Composition Operators Associated with Linear Fractional Transformations in Complex Spaces Fiana Jacobzon, Simeon Reich and David Shoikhet ORT BRAUDE COLLEGE & TECHNION

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 model Historical 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... III This stochastic process is called the Galton-Watson branching process. The Galton-Watson model

Since p m (1) is distribution of probabilities, then it generates function Let ∆ be the open unit disk in C. Obviously, F : ∆ → ∆ is an analytic function. 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 with The Galton-Watson model

III 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 model

III The 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 is n -iterate of F, i.e., then It can be shown that there exists the limit the extinction probability of the branching process which is called the extinction probability of the branching process. The Galton-Watson model F 2 (0) 0 F 5 (0) F 4 (0) F 3 (0) F 1 (0) p

One parameter semigroups of analytic mappings Consider the family S ={ F 0, F 1, F 2,...} of iterates of F i.e., F 0 = I, F 1 =F, F 2 = F o F=F 2,... In other words, F 0 (z)=z, F 1 (z)=F(z), F 2 (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 a aa a self-mapping of the unit disk ∆. The pair (∆, S ) is called a aa a discrete dynamic system.

For any z Є ∆ we can construct the sequence {F n (z)} n Є N (N={ 0,1,2 …}) of points in ∆. F1(z)F1(z) z F4(z)F4(z) F3(z)F3(z) F2(z)F2(z) F5(z)F5(z) x y 1 ∆ {F (n) (z)} n Є N

The iteration problem Koenigs embedding process This problem can be solved by using the so-called Koenigs embedding process. Consider a family of the functions: S={F 0, F 1, F 2,...} such that F 0 = I, F 1 =F, F 2 = F (2),..., F n = F (n),... F 0 = I, F 1 =F, F 2 = F (2),..., F n = F (n),... Find F (n) explicitly for all n = 1,2,3,… i.F 1 = F ii.F t preserves iteration property for all t ≥ 0 a continuous semigroup A family of functions that satisfies both these properties is called a continuous semigroup. To do this we first should find a continuous function u (t, z) = F t (z) in parameter t, such that

Continuous semigroups of analytic functions A family S={F t } t≥0 is called a one-parameter continuous semigroup (flow) in ∆ if For integer t we get by property ( i ) than F 1 is F, than F 2 is F (2) and The pair (∆, S ) is called a continuous dynamic system. F1(z)F1(z) z F4(z)F4(z) F3(z)F3(z) F2(z)F2(z) F5(z)F5(z) F½(z)F½(z) F 1¾ (z) F 4⅔ (z)

Embedding problem A classical problem of analysis is given an analytic self-mapping F of the open unit disk ∆, to find a continuous semigroup S={F t } t≥0 in ∆ such that F 1 =F. If such a semigroup exists then F is said to be e ee embeddable. In general there are those self-mappings which are not embeddable. So, the problem becomes: describe the class of self-mappings which are embeddable. F 2 (0) 0 F 5 (0) F 4 (0) F 3 (0) F 1 (0)

Continuous semigroups of Linear Fractional mappings Linear Fractional Mappings(LFM) Most of those discrete applications based on semigroups produced from the so-called Linear Fractional Mappings (LFM), i.e., analytic functions in the complex plane of the form: 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 ensure that it preserves the open disk Δ. Coefficients Problem Find the condition on the coefficients of LFM which ensure the existence of a continuous semigroup S={F t } t≥0 in ∆, such that F 1 =F.

Solution The 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)} n Є N converge to τ uniformly on compact subsets of ∆. Denjoy –Wolff point The point τ is called the Denjoy –Wolff point of the semigroup and it is a common fixed point of {F (n) (z)} n Є N. 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 ∆. F 2 (z) z F 5 (z) F 4 (z) F 3 (z) F 1 (z) τ

1. Dilation case (rotation + shrinking): - the common fixed point (a) Re c = 0 (group of rotations) Examples (b) Re c ≠ 0

2. Hyperbolic case (shrinking the disk to a point): - the common DW pointExamples

3. Parabolic case : - the common DW pointExamples

with different features and properties, we consider these classes separately. Classification Since the class of analytic self-mappings comprises three subclasses:  dilation τ є Δτ є Δ  hyperbolic τ є ∂∆, 0< F ’ ( τ )<1  parabolic τ є ∂∆, F ’ ( τ )=1

Results Dilation case Proposition 1 (Elin, Reich and Shoikhet, 2001) Let F : Ĉ → Ĉ be an LFM of the form 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 In particular, if a є R, then F is always embeddable into a continuous semigroup of analytic self-mappings of Δ.

Results Hyperbolic case Proposition 2 Let F : Ĉ → Ĉ be an LFM of the form 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.

Results Proposition 3 Let F : Ĉ → Ĉ be an LFM of the form 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. Parabolic case

The Method Koenigs function Our method of proof based on the so-called Koenigs function, which is a powerful tool to solve also many other problems as well as computational problems. with F (τ) = τ and 0<|F ’ (τ)| <1. functional equation of Shcroeder It was proven by Koenigs and Valiron that there exist a solution of the following functional 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 F t as F t (z) = h -1 ( λ t h(z)), where λ = F ’ (τ) Namely, consider for example an LFM of dilation or hyperbolic case, that is

Example Let us consider an LFM of dilation type The Koenigs function associated with F is so Thus, F can be represented in the form F(z) = h -1 ( λ h(z)), where λ = F ’ ( τ) = ½, hence F t (z) = h -1 ( λ t h(z)). Direct calculations show: In particular, substituting here t = n, we get explicitly all iterates F ( n ) = F n

Thank you III