ϕ| ≡ 0, where F (G ) denotes the connected component of
τ F∞(Gτ) Kˆ(Gτ) ∞ τ
the FatousetF(G )ofG containing∞(Proposition3.26). Fromthese results,wecanshowthat
τ τ
T has a kind of “monotonicity,” and applying it, we get information regarding the structure
,τ
∞
of the Julia set J(G ) of G (Theorem 3.31). We call the function T a devil’s coliseum,
τ τ ,τ
∞
especially when int(J(G )) = ∅ (see Example 6.2, Figures 2, 3, and 4). Note that for any h ∈ P,
τ
T is not continuous at any point of J(h)6=∅. Thus the above results deal with a phenomenon
∞,δh
which canholdin the randomcomplex dynamics,but cannotholdin the usualiterationdynamics
of a single polynomial.
It is a naturalquestion to ask about the regularity of non-constantϕ∈U (e.g., ϕ=T ) on
τ ,τ
∞
theJuliasetJ(Gτ).ForarationalsemigroupG,wesetP(G):= h G{all critical values of h:Cˆ →Cˆ},
wheretheclosureistakeninCˆ,andwesaythatGishyperbolicif∈P(G)⊂F(G).IfGisgenerated
S
by {h ,...,h } as a semigroup, we write G=hh ,...,h i. We prove the following theorem.
1 m 1 m
Theorem 1.8 (see Theorem 3.82 and Theorem 3.84). Let m≥2 and let (h ,...,h )∈Pm. Let
1 m
G = hh ,...,h i. Let 0 <p ,p ,...,p <1 with m p = 1. Let τ = m p δ . Suppose that
1 m 1 2 m i=1 i i=1 i hi
h−i 1(J(G))∩h−j1(J(G))=∅ for each (i,j) with i6=Pj and suppose also thPat G is hyperbolic. Then
we have all of the following statements.
(1) J (G ) = ∅, int(J(G )) = ∅, and dim (J(G)) < 2, where dim denotes the Hausdorff
ker τ τ H H
dimension with respect to the spherical distance on Cˆ.
(2) Suppose further that at least one of the following conditions (a)(b)(c) holds.
m
(a) p log(p deg(h ))>0.
j=1 j j j
(b) PP(G)\{∞} is bounded in C.
(c) m=2.
Then there exists a non-atomic “invariant measure” λ on J(G) with suppλ =J(G) and an
uncountable dense subset A of J(G) with λ(A) = 1 and dim (A) > 0, such that for every
H
z ∈ A and for each non-constant ϕ ∈U , the pointwise Ho¨lder exponent of ϕ at z, which is
τ
defined to be
|ϕ(y)−ϕ(z)|
inf{α∈R|limsup =∞},
|y−z|α
y z
→
is strictly less than 1 and ϕ is not differentiable at z (Theorem 3.82).
(3) In(2)above,thepointwiseHo¨lderexponentofϕatz canberepresentedintermsofp ,log(deg(h ))
j j
and the integral of the sum of the values of the Green’s function of the basin of ∞ for the
N
sequence γ =(γ ,γ ,...)∈{h ,...,h } at the finite critical points of γ (Theorem 3.82).
1 2 1 m 1
(4) Undertheassumptionof(2),foralmosteverypointz ∈J(G)withrespecttotheδ-dimensional
Hausdorff measure Hδ where δ = dim (J(G)), the pointwise Ho¨lder exponent of a non-
H
constant ϕ∈U at z can be represented in terms of the p and the derivatives of h (Theo-
τ j j
rem 3.84).
Combining Theorems 1.5, 1.6, 1.8, it follows that under the assumptions of Theorem 1.8, the
chaosoftheaveragedsystemdisappearsintheC0 “sense”,butitremainsintheC1 “sense”. From
Theorem1.8,wealsoobtainthatifp issmallenough,thenforalmosteveryz ∈J(G)withrespect
1
to Hδ and for each ϕ ∈ U , ϕ is differentiable at z and the derivative of ϕ at z is equal to zero,
τ
even though a non-constant ϕ ∈ U is not differentiable at any point of an uncountable dense
τ
subsetofJ(G)(Remark3.86). Toprovetheseresults,weuseBirkhoff’sergodictheorem,potential
6
theory, the Koebe distortion theorem and thermodynamic formalisms in ergodic theory. We can
constructmanyexamplesof(h1,...,hm)∈Pm suchthath−i 1(J(G))∩h−j1(J(G))=∅foreach(i,j)
with i 6= j, where G = hh ,...,h i, G is hyperbolic, Kˆ(G) 6= ∅, and U possesses non-constant
1 m τ
m
elements (e.g., T ) for any τ = p δ (see Proposition 6.1, Example 6.2, Proposition 6.3,
∞,τ i=1 i hi
Proposition 6.4, and Remark 6.6).
