Semigroup representations in holomorphic dynamics

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.


Introduction
One of the motivations of this paper is to add a new entry to Sullivan's dictionary between holomorphic dynamics and Kleinian groups. This entry consist of the algebraic part of a holomorphic dynamical system, that corresponds to the algebraic structure of a Kleinian group. From this point of view, the natural object is a semigroup. This investigation was initiated by the following question ofÉtienne Ghys: • Are there multiplicative characters, defined on the semigroup of polynomials with composition, which do not arises as a multiplicative function of the degree?
In general, multiplicative characters play an important role in representation theory, which is also the subject of this work. We give a positive answer to Ghys' question, and suggest a general method to construct multiplicative characters on polynomials. Then, we study the automorphism groups of several semigroups of interest in holomorphic dynamical systems. In the second section, we show that the semigroup of polynomials and rational maps are generated by linear automorphisms and the Galois group action on C. A result, due to Hinkkanen, states that the automorphism group of entire functions consist of the group of continuous inner automorphisms. Using Hinkkanen's Theorem, we show that, the group of automorphisms of the semigroup of meromorphic functions also consists of the group of continuous inner automorphisms. We give algebraic conditions, using sandwich semigroups, that characterize when two given polynomials, or rational maps, are conformally conjugated. The main tool is a theorem, due to Schreier, which states that any representation of a semigroup of maps S is geometric, whenever S contains constant maps. This result is useful to study the representation space of semigroups of maps and correspondences. Hence, this theorem remarks the importance of considering semigroups of maps together with constants. This point of view is also adopted in Eremenko's paper [5].
On the third section we provide several examples of semigroups representations that appear in holomorphic dynamical systems. Among these, there is a connection with the deformation space of a given rational map. The topology of the deformation space of a rational space is discussed in [2].
On the last section, we generalize the results of Section 2, to the setting of holomorphic correspondences. In particular, we prove a generalized Schreier Lemma for correspondences and holomorphic correspondences. This allow us to characterize the Galois group Gal(C), the group of all field automorphisms of C, as the subgroup of Bij(C) that under conjugation functionally preserves the finite holomorphic world.

Semigroups.
A semigroup is a set S together with a binary operation which is associative. Given any set X, consider the semigroup M ap(X) of all maps φ : X → X, with composition as semigroup operation. It contains the group of bijections Bij(X). A subset I ⊂ S is called a left (respectively right) ideal, if si ∈ I (respectively is ∈ I), for all s ∈ S and i ∈ I.
For every element s in S, let τ s be the left translation by s. The map s → τ s induces a representation φ from S into M ap(S). In fact, the same map gives a representation from S into M ap(I) for every left ideal I in S. Note that φ is a faithful representation only in the case when, for every pair of elements g, h in S, there is i ∈ I such that gi = hi.
Let X be an abstract set, then there is a canonical inclusion of X into M ap(X), sending every x in X to the constant map x. The image of this map is a left ideal I in M ap(X), the ideal of constants of X. An element x in M ap(X) belongs to I if, and only if, for every g ∈ M ap(X) we have : The ideal of constants is contained in every left ideal of M ap(X). In this sense, the ideal of constants is the smallest left ideal in M ap(X). Given any semigroup S, we can use properties (i) and (ii) to define the ideal of constants, whenever it exists.
Example. In general, the ideal of constants in M ap(X) is not prime. Assume that X has at least three points x 1 , x 2 and x 3 , let g 1 and g 2 in M ap(X), such that, Image(g 1 ) = {x 1 , x 2 }, g 2 (x 1 ) = g 2 (x 2 ) = g 2 (x 3 ), then g 2 • g 1 is constant but g 2 is not. However, if we restrict to the space of continuous maps C 0 (X) in a topological space X with enough regularity, then the ideal of constants is prime.
From now on, we will consider the special case where X is either the complex plane C or the Riemann sphereC. Given two subsets S 1 and S 2 in M ap(X), we denote by S 1 , S 2 the semigroup generated by S 1 and S 2 .

Multiplicative characters of semigroups.
Let P ol(C) denote the semigroup of complex polynomials with composition as semigroup multiplication. Let us consider the set Hom(P ol(C), C) of all multiplicative characters, that is, the set of homomorphisms χ in satisfying for all P 1 , P 2 ∈ P ol(C).
The degree function deg, is a basic example of a multiplicative character in P ol(C). Any multiplicative function of deg induces a multiplicative character. It was a question ofÉ. Ghys whether there are other characters apart from these examples. We give a positive answer to this question and give a description of how to construct multiplicative characters on P ol(C). To do so, first let us recall a theorem due to Ritt, see [13].
