Composition Operators Associated with Linear Fractional Transformations

in Complex Spaces

Fiana Jacobzon, Simeon Reich and David Shoikhet

ORT BRAUDE COLLEGE & TECHNION

1

tt

There was concern amongst the Victorians that aristocratic surnames were becoming extinct. Galton originally posed the question regarding the probability of such an event in the Educational Times of 1873, and the Reverend Henry William Watson replied with a solution. Together, they then wrote in 1874 paper entitled “On the probability of extinction of families. “

The Galton-Watson modelThe Galton-Watson model Historical backgroundHistorical background

One of the first applicable models of the complex dynamical systems on the unit disk arose more than a hundred years ago in studies of dynamics of stochastic branching processes.

Let us consider a process starting with a single particle which splits

to an unknown number m of new identical particles in the first generation.

Then in the next generation each one of m particles splits to an

unknown number of new identical particles and so on . . .

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This stochastic process is called the Galton-Watson branching process.This stochastic process is called the Galton-Watson branching process.

The Galton-Watson modelThe Galton-Watson model

Since pm(1) is distribution of probabilities, then it generates function

10

1

mmp

212

11

10

0

)1( zpzppzpzFm

mm

Let ∆ be the open unit disk in C. Obviously, F : ∆ → ∆ is an analytic function.

10

1

0

1

0

1

mm

m

m

mm

mm pzpzpzF

I

So we start at time t = 0 with a single particle (Z(0)=1)

The first generation Z(1) is a random variable with distribution of probabilities

,2,1,0,10/1)1( mZmZppm

with

The Galton-Watson modelThe Galton-Watson model

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The question is:

What is the distribution of probabilities of a random variable Z(t) in the t-th generation?

In other words, what is the probability p(n)k that after t = n

generation the number of particles will be k?

The Galton-Watson modelThe Galton-Watson model

I II

The Galton-Watson modelThe Galton-Watson model

The question is very complicated because we do not know the situation in the previous generations.

• So one can define the needed distribution of probabilities as

the coefficients of Taylor extension of F(n)

• It turns out, that if times

)( ))(((n

n zFFFzF is n-iterate of F,

i.e.,

0!

1

z

k

nkn

k z

F

kp

2210

0

)( zpzppzpzF nnn

k

knk

nthen

It can be shown that there exists the limit n

n

n

npFp 0lim0lim

which is called the extinction probability of the branching processthe extinction probability of the branching process.

The Galton-Watson modelThe Galton-Watson model

F2(0)

0

F5(0)F4(0)

F3(0)

F1(0)p

One parameter semigroups of One parameter semigroups of analytic mappingsanalytic mappings

Consider the family S={F0, F1, F2,...} of iterates of F i.e.,

F0= I, F1=F, F2 = FoF=F2, ...

In other words, F0(z)=z, F1(z)=F(z), F2(z)=F(F(z)),…

Let ∆ be the open unit disk in the complex plane C,

and F : ∆ → ∆ be an analytic function in ∆ with values in ∆.

In this case one says that F is aa self-mapping of the unit diskself-mapping of the unit disk ∆∆.

The pair (∆, The pair (∆, SS ) is called ) is called aa discrete dynamic systemdiscrete dynamic system..

One parameter semigroups of One parameter semigroups of analytic mappingsanalytic mappings

For any z Є ∆ we can construct the sequence {Fn(z)}nЄN (N={0,1,2…}) of points in ∆.

F1(z)

zF4(z)

F3(z)

F2(z)

F5(z)x

y

1

∆

{F(n)(z)}nЄN

The iteration problemThe iteration problem

This problem can be solved by using the so-called Koenigs embedding processKoenigs embedding process.

Consider a family of the functions: Consider a family of the functions: SS={={FF00, , FF11, , FF22,...},...} such that such that

FF00= = II, , FF11==FF, , FF22 == FF(2)(2),,...... , , FFnn = = FF((nn)),... ,...

Find Find FF((nn)) explicitly for allexplicitly for all n = n = 1,2,3,…1,2,3,…

i. F1 = F

ii. Ft preserves iteration property for all t ≥ 0

A family of functions that satisfies both these properties is called a continuous semigroupa continuous semigroup.