We also investigate the topologPy of the Julia sets J of sequences γ ∈ (suppτ)N, where τ is
γ
a Borel probability measure on P with compact support. We show that if P(G )\{∞} is not
τ
bounded in C, then for almost every sequence γ with respect to τ˜:=⊗∞j=1τ, the Julia set Jγ of γ
has uncountably many connected components (Theorem 3.38). This generalizes [2, Theorem 1.5]
and [4, Theorem 2.3]. Moreover, we show that Kˆ(G ) = ∅ if and only if T ≡ 1, and that if
τ ,τ
Kˆ(G ) = ∅, then for almost every γ with respect to τ˜, the 2-dimensional L∞ebesgue measure of
τ
filled-inJuliasetK (seeDefinition3.40)ofγ isequaltozeroandK =J hasuncountablymany
γ γ γ
connected components (Theorem 3.41 and Example 3.59). These results generalize [4, Theorem
2.2] and one of the statements of [2, Theorem 2.4].
Another matter of considerable interest is what happens when J (G ) 6= ∅. We show that
ker τ
if τ is a Borel probability measure on Rat with compact support and G is “semi-hyperbolic”
+ τ
(see Definition 2.12), then J (G ) 6= ∅ if and only if J (τ) 6= ∅ (Theorem 3.71). We define
ker τ meas
several types of “smaller Julia sets” of M . We denote by J0(τ) the “pointwise Julia set” of M
τ∗ pt τ∗
restricted to Cˆ (see Definition 3.44). We show that if Gτ is semi-hyperbolic, then dimH(Jp0t(τ))<
2 (Theorem 3.71). Moreover, if J (G ) 6= ∅, G is semi-hyperbolic, and ♯suppτ < ∞, then
ker τ τ
J0(τ) = J(G ) (Theorem 3.71). Thus the dual of the transition operator of the Markov process
pt τ
induced by τ can detect the Julia set of G . To prove these results, we utilize some observations
τ
concerning semi-hyperbolic rational semigroups that may be found in [29, 32]. In particular, the
continuity of γ 7→J is required. (This is non-trivial, and does not hold for an arbitrary rational
γ
semigroup.)
Moreover, even when J (G ) 6= ∅, it is shown that if J (G ) is included in the unbounded
ker τ ker τ
componentofthecomplementoftheintersectionofthesetofnon-semi-hyperbolicpointsofG and
τ
J(G ),thenforalmosteveryγ ∈PN withrespecttoτ˜,the2-dimensionalLebesguemeasureofthe
τ
JuliasetJ ofγ isequaltozero(Theorem3.48). Toprovethisresult,weagainutilizeobservations
γ
concerning the kernel Julia set of G , and non-constant limit functions must be handled carefully
τ
(Lemmas 4.6, 5.32 and 5.33).
As pointed out in the previous paragraphs, we find many new phenomena which can hold in
random complex dynamics and the dynamics of rationalsemigroups, but cannothold in the usual
iteration dynamics of a single rational map. These new phenomena and their mechanisms are
systematically investigated.
In the proofs of all results, we employ the skew product map associated with the support of τ
(Definition 3.46), and some detailed observations concerning the skew product are required. It is
a new idea to use the kernel Julia set of the associated semigroup to investigate random complex
dynamics. Moreover,itisbothnaturalandnewtocombinethetheoryofrandomcomplexdynamics
and the theory of rational semigroups. Without considering the Julia sets of rational semigroups,
we are unable to discern the singular properties of the non-constant finite linear combinations ϕ
(e.g., ϕ=T , a devil’s coliseum) of the unitary eigenvectors of M .
,τ τ
∞
In section 2, we give some fundamental notations and definitions. In section 3, we present the
mainresultsofthispaper. Insection4,weintroducethebasictoolsusedtoprovethemainresults.
In section 5, we provide the proofs of the main results. In section 6, we give many examples to
which the main results are applicable.
In the subsequent paper [40], we investigate the stability and bifurcation of M (some results
τ
of [40] are announced in [41]).