Definition. A polynomial P , is called prime, or indecomposable, if whenever we have P = Q • R, where Q and R are polynomials, then either deg(Q) = 1 or deg(R) = 1. A decomposition of P = P 1 • P 2 • ... • P n is called a prime decomposition if, and only if, each P i is a prime polynomial of degree at least 2 for all i.
Given a prime decomposition of a polynomial P = P 1 • ... • P n , a Ritt transformation, say in the j place, is the substitution of the pair P j • P j+1 , in the prime decomposition of P , by the pair Q j • Q j+1 . Where Q j and Q j+1 are prime polynomials satisfying P j • P j+1 = Q j • Q j+1 . Now we can state Ritt's theorem.
Theorem 1 (Ritt). Let P = P 1 • P 2 • ... • P m and P = Q 1 • ... • Q n be two prime decompositions of P , then n = m. Moreover, any two given prime decompositions of P are related by a finite number of Ritt transformations.
In [13], Ritt showed that there are three types of Ritt transformations, namely, see also [1]: 2. Substitute P i • P i+1 by P i+1 • P i , when P i and P i+1 are Tchebychev polynomials.
3. If P i (z) = z k and P i+1 (z) = z r P (z k ) for some polynomial P and natural numbers r and k. Define Q i+1 (z) = z r (P (z)) k , then substitute In particular, there are two invariants of a prime decomposition, the length of a prime decomposition, and the set of degrees in the prime decomposition. Hence, for every P ∈ P ol(C) the length of a prime decomposition of P is a well defined additive character l(P ). That is, it satisfies l(P 1 • P 2 ) = l(P 1 ) + l(P 2 ). Now, define the function χ by χ(P ) = e l(P ) . Then, χ is a multiplicative character which is not a multiplicative function of the degree.
The following theorem gives a method to generate multiplicative characters in P ol(C).
Theorem 2. Let φ be a complex function, defined on the set of prime polynomials, satisfying: (i) φ(c) = 0 for every constant c.
(ii) If P 1 , P 2 , P 3 , P 4 are prime polynomials with P 1 • P 2 = P 3 • P 4 , then Then, φ generates a multiplicative character Φ. Conversely, if Φ is a multiplicative character in P ol(C), which is not the constant map 1, then Φ satisfies the conditions above.
Proof. Let P be a composite polynomial and P = P 1 • P 2 • ... • P n be a prime decomposition of P , define Let us check that Φ is well defined. By Theorem 1, it is enough to consider a step modification of P 1 • P 2 • ... • P n . Let Q j and Q j+1 be two polynomials such Hence Φ is invariant under step modifications and, by Theorem 1, it is independent of the prime decomposition of P . It follows, from the definition, that Φ is a multiplicative character.
Conversely, let Φ be a multiplicative character. For any pair of constants c 1 and c 2 , the equations c 1 • c 2 = c 1 and Then, for every constant c, either we have Φ(c) = 1 or Φ(c) = 0. If Φ(c) = 1, the equation P (c) = P • c implies that Φ(P ) = 1 for all P . Hence if Φ is not constantly 1, then we have Φ(c) = 0 for every constant c. The second condition follows from the fact that Φ is a multiplicative character.
Example (Affine characters.). Let H be the ideal of non injective polynomials. Any multiplicative character χ : Aff(C) → C admits an extension to a multiplicative character defined in P ol(C). For instance, put χ(c) = 0 for all constant c, and χ(h) = 0, for all other h in H.
In the same way, we can extend affine characters to other semigroups containing Aff(C), such as Rat(C), Ent(C) or the semigroup of holomorphic correspondences discussed at the end of this work. Now let us construct non-trivial extension of the constant affine character equal to 1. In order to do so, we have to consider the bi-action, left and right, of Aff(C) on P ol(C). The bi-orbit of a polynomial P is the set of all polynomials of the form A • P • B, where A, B belong to Aff(C). We say that a polynomial has no symmetries if, there are no elements A, B, in Aff(C), such that P = A•P •B.
Lemma 3. Let P be a prime polynomial, and let AF (P ) be the semigroup generated by the bi-orbit of the Affine group of the set of iterates {P n }. Let Q and R be a pair of polynomials, of degree at least 2, such that Q • R ∈ AF (P ), then Q ∈ AF (P ) and R ∈ AF (P ).
Proof. Since Q • R belong to AF (P ), there is a prime decomposition of Q • R whose elements are of the form A • P • B. By Ritt's Theorem, any other prime decomposition of Q • R is obtained by a finite number of Ritt's transformations. But, Ritt's transformations are either permutations, or substitution by a pair of elements in the bi-affine orbit. Hence, all prime decompositions of Q • R have prime elements in AF (P ). Then the conclusion of the Lemma follows.