To do this we first should find a continuous function u (t , z) = Ft (z) in

parameter t , such that

Continuous semigroups of analytic Continuous semigroups of analytic functionsfunctions

A family A family S={FS={Ftt}}t≥0t≥0 is called a one-parameter continuous semigroup is called a one-parameter continuous semigroup

(flow) in(flow) in ∆ ∆ ifif

zandstallforzFFzFi stst ,0,

zallforzzFii tt 0lim

For integer t we get by property (i) than F1 is F, than F2 is F(2) and

,,,33

nn FFFF

The pair (∆,The pair (∆, SS) is called a continuous dynamic system.) is called a continuous dynamic system.

F1(z)

zF4(z)

F3(z)

F2(z)F5(z)F½(z)

F1¾(z)

F4⅔(z)

Embedding problemEmbedding problem

A classical problem of analysis is given an analytic self-mapping F of the open unit disk ∆, to find a continuous semigroup S={Ft}t≥0

in ∆ such that F1=F.

If such a semigroup exists then F is said to be embeddableembeddable.

In general there are those self-mappings which are not embeddable.

So, the problem becomes: describe the class of self-mappings So, the problem becomes: describe the class of self-mappings which are embeddable.which are embeddable.

F2(0)

0

F5(0)F4(0)

F3(0)

F1(0)

Continuous semigroups of Continuous semigroups of Linear Fractional mappingsLinear Fractional mappings

Most of those discrete applications based on semigroups produced from the so-called Linear Fractional MappingsLinear Fractional Mappings (LFM)(LFM), i.e., analytic functions in the complex plane of the form:

dcz

bazzF

The interest of the Galton-Watson model has increased because of connections with chemical and nuclear chain reactions, the theory of cosmic radiation, the dynamics of disease outbreaks in their generations of spread.

Another important problem is finding conditions on a self-mapping F which ensure that it can be embedded into a continuous semigroup.

In particular, for LFM this problem can be reformulated as follows:

Find the conditions on the coefficients of an LFM which Find the conditions on the coefficients of an LFM which

ensure that it preserves the open disk ensure that it preserves the open disk ΔΔ..

Coefficients ProblemCoefficients Problem

Find the condition on the coefficients of LFM which ensure Find the condition on the coefficients of LFM which ensure thethe

existence of a continuous semigroupexistence of a continuous semigroup SS={={FFtt}}tt≥≥00 in in ∆, ∆,

such thatsuch that FF11=F=F..

SolutionSolutionThe important key to solve our problem is the asymptotic behavior of the semigroup in both discrete and continuous cases. It described in the well-known Theorem of Denjoy and Wolff.

Theorem (Denjoy-Wolff, 1926)

Let ∆ be the open unit disk in the complex plane C. If an analytic self-

mapping F is not an elliptic automorhpism of ∆, then there is an unique

point τ in ∆U∂∆ such that the iterates {F(n)(z)}nЄN converge to τ

uniformly on compact subsets of ∆.

The point τ is called the Denjoy –Wolff pointDenjoy –Wolff point of the semigroup and it is a

common fixed point of {F(n)(z)}nЄN .

If, in particular, F is a producing function of a Galton-Watson branching

process, then τ is exactly the extinction probability of this process.

E. Berkson and H. Porta (1981) established a continuous analog of Denjoy-Wolff theorem for continuous semigroups of analytic self-mapping of ∆.

F2(z)

z

F5(z)F4(z)

F3(z)

F1(z)τ

z 1)(1 zF

)(2 zF )(3 zF

1. Dilation case (rotation + shrinking): 0Re,)( czezF ctt

0 - the common fixed point

)(1 zF

)(2 zF

)(3 zF)(4 zF

)(1 zF

)(2 zF

)(3 zF )(4 zF

z

1

(a) Re c = 0 (group of rotations)

1

z

...,2,1,0,)()(),()(,)( 110 ntzFFzFzFzFzzF nn

ExamplesExamples

(b) Re c ≠ 0

1

z

2. Hyperbolic case (shrinking the disk to a point):tt

t ezezF 1)(

1 - the common DW point

)(zFt

ExamplesExamples

...,2,1,0,)()(),()(,)( 110 ntzFFzFzFzFzzF nn

3. Parabolic case : tzt

tztzFt

1221

221)(

1 - the common DW point

ExamplesExamples

...,2,1,0,)()(),()(,)( 110 ntzFFzFzFzFzzF nn

)(1 zF

)(2 zF

)(3 zF)(4 zF

z

1

)(1 wF

)(2 wF

)(3 wF)(4 wF

w

with different features and properties, we consider these classes separately.