Acknowledgment: The author thanks Rich Stankewitz for valuable comments. This work was
supported by JSPS Grant-in-Aid for Scientific Research(C) 21540216.
7
2 Preliminaries
In this section, we give some basic definitions and notations on the dynamics of semigroups of
holomorphic maps and the i.i.d. random dynamics of holomorphic maps.
Notation: Let (X,d) be a metric space, A a subset of X, and r > 0. We set B(A,r) := {z ∈
X | d(z,A) < r}. Moreover, for a subset C of C, we set D(C,r) := {z ∈ C | infa C|z−a| < r}.
∈
Moreover, for any topological space Y and for any subset A of Y, we denote by int(A) the set of
all interior points of A.
Definition 2.1. Let Y be a metric space. We set CM(Y) := {f : Y → Y | f is continuous} en-
dowedwiththecompact-opentopology. Moreover,wesetOCM(Y):={f ∈CM(Y)|f is an open map}
endowed with the relative topology from CM(Y). Furthermore, we set C(Y) := {ϕ : Y → C |
ϕ is continuous }. When Y is compact, we endow C(Y) with the supremum norm k·k . More-
∞
over, for a subset F of C(Y), we set F :={ϕ∈F |ϕ is not constant}.
nc
Definition 2.2. LetY beacomplexmanifold. WesetHM(Y):={f :Y →Y |f is holomorphic}
endowedwiththecompactopentopology. Moreover,wesetNHM(Y):={f ∈HM(Y)|f is not constant}
endowed with the compact open topology.
Remark 2.3. CM(Y), OCM(Y), HM(Y), and NHM(Y) are semigroups with the semigroup op-
eration being functional composition.
Definition 2.4. Arational semigroupisasemigroupgeneratedbyafamilyofnon-constantra-
tionalmapsontheRiemannsphereCˆ withthesemigroupoperationbeingfunctionalcomposition([13,
11]). Apolynomialsemigroup isasemigroupgeneratedbyafamilyofnon-constantpolynomial
maps. We set Rat : ={h:Cˆ →Cˆ |h is a non-constant rational map} endowed with the distance
κ which is defined by κ(f,g):=supz Cˆd(f(z),g(z)), where d denotes the spherical distance on Cˆ.
Moreover, we set Rat := {h ∈ Rat∈| deg(h) ≥ 2} endowed with the relative topology from Rat.
+
Furthermore, we set P := {g : Cˆ → Cˆ | g is a polynomial,deg(g) ≥ 2} endowed with the relative
topology from Rat.
Definition 2.5. Let Y be a compact metric space and let G be a subsemigroup of CM(Y). The
Fatou set of G is defined to be F(G):=
{z ∈Y |∃ neighborhood U of z s.t.{g| :U →Y} isequicontinuousonU}.(Forthedefinition
U g G
∈
ofequicontinuity,see[1].) TheJulia set ofGis definedtobe J(G):=Y \F(G). IfGisgenerated
by {g } , then we write G=hg ,g ,...i. If G is generated by a subset Γ of CM(Y), then we write
i i 1 2
G=hΓi. For finitely many elements g ,...,g ∈CM(Y), we setF(g ,...,g ):=F(hg ,...,g i)
1 m 1 m 1 m
and J(g ,...,g ) := J(hg ,...,g i). For a subset A of Y, we set G(A) := g(A) and
1 m 1 m g G
G 1(A):= g 1(A). We set G :=G∪{Id}, where Id denotes the identity map.∈
− g G − ∗ S
∈
By usingSthe method in [13, 11], it is easy to see that the following lemma holds.
Lemma 2.6. Let Y be a compact metric space and let G be a subsemigroup of OCM(Y). Then
for each h ∈ G, h(F(G)) ⊂ F(G) and h 1(J(G)) ⊂ J(G). Note that the equality does not hold in
−
general.
The following is the key to investigating random complex dynamics.
Definition 2.7. Let Y be a compact metric space and let G be a subsemigroup of CM(Y). We
set J (G):= g 1(J(G)). This is called the kernel Julia set of G.
ker g G −
∈
Remark 2.8.TLet Y be a compact metric space and let G be a subsemigroup of CM(Y). (1)
J (G) is a compact subset of J(G). (2) For each h ∈ G, h(J (G)) ⊂ J (G). (3) If G is a
ker ker ker
rationalsemigroupandifF(G)6=∅,thenint(J (G))=∅.(4)IfGisgeneratedbyasinglemapor
ker
if G is a group,then J (G)=J(G). However,for a generalrationalsemigroupG, it may happen
ker
that ∅=J (G)6=J(G) (see subsection 3.5 and section 6).
ker
8
The following postcritical set is important when we investigate the dynamics of rational
semigroups.