Example. Let χ be the constant multiplicative character equal to 1 defined on Aff(C), and P be a prime polynomial of degree at least 2. Let us extend χ to all P ol(C) defining χ(Q) = 1 for all Q in the bi orbit by Aff(C) of the set {P n } n∈P , and χ(Q) = 0 for all the other polynomials Q in P ol(C). By Lemma 3 and Theorem 2, this is a well defined character. In fact for any number a, defining χ(A • P n • B) = a n where A, B are elements in Aff(C) gives other extensions of χ in P ol(C).
To extend arbitrary multiplicative characters defined on Aff(C), the construction of the character is more involved. At least in the case where P is a prime polynomial, such that every iterate P n is without symmetries, it is possible to extend any multiplicative character χ on Aff(C).
The ideal of constants is very useful to understand the structure of M ap(X).
. Now we recall a result due to Schreier that describes the semigroup M ap(X) using the ideal of constants. For further details see [14], and also the discussions in Eremenko's paper [5] and Magill's survey [9].
Proof. Consider the restriction f := φ |X to the ideal of constants. Since φ is a homomorphism, it maps ideals into ideals, it also preserves the properties of the ideal of constants, hence f sends constants to constants. So f is a map from X to Y . Moreover, If φ ∈ Aut(M ap(X)), then f is a map from X to itself. Moreover, since φ is an automorphism, we can apply the argument to φ −1 , so we get that f is invertible.
Which implies that f is a bijection and the formula φ(P ) = f • P • f −1 .
In fact, there is no need that the homomorphism in Lemma 4 is defined in all M ap(X), the same proof above shows.
Corollary 5. Let S 1 and S 2 be subsemigroups of M ap(X) and M ap(Y ), respectively, and such that X 1 = S 1 ∩ X and Y 1 = S 2 ∩ Y are both non empty sets.
• the homomorphism φ is injective, or surjective, if and only if, the map f is injective or surjective. In particular, φ is an isomorphism if, and only if, f is a bijection.
• When S 1 and S 2 are topological semigroups, then Φ is continuous if, and only if, f is continuous.
Along with the ideal of constants, the affine group Aff(C) plays an important role in the description of automorphisms of polynomials. Later on, we will consider generalizations to semigroups generated by correspondences. A particular case of Lemma 4 is the following Corollary 6. For any set X, the group Aut(M ap(X)) is isomorphic to Bij(X).
Let Gal(C) denote the absolute Galois group of C, that is, the full group of field automorphisms of C. Remind that since every orientation preserving element in Gal(C) must fix the complex rationals, the identity and complex conjugation are the only continuous elements in Gal(C). The action of Gal(C) extends to an action in Rat(C), the semigroup of rational functions in C. In particular, the action of Gal(C) in C extends to an action in P ol(C).
Proof. Let φ be an element of Aut(P ol(C)). By Lemma 4, the restriction f = φ | C is a bijection from C to C, and φ(P ) = f •P •f −1 . First, let us check that φ = Id if, and only if, f = Id. Note that we can realize evaluation as composition with a constant function. If f = Id, then we have for every polynomial P and z ∈ C, that is φ = Id. The converse is clear.
Since, by Lemma 4, φ is a conjugation, then φ(P ) and P have the same degree. In particular, φ leaves the affine group Aff(C) invariant, so φ(Aff(C)) = Aff(C). This fact also follows from the characterization of Aff(C) as the set of injective polynomials. In particular, Aut(P ol(C)) ⊂ Aut(Aff(C)), the converse is also true by Lemma 4, since any conjugacy in the Aff(C) extends to a conjugacy in P ol(C).
The group of translations T is the commutator of Aff(C), hence T is invariant under φ. The value of a translation at one point, determines the translation. Let τ c denote the translation z → z + c, since , then g is a bijection of C which is the restriction to the constants of the mapφ = τ −f (0) • φ and g(0) = 0. By definition,φ ∈ Aut(P ol(C)) and φ(τ c ) = τ g(c) , it follows that Let A 0 be the group of injective polynomials fixing 0, sinceφ(0) = 0, theñ φ(A 0 ) = A 0 . Now we repeat the argument above, this time in multiplicative terms, to show that h(c) = g(c) g (1) is a bijection of C preserving multiplication and h(1) = 1. By definition, h also preserves addition with h(0) = 0, hence h ∈ Gal(C). Note that h is the restriction to constants of the map (g(1 The proof of Proposition 7 can be adapted to show Proposition 8. Let Rat(C) denote the semigroup of rational maps in the Riemann sphere, then Aut(Rat(C)) = Gal(C), P SL(2, C) Proof. Since φ(Id) = Id, and using the formula R • R −1 = Id, one can check that φ sends P SL(2, C), the group of invertible rational maps, into P SL(2, C). Post composing φ with an element of P SL(2, C) we can assume that φ(∞) = ∞. In this case, it follows that φ(Aff(C)) ⊂ Aff(C), hence if φ(∞) = ∞ then φ ∈ Aut(Aff(C)). Since every element in Gal(C), P SL(2, C) induces a conjugation in Rat(C), we have the claim of the proposition. Now we want to study the semigroup of meromorphic functions M er(C). This semigroup contains the semigroup of entire functions Ent(C). We recall a theorem by Hinkkanen [6].