ClassificationClassification

Since the class of analytic self-mappings comprises three subclasses:

)(1 zF

)(2 zF

)(3 zF )(4 zF

z

1 dilation

τ є Δ

1

z)(zFt

hyperbolicτ є ∂∆, 0<F’ (τ)<1

)(1 zF

)(2 zF

)(3 zF)(4 zF

1

z)(1 wF

)(2 wF

)(3 wF)(4 wF

w

parabolic τ є ∂∆, F’ (τ)=1

ResultsResults

Dilation caseDilation case

Proposition 1 (Elin, Reich and Shoikhet, 2001)

Let F : Ĉ → Ĉ be an LFM of the form .1,1

acz

azzF

The following assertions hold:

i. F analytic self-mapping of Δ if and only if | a |+| c | ≤ 1;

ii. If i. holds and a ≠ 0 then F is embeddable into continuous semigroup of analytic self mappings of Δ if and only if

.1

1logargcos

a

c

a

In particular, if a є R, then F is always embeddable into a continuous semigroup of analytic self-mappings of Δ.

)(1 zF

)(2 zF

)(3 zF)(4 zF

z

1

ResultsResults

Hyperbolic caseHyperbolic case

Proposition 2

Let F : Ĉ → Ĉ be an LFM of the form dcz

bazzF

with F (1) = 1 and 0<F’ (1) <1.

The following assertion are equivalent:

i. |c / b| ≤ 1 and c ≠ -b;

ii. F analytic self-mapping of Δ of hyperbolic type;

iii. F is embeddable into continuous semigroup of analytic self-mappings of Δ.

c

dz ?

1

z)(zFt

ResultsResults

Proposition 3

Let F : Ĉ → Ĉ be an LFM of the form dcz

bazzF

with F (1) = 1 and F’ (1) =1.

The following assertion are equivalent:

i. Re(d / c) ≤ -1 and d ≠ -c;

ii. F analytic self-mapping of Δ of parabolic type;

iii. F is embeddable into continuous semigroup of analytic self-mappings of Δ.

)(1 zF

)(2 zF

)(3 zF)(4 zF

1

z)(1 wF

)(2 wF

)(3 wF)(4 wF

w

Parabolic caseParabolic case

c

dz ?

The MethodThe Method

Our method of proof based on the so-called Koenigs functionKoenigs function, which is a powerful tool to solve also many other problems as well as computational problems.

dcz

bazzF

with F (τ) = τ and 0<|F’ (τ)| <1.

It was proven by Koenigs and Valiron that there exist a solution of the following functional equation of Shcroederfunctional equation of Shcroeder:

h ( F(z)) = λ h(z), where λ = F’ (τ)

Thus, F can be represented in the form

F(z) = h-1 ( λ h(z)), where λ = F’ (τ)

Then for all t ≥ 0 we can write Ft as

Ft (z) = h-1 ( λt h(z)), where λ = F’ (τ)

Namely, consider for example an LFM of dilation or hyperbolic case, that is

ExampleExample

Let us consider an LFM of dilation type 13

1

21

z

zzF

The Koenigs function associated with F is

,3

2

1where

1 21

31

kkz

zzh so

11

kz

zzh

Thus, F can be represented in the form

F(z) = h-1 ( λ h(z)), where λ = F’ (τ) = ½, hence Ft (z) = h-1 ( λt h(z)).

Direct calculations show:

z

zzF t

t

t

123

3

21

21

In particular, substituting here t = n, we get explicitly all iterates F(n)=Fn

Thank youThank you

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1

z)(zFt