Definition 2.9. For a rational semigroup G, let P(G) := {all critical values of g :Cˆ →Cˆ}
g G
where the closure is taken in Cˆ. This is called the postcritica∈l set of G.
S
Remark 2.10. If Γ ⊂ Rat and G = hΓi, then P(G) = G ( {all critical values of h}). From
∗ h Γ
this one may know the figure of P(G), in the finitely generated∈case, using a computer.
S
Definition 2.11. Let G be a rational semigroup. Let N be a positive integer. We denote by
SHN(G) the setofpoints z ∈Cˆ satisfying thatthere exists a positivenumber δ suchthatfor each
g ∈G, deg(g :V →B(z,δ))≤N, for each connected component V of g 1(B(z,δ)). Moreover,we
−
set UH(G):=Cˆ \ N NSHN(G).
∈
Definition 2.12. SLet G be a rational semigroup. We say that G is hyperbolic if P(G)⊂F(G).
We say that G is semi-hyperbolic if UH(G)⊂F(G).
Remark 2.13. We have UH(G)⊂P(G). If G is hyperbolic, then G is semi-hyperbolic.
It is sometimes important to investigate the dynamics of sequences of maps.
Definition 2.14. Let Y be a compact metric space. For each γ = (γ ,γ ,...) ∈ (CM(Y))N and
1 2
each m,n∈N with m≥n, we set γm,n =γm◦···◦γn and we set
Fγ :={z ∈Y |∃ neighborhood U of z s.t. {γn,1}n N is equicontinuous on U}
∈
and J := Y \F . The set F is called the Fatou set of the sequence γ and the set J is called
γ γ γ γ
the Julia set of the sequence γ.
Remark 2.15. Let Y = Cˆ and let γ ∈ (Rat+)N. Then by [1, Theorem 2.8.2], Jγ 6= ∅. Moreover,
if Γ is a non-empty compact subset of Rat and γ ∈ ΓN, then by [29], J is a perfect set and J
+ γ γ
has uncountably many points.
We now give some notations on random dynamics.
Definition 2.16. For a topological space Y, we denote by M (Y) the space of all Borel prob-
1
ability measures on Y endowed with the topology such that µ → µ in M (Y) if and only if
n 1
for each bounded continuous function ϕ : Y → C, ϕ dµn → ϕ dµ. Note that if Y is a
compact metric space, then M (Y) is a compact metric space with the metric d (µ ,µ ) :=
1 0 1 2
R R
M∞j=(Y1)21,jw1+|eR|sRφejφtdjµsdu1µ−p1−RpRφτjφd:j=µd2µ|{2|z,∈wYher|e∀{nφejig}hj∈bNorihsooaddUensoef zs,ubτs(eUt)o>f C0}(.YN).otMeothreaotvseurp,pfoτriseaachcloτse∈d
P1
subset of Y. Furthermore, we set M (Y):={τ ∈M (Y)|suppτ is compact}.
1,c 1
For a complex Banach space B, we denote by B the space of all continuous complex linear
∗
functionals ρ:B →C, endowed with the weak∗ topology.
For any τ ∈M (CM(Y)), we will consider the i.i.d. randomdynamics on Y such that atevery
1
step we choose a map g ∈ CM(Y) according to τ (thus this determines a time-discrete Markov
process with time-homogeneous transition probabilities on the phase space Y such that for each
x∈Y and each Borel measurable subset A of Y, the transition probability p(x,A) of the Markov
process is defined as p(x,A)=τ({g ∈CM(Y)|g(x)∈A})).
Definition 2.17. Let Y be a compact metric space. Let τ ∈M (CM(Y)).
1
1. We set Γ := suppτ (thus Γ is a closed subset of CM(Y)). Moreover, we set X := (Γ )N
τ τ τ τ
(= {γ = (γ ,γ ,...) | γ ∈ Γ (∀j)}) endowed with the product topology. Furthermore,
1 2 j τ
we set τ˜ := ⊗ τ. This is the unique Borel probability measure on X such that for each
∞j=1 τ
n
cylinder setA=A ×···×A ×Γ ×Γ ×··· in X , τ˜(A)= τ(A ). We denote by G
1 n τ τ τ j=1 j τ
the subsemigroup of CM(Y) generated by the subset Γ of CM(Y).