In other words, except for the identity, no element in Gal(C) leaves the semigroup Ent(C) invariant in the space of formal series. The following are immediate consequences of Lemma 4 and Hinkkanen's Theorem.
Proposition 10. The group of automorphisms of M er(C) is isomorphic to P SL(2, C).
Proof. Let φ be an element in Aut(M er(C)), and γ be an element in P SL(2, C) so that γ(φ(∞)) = ∞. By Lemma 4, γ • φ is a geometric automorphism in M er(C). Now, a meromorphic map g is entire if, and only if, g has no finite poles. Since infinity is fixed by γ • φ, the map γ • φ sends entire functions into entire functions. By Theorem 9, we have γ • φ ∈ Aff(C) and φ ∈ P SL(2, C).
Corollary 11. Every automorphism of M er(C) is continuous.
is continuous if, and only if, φ extends to a map in Aut(M er(C)).
All above gives a characterization of elements in Bij(C) that belong to the Galois group Gal(C).
Theorem 13. Let F be an element in Bij(C), let us assume it fixes three points in C, then the following are equivalent.
i) The map F belongs to Gal(C).
ii) The induced map in M ap(C) sends Rat(C) into itself.
iii) The induced map in M ap(C) sends P ol(C) into itself.
iv) The induced map in M ap(C) sends Aff(C) into itself.

Sandwich semigroups.
Here, we give an algebraic condition for when two rational maps are Möbius conjugated, for this we do not require any dynamical restrictions on the rational maps. We start with the polynomial case, where the action of P SL(2, C) is replaced by the action of Aff(C).
Given a map g : Y → X, let us define on M ap(X, Y ) the following op- In particular, if S is a subsemigroup of M ap(X) and g ∈ M ap(X), the set S g := (S, * g ) is also a semigroup. In particular, given a polynomial P , let us consider the semigroup P ol P (C).
By substituting Aff(C) by P SL(2, C), and P ol(C) by Rat(C) in the proof of previous theorem, we obtain the following Theorem 16. Let R 1 and R 2 be two complex rational maps, and consider an automorphism of semigroups Φ : Rat R1 (C) → Rat R2 (C). Then there is f ∈ Bij(C) and B ∈ P SL(2, C) such that In particular, if Φ is continuous with Φ(Id) = Id, then Φ is conjugation by an element of P SL(2, C).

Which implies the following
Corollary 17. Two rational maps R 1 and R 2 are conjugate by a map in P SL(2, C) if, and only if, the semigroups Rat R1 (C) and Rat R2 (C) are continuously isomorphic with an isomorphism φ, such that φ(Id) = Id.
By Theorem 14, the condition φ(Id) = Id is equivalent to require that φ(R 1 ) = R 2 . Every automorphism of Rat(C) induces an isomorphism of sandwich semigroups. Indeed, if φ ∈ Aut(Rat(C)) take Q, R rational maps such that φ(Q) = R, then φ is an isomorphism between Rat Q (C) and Rat R (C). Let ψ be an isomorphism of sandwich semigroups in Rat(C). By Theorem 14 and Lemma 4, ψ induces an automorphism of Rat(C) if, and only if, ψ(Id) = Id. Let us now discuss the situation of sandwich isomorphisms for small semigroups. Let Q y R be two non-constant rational maps, and consider the semigroup S = Q, R, C . Take R 1 and R 2 in S and consider an isomorphism φ between S R1 and S R2 . Since Q and R are the non constant elements in S with smaller degree, then we have either In any case, φ 2 fixes Q and R. Then the restriction of φ to constants is a non trivial bijection of C, which commutes with Q and R.

Semigroup representations.
In We can generalize the previous example to rational functions R : C → C. In this case, we obtain a family of semigroups R, a parametrized by a point a in the plane C. In this way, the set D R = { R, a : a ∈ C} inherits the usual topology from C. Let X R ⊂ D R be the set of finite semigroups, we call the set J R = X R \ {isolated points}, the algebraic Julia set of R. The complement F R = { R, a : a ∈ C} \ J R will be called the algebraic Fatou set of R in D R . In this setting, the algebraic Fatou set is the interior of the set of free semigroups in D R . These definitions reflect the dynamical Julia set J(R), which is the closure of the repelling periodic points in C and, the dynamical Fatou set F (R) which is the complement of J(R) in C.

Representations of semigroups of polynomials.
Let P be a partition of P ol(C), we say that P is a compatible partition if for A, B ∈ P, and a pair of points a ∈ A, b ∈ B, the composition a • b belongs to a component C in P which do not depend on the representatives a and b. A graduation is a partition of P ol(C) which is compatible with composition.
As we discussed earlier in the paper, P ol(C) has a non empty set of multiplicative characters. Each multiplicative character in P ol(C) induces a graduation in P ol(C). The fibers of multiplicative characters induce compatible partitions. In particular, the degree of a polynomial induces a compatible partition of P ol(C). In this case, the classes of this partition are P ol d (C), the set of polynomials of given degree d. We will describe now some examples of representations of semigroups of the form P, A into P ol d (C).
Note that since we are including an ideal of constants A in the domain, then we have to include the constants in P ol d (C) as well. Otherwise, there is no representation from P, A into P ol d (C). Nevertheless, including constants, in both domain and range, is consistent with the philosophy of Lemma 4. In this setting, every representation of S in P ol(C) is geometric, and realized by a map defined in the complex plane. Let S be a subsemigroup in P ol(C) containing Id, and let us consider representations of S into P ol 0 (C), the semigroup of constant polynomials. Let φ : S → P ol 0 (C) be a homomorphism, since φ(Id) is constant, then φ(R) = φ(Id) • φ(R) = φ(Id). Hence, any representation of S into the constant polynomials is a constant map.
The theory of representation of semigroups of the form P, J(P ) into P ol(C) is widely discussed in holomorphic dynamics in other terms. For example, the theory of the continuous representations of P, J(P ) into P ol(C) is parameterized by the J-stable components of P . For example, see [2].
Another important situation is representations of semigroups P, P(P ) into P ol(C), here P(P ) is the postcritical set of P . Interior components of the representation space can be parameterized by combinatorially equivalent polynomials. Uniformization of these components by suitable geometric objects (like suitable Teichmüller spaces), shed light on many problems in holomorphic dynamics. In this direction, important advances were made by Douady, Hubbard, Lyubich, McMullen, Sullivan and Thurston, among many others. See for example [4] and [10]. Now, let us consider the space of representations of affine semigroups into the space of polynomials of degree d. This space includes all linearizations around periodic orbits. Here, we review the repelling case. A complete treatment of linearization theory in holomorphic dynamics can be found in Milnor's book [12].
Let A λ in Aff(C) of the form z → λz. Let P be a polynomial such that there exist a repelling cycle O = {z 0 , z 1 , ..., z n } with multiplier λ. The Poincaré function associated to z 0 , is a map φ : C → C sending 0 to z 0 which locally conjugates A λ to P n around z 0 . This construction induces a representation of A λ , C into P, U 0 for a suitable neighborhood U 0 of z 0 . Moreover, since Poincaré functions turn out to be meromorphic functions, it also induces a representation of affine semigroups into the semigroups of meromorphic functions. Similar constructions apply to other kind of linearizations. In the attracting case, the inverse of the Poincaré function, defined on a neighborhood U 0 of z 0 , is known as König's coordinate and gives a representation of P, U 0 into A λ , D r , where D r denotes the disk of radius r and r < 1. This construction can also be applied to the parabolic case.
The process of renormalization, in holomorphic dynamics, gives examples of semigroups of the form P, U that admit representations into themselves.
Let P be a polynomial P (z), of degree n, with connected and locally connected Julia set. Then, ∞ is a superattracting fixed point of P . If A 0 (∞) denotes the basin of ∞ of P , by Böttcher's theorem, there is a homeomorphism φ :C \D → A 0 (∞), that conjugates z → z n inC \D with P in A 0 (∞). Since J(P ) is connected and locally connected, the map φ extends to the boundaries by Caratheodory's theorem. The map on the boundaries induces a representation of z n , S 1 into P, J(P ) .
It would be interesting to have results, analogous to Theorem 13 or Theorem 14, that characterizes the action of quasiconformal maps in C. This would allow us to determine quasiconformal conjugation in terms of semigroup representations.

When Julia set is homeomorphic to a Cantor set.
Now consider the special case where J(R) is homeomorphic to a Cantor set. For simplicity in the arguments, let us assume that deg(R) = 2.
Let us consider a Jordan curve γ containing in its interior the Julia set and a critical point; while the other critical point and all critical values lie outside γ. Choosing a suitable γ, we assume that R −1 (γ) is contained in the interior of γ and consists of two Jordan curves γ 1 and γ 2 . We get an scheme similar to the one sketched in Figure 1. Let us call D 1 and D 2 the interiors of γ 1 and γ 2 , respectively. With this scheme, we obtain representations of R, D into other semigroups. To do so, let us modify topologically the Cantor scheme, and instead of the restrictions of R on D i , consider affine maps A i sending the modified γ i to γ. This induces a representation of R, J(R) into Aff(C).