τ
Q
9
2. Let M be the operator on C(Y) defined by M (ϕ)(z) := ϕ(g(z)) dτ(g). M is called
τ τ Γτ τ
the transition operator of the Markovprocess induced by τ. Moreover,let M :C(Y) →
R τ∗ ∗
C(Y) be the dual of M , which is defined as M (µ)(ϕ) = µ(M (ϕ)) for each µ ∈ C(Y)
∗ τ τ∗ τ ∗
and each ϕ ∈ C(Y). Remark: we have M (M (Y)) ⊂ M (Y) and for each µ ∈ M (Y) and
τ∗ 1 1 1
each open subset V of Y, we have M (µ)(V)= µ(g 1(V)) dτ(g).
τ∗ Γτ −
3. We denote by F (τ) the set of µ∈M (Y) saRtisfying that there exists a neighborhood B
meas 1
of µ in M1(Y) such that the sequence {(Mτ∗)n|B :B →M1(Y)}n N is equicontinuous on B.
We set J (τ):=M (Y)\F (τ). ∈
meas 1 meas
4. WedenotebyF0 (τ)thesetofµ∈M (Y)satisfyingthatthesequence{(M )n :M (Y)→
meas 1 τ∗ 1
M1(Y)}n N is equicontinuous at the one point µ. We set Jm0eas(τ):=M1(Y)\Fm0eas(τ).
∈
Remark 2.18. We have F (τ)⊂F0 (τ) and J0 (τ)⊂J (τ).
meas meas meas meas
Remark 2.19. Let Γ be a closed subset of Rat. Then there exists a τ ∈ M (Rat) such that
1
Γ = Γ. By using this fact, we sometimes apply the results on random complex dynamics to the
τ
study of the dynamics of rational semigroups.
Definition 2.20. Let Y be a compact metric space. Let Φ : Y → M (Y) be the topological
1
embeddingdefinedby: Φ(z):=δ ,whereδ denotestheDiracmeasureatz.Usingthistopological
z z
embedding Φ:Y →M (Y), we regardY as a compact subset of M (Y).
1 1
Remark 2.21. If h ∈ CM(Y) and τ = δ , then we have M ◦Φ = Φ◦h on Y. Moreover, for
h τ∗
a general τ ∈ M (CM(Y)), M (µ) = h (µ)dτ(h) for each µ ∈ M (Y). Therefore, for a general
1 τ∗ 1
τ ∈M (CM(Y)), the map M :M (Y)→∗M (Y) can be regarded as the “averagedmap” on the
extensi1on M (Y) of Y. τ∗ 1 R 1
1
Remark 2.22. If τ = δ ∈ M (Rat ) with h ∈ Rat , then J (τ) 6= ∅. In fact, using the
h 1 + + meas
embedding Φ:Cˆ →M1(Cˆ), we have ∅6=Φ(J(h))⊂Jmeas(τ).
The following is an important and interesting object in random dynamics.
Definition2.23. LetY beacompactmetricspaceandletAbeasubsetofY.Letτ ∈M (CM(Y)).
1
For each z ∈ Y, we set T (z):= τ˜({γ = (γ ,γ ,...) ∈X | d(γ (z),A) →0 as n →∞}). This
A,τ 1 2 τ n,1
is the probability of tending to A starting with the initial value z ∈ Y. For any a ∈ Y, we set
T :=T .
a,τ a ,τ
{ }
3 Results
In this section, we present the main results of this paper.
3.1 General results and properties of M
τ
In this subsection, we present some general results and some results on properties of the iteration
of Mτ :C(Cˆ)→C(Cˆ) andMτ∗ :C(Cˆ)∗ →C(Cˆ)∗. The proofs are givenin subsection5.1. We need
some notations.
Definition 3.1. Let Y be a n-dimensional smooth manifold. We denote by Leb the two-
n
dimensional Lebesgue measure on Y.
Definition 3.2. Let B be a complex vector space and let M : B → B be a linear operator. Let
ϕ ∈ B and a ∈ C be such that ϕ 6= 0,|a| = 1, and M(ϕ) = aϕ. Then we say that ϕ is a unitary
eigenvector of M with respect to a, and we say that a is a unitary eigenvalue.
10