If we modify the curves γ, γ 1 and γ 2 to circles and considering Möbius transformations g i , instead of the maps that send γ i to γ. We get a representation Φ of R, J(R) into a "half" classical Schottky group Γ with two generators. This is an example of a representation of non cyclic Kleinian groups in rational semigroups. The conjugating map of Φ may be taken quasiconformal, hence the Hausdorff dimension of the limit set of the Schottky group can be estimated in terms of the Hausdorff dimension of J(R). In particular, let R(z) be a quadratic polynomial of the form z 2 + c, such that J(R) is a Cantor set. In this case, the parameter c belongs to the complement of the Mandelbrot set. A theorem of Shishikura shows that there are sequences of quadratic polynomials R cn (z) = z 2 + c n , with parameters c n tending to the boundary of the Mandelbrot set, and such that the Hausdorff dimension of the Julia sets tends to 2. With this result, Shishikura showed that the Hausdorff dimension of the boundary of the Mandelbrot set is 2. Perhaps, using the representation above is possible to get a result analogous of Shishikura's theorem for the boundary of the Classical Schottky space.
It is interesting to solve the extremal problem between these two objects from holomorphic dynamics. In case there exist an extremal map from R, J(R) into the Classical Schottky space, there would be a sort of estimate from above of the distance between this two pieces of Sullivan's dictionary.
The problem to describe the set of representations of P, A , for an invariant set A, into Aff(C), C is difficult, still remain many questions. In the case where S ∈ Rat(C), it is interesting to understand the space of representations of S, A into P SL(2, C).

Binding semigroups of maps with constants.
Let us consider two semigroups of the form S 1 = g 1 , A 1 and S 2 = g 2 , A 2 , in this case the categorical sum, or coproduct, S 1 S 2 , is defined as Analogously, we define the binding of a countable family of semigroups of the form g i , A i . A classical example of such construction in dynamics is the process of mating of quadratic polynomials, first described by Douady in [3]. We start with two mateable polynomials S 1 = P 1 , C and S 2 = P 1 , C . Using a topological construction, the mating P 1 P 2 is a quadratic rational map R, C . Thus we have a representation of S 1 S 2 into the space of rational maps of degree 2.

Simultaneous linearizations and deformation spaces.
Let us now discuss a more elaborated example, associated to a fixed rational map R 0 of degree d. Let {a 0 , a 1 , ..., a n−1 } be a periodic cycle of R 0 , of period n and multiplier λ, with |λ|> 1. Let us denote by A λ the map z → λz, and φ the Poincaré function associated to R n 0 and a 0 . As we discussed above φ induces a representation of A λ , C into R 0 , C . The same is true for R i 0 • φ, for each i = 0, ..., n − 1, all together, induce a representation of the binding n−1 i=0 A λ , ⊔ n−1 i=0 C × {i} into R 0 , C , here we put a component A λ , C for each periodic point in the cycle {a 0 , a 1 , ..., a n−1 }. Let us carry this construction further considering all repelling periodic cycles of R 0 , we obtain a countable binding of semigroups of the form A λ , C associated to all Poincaré functions of R 0 . Let us call A(R 0 ) this countable binding, so we have a representation Ψ : A(R 0 ) → R 0 , C . Taking instead of R 0 , C , the corresponding Poincaré functions, we obtain a representationΨ from A(R 0 ) into M er(C). The image φ(A(R 0 )) has a compactification which is related to Lyubich-Minsky laminations discussed in Section 7 of [8].
Let us assume that R 0 is hyperbolic of degree d. Since R 0 , C is a subsemigroup of Rat(C), let us now regard Ψ as a homomorphism from A(R 0 ) into Rat(C). Let X (R 0 ) be the space of representations from A(R 0 ) into Rat(C), whose image is of the form R, C for some R of degree d. In other, words we are considering all graduated representations that arise by deformations of the semigroup R 0 , C . Let us define the map P : X (R 0 ) → Rat d (C), such that for every Φ ∈ X (R 0 ), let P (Φ) = R where R is the non constant rational map generating P (Φ).
Let P ar d (C) be the set of all rational maps, of degree d, that admit a parabolic periodic point. Then P (X (R 0 )), in Rat d (C), is equal to Rat d (C) \ P ar d (C). By a result of Lyubich, see [7], the space Rat d (C) \ P ar d (C) consists of the union of J-stable components in Rat d (C).
In [2], the authors construct a dynamical Teichmüller space T 2 (R 0 ), which uniformize the J-stable components of R 0 . It turns out that the space T 2 (R 0 ) is isomorphic to X (R 0 ).

Correspondences.
Let A and B be two sets, let G be a subset of A × B. A correspondence is a triple (G, A, B). If (a, b) ∈ G we say that b corresponds to a under G. The notion of correspondences generalizes, in a way, the notion of functions. Indeed, for every map f : X → Y , the graph of f induces a correspondence in X × Y . Borrowing notation from Function Theory, we define the set Im(G) = {b ∈ B : ∃a ∈ A such that (a, b) ∈ G} is called the image of G, analogously the domain of G is defined by Similarly, the image of an element a ∈ A is the set G(a) = {b ∈ B : (a, b) ∈ G}. Given a set G ⊂ A×B, the set G −1 = {(b, a) ∈ B ×A : (a, b) ∈ G} is called the inverse of G. Let G 1 ⊂ A×B and G 2 ⊂ B×C be two correspondences, the composition G 1 • G 2 of G 1 and G 2 is the correspondence induced by the set Let X be a set, a correspondence K in X is a correspondence of the form (K, X, X), additionally we require that Dom(K) = X. A correspondence K in X is called surjective if Im(K) = X and, finite if every image is a finite set. In particular, constant maps are finite correspondences.
If G is a finite correspondence, the degree of the image of G is the maximum of the cardinalities of its images.

Schreier's Lemma for correspondences.
With composition, the set of correspondences Corr(X) in X is a semigroup. Since functions are special cases of correspondences, the semigroup of correspondences of X contains M ap(X). The proof of the following lemma is immediate by contradiction.
Lemma 18. Let K 1 and K 2 be two correspondences in Corr(X) such that g = K 1 • K 2 is a map and K 2 is surjective, then K 1 is a map.
We will start by generalizing Schreier's lemma restricted to correspondences generated by maps. Let us start with some definitions, where R 1 and R 2 belong to M ap(X) and R 2 is surjective. We denote by BL(X) the subsemigroup of Corr(X), generated by all block correspondences.
Theorem 19 (Schreier Lemma for blocks). Let φ : BL(X) → BL(Y ) be an homomorphism then, there exist f ∈ M ap(X, Y ) such that for every K ∈ BL(X) Proof. The identity Id is characterized among Corr(X) by the properties Id • Id = Id and that for every C ∈ Corr(X) we have Id • C = C • Id = C. Since this properties are preserved by homomorphisms we have Φ(Id) = Id. Let R ∈ Corr(X) be any map, then R • R −1 = Id. But then Φ(R • R −1 ) = Φ(R) • φ(R −1 ) = Id is a map, by Lemma 18 then Φ(R) is a map. Hence Φ sends maps into maps, so φ restricted to M ap(X) is a homomorphism of semigroups. By Lemma 4, there exist f ∈ M ap(X, Y ) such that, for every map Since blocks generate BL(X), it is enough to check that the theorem holds for every correspondence of the form • f −1 and then for every block K, Φ(K) = f • K • f −1 as we wanted to prove.
We now include in the discussion the constant maps in Corr(X), these are no longer an ideal, but we can consider the unique minimal left ideal I in Corr(X), which is generated by all constant maps. The semigroup of correspondences acts on I. That is, there is a map α : Corr(X) → M ap(I) that sends every correspondence K ∈ Corr(X) to the left translation by K in M ap(I).
Lemma 20. The map α : Corr(X) → M ap(I) is a one-to-one map. Moreover, for every c ∈ I, we have α(c) = c.
Proof. Suppose that K 1 and K 2 are correspondences in Corr(X) such that α(K 1 ) = α(K 2 ). In particular, for every constant c ∈ I, we have However, a correspondence is characterized by the set of images, then K 1 = K 2 . The second part of the Lemma follows from the equation c • K = c for all c ∈ I. Now we are set to prove: be a homomorphism of semigroups. Then, there is a map f ∈ M ap(X, Y ), such that, for every K ∈ Corr(X) we have Proof. By the same argument in the proof of Theorem 19, the map Φ sends maps to maps. Moreover, the restriction of Φ to BL(X), is an homomorphism from BL(X) to BL(Y ), by Theorem 19 there is f ∈ M ap(X, Y ) such that for every K ∈ BL(X), we have that φ(K) = f • K • f −1 .
Let I and J denote the minimal ideals in Corr(X) and Corr(Y ), respectively. Let us consider the maps α X : Corr(X) → M ap(I), and α Y : Corr(Y ) → M ap(J ) as in Lemma 20, and define S X = α(Corr(X)) and S Y = α(Corr(Y )). By Lemma 20, the maps α X and α Y are bijections to their images. Moreover, α X and α Y send constants to constants. Hence the map α Y • φ • α −1 X is a homomorphism between the semigroups G X and G Y , sending constants to constants. By Corollary 5, there exist F ∈ M ap(I, J ) such that for every g ∈ G, we have Since α, restricted to minimal ideal is the identity, then for every c in I and every correspondence K ∈ Corr(X) we have F (c) = f (c), also and α −1 X (c) = c evaluating in c the equation above, we get Note that in the proof of Theorem 21, we need the theorem on block correspondences to get the existence of the map f . Once we have Schreier's lemma for the whole semigroup of correspondences, we can generalize it for subsemigroups of correspondences, as long as they contain the minimal ideal of constants.
Corollary 22. Let S 1 and S 2 be subsemigroups of Corr(X) and Corr(Y ), respectively, such that X 1 = S 1 ∩ X and Y 1 = S 2 ∩ Y are both non empty. If φ : S 1 → S 2 is a homomorphism of semigroups, then there is f : • the homomorphism φ is injective, or surjective, if and only if, the map f is injective or surjective. In particular, φ is an isomorphism if, and only if, f is a bijection.
• When S 1 and S 2 are topological semigroups, then Φ is continuous if, and only if, f is continuous.

Holomorphic correspondences.
A correspondence K in C is holomorphic if, as a set of C × C, K can be decomposed as a countable union of analytic varieties, see McMullen's book [11]. However recall that, in our setting, we require that Dom(K) = C. Moreover, we assume that the preimage of every point admits an analytic extension to the whole Riemann sphere, with the exception of finitely many points. Let us denote by HCorr(C) the semigroup of holomorphic correspondences, which includes the semigroup of entire maps and constants. We denote by F HCorr(C), the semigroup of finite correspondences on the Riemann sphere. By definition F HCorr(C) contains the semigroup of rational maps Rat(C) together with all constant maps. Hence, there exist a minimal left ideal of finite holomorphic correspondences. Since rational maps are onto the Riemann sphere, if R 1 and R 2 are rational maps, the block R −1 1 • R 2 belongs to F HCorr(C). Let K ∈ F HCorr(C), a holomorphic correspondence with degree d. That is there is z such that K(z) consists of d points. Let S 1 , S 2 , ..., S d denote all the symmetric polynomials with d variables. For every i, S i (K) induces a holomorphic map fromC toC, it follows that S i (K) is a rational map. Moreover, for every z the image K(z) are the roots of the polynomial S 1 (K(z)) + S 2 (K(z))Z + ... + S d (K(z))Z d−1 + Z d .
Reciprocally any polynomial in Z, whose coefficients are rational maps in z, defines a finite holomorphic correspondence inC. From this discussion, we have the following known fact.
Proposition 23. The space F HCorr(C) is equivalent to the space of monic polynomials with coefficients in Rat(C).
The proof of Theorem 21, can be repeated in the setting of holomorphic correspondences. In this case, every automorphism of HCorr(C) or F HCorr(C) is induced by conjugation of some function in Bij(C) or Bij(C). Nevertheless, the holomorphic structure imposes holomorphic conditions in such bijections.
Proof. The semigroup of maps in HCorr(C) coincides with the semigroup of entire maps. The first part of the theorem is a consequence of Corollary 11. Since the semigroup of maps in F HCorr(C) is equal to Rat(C). By restriction, any automorphism of F HCorr(C) induces an automorphism of Rat(C). But every automorphism of Rat(C) is generated by P SL(2, C) and Gal(C). Now let us see that, in fact, Gal(C) also acts on F HCorr(C). Let γ an element in Gal(C), and let K ∈ F HCorr(C), then γ • K • γ −1 is a finite correspondence in Corr(C). Let d be the maximum cardinality of a fiber of K. Remind that K is holomorphic in the Riemann sphere if, and only if, there is a symmetric polynomial S d in d variables, such that S d (K) is a rational map inC. Since γ acts on symmetric polynomials, there is a symmetric polynomialS d such that But the second equality is the conjugation of a rational map by a Galois map, hence is rational. This implies that γ • K • γ −1 is a holomorphic correspondence. It follows that the group of automorphisms of F HCorr(C) is isomorphic to the group of automorphisms of Rat(C), which by Proposition 8 is isomorphic to P SL(2, C), Gal(C) .
The central argument for Theorem 14 is Schreier's Lemma, with some modifications we can prove the corresponding theorem for holomorphic correspondences.
It is not clear whether the Galois group action acts on holomorphic correspondences. Perhaps there is a generalization to Hinkkanen's argument in this setting. Now, we can state an analogous statement to Theorem 13 for F HCorr(C).
Corollary 26. Let F be an element in Bij(C), that fixes 0, 1 and ∞. Then F belongs to Gal(C) if, and only if, F induces an automorphism of F HCorr(C).