A survey on the hypertranscendence of the solutions of the Schröder's, Böttcher's and Abel's equations
AA survey on the hypertranscendence of the solutions of theSchröder’s, Böttcher’s and Abel’s equations
Gwladys Fernandes
Abstract.
In 1994, P.-G. Becker and W. Bergweiler [8] listed all the differentially algebraicsolutions of three famous functional equations: the Schröder’s, Böttcher’s and Abel’s equations.The proof of this theorem combines various domains of mathematics. This goes from the theory ofiteration, which gave birth to these equations, to the algebro-differential notion of coherent familiesdeveloped by M. Boshernitzan and L. A. Rubel. This survey is an excursion into the history ofthese equations, in order to enlighten the different pieces of mathematics they bring together andhow these parts fit into the result of P.-G. Becker and W. Bergweiler.
This survey is about three famous functional equations, named after the mathematicians ErnstSchröder, Lucien Böttcher and Niels Henrik Abel. These equations are linked to the local study ofthe iteration of a rational function R ( z ) ∈ C ( z ) with complex coefficients around a fixed point α .That is a point of C ∪ {∞} such that R ( α ) = α . We can assume, without any loss of generality,that α = 0 (see Section 2.1 and (8)). Moreover, to avoid the trivial case of Möbius transformations(see (7) for the definition), which is well-understood, we assume that the degree of R ( z ), that isthe maximum of the degrees of the coprime polynomials on its numerator and denominator, is atleast 2. Then, the equations we are interested in are the following:1. Let s = R (0). If s = 0, then, the Schröder’s equation is: f ( sz ) = R ( f ( z )) , (S)2. If R ( z ) = P + ∞ n = d a n z n , where d ≥
2, then the Böttcher’s equation is: f ( z d ) = R ( f ( z )) , (B)3. The Abel’s equation is: f ( R ( z )) = f ( z ) + 1 . (A)In 1994, P.-G. Becker and W. Bergweiler [8] listed all the differentially algebraic solutions ofEquations ( S ), ( B ) and ( A ). The aim of this survey is to present the proof of this result as atestimony of the richness of the interactions these three equations centralize between various areas1 a r X i v : . [ m a t h . N T ] F e b f mathematics and how the two authors combine them to get their beautiful result. For morehistorical details on the theory of iteration, besides the ones given below, we refer to [5] and [2].Before diving into the history of these equations, let us remind some definitions. First, a formalpower series f ( z ) with coefficients in the complex plane C is said to be differentially algebraic over C ( z ) if there exists a non-zero polynomial P ( z, X , . . . , X n ) with coefficients in C such that: P ( z, f ( z ) , f ( z ) , . . . , f ( n ) ( z )) = 0 , (1)where f ( n ) ( z ) is the n -th derivative of f with respect to z . We say that f is differentiallytranscendental or hypertranscendental over C ( z ) if it is not differentially algebraic over C ( z ).This notion of differential algebraicity generalises the one of algebraicity. Indeed, a formal powerseries f with coefficients in C is said to be algebraic over C ( z ) if there exists a non-zero polynomial P ( z, X ) with coefficients in C such that: P ( z, f ( z )) = 0 , (2)and it is said to be transcendental over C ( z ) otherwise. Moreover, formal power series f ( z ) , . . . , f n ( z )with coefficients in C are said to be algebraically dependent over C ( z ) if there exists a non-zeropolynomial P ( z, X , . . . , X n ) with coefficients in C such that: P ( z, f ( z ) , . . . , f n ( z )) = 0 . (3)If they are not algebraically dependent over C ( z ), we say that these functions are algebraicallyindependent over C ( z ). Thus, a hypertranscendental function is transcendental, and all its deriva-tives are algebraically independent over C ( z ).Coming back to our topic, the three equations ( S ) , ( B ) and ( A ) are introduced for the need ofthe iteration theory, for which Newton’s method is a famous representative, and the research offixed points of a rational fraction. They reach their zenith with the development of the theory of P.Fatou and G. Julia around 1918, which divides the complex plane into different domains accordingto the local behaviour of the iterates of a rational fraction, in terms of convergence or divergenceto a fixed point. Thus, the rather algebraic problem of iteration of a rational fraction and thedetermination of its fixed points is linked to the rather analytic theory developed by P. Fatou andG. Julia. The interface between these different domains of mathematics grows in the following yearswith the work of J. F. Ritt [63] in the 20’s, who furnishes the list of all the differentially algebraicsolutions of the Schröder’s equation near a repelling fixed point (see the definition in Section 2.1).This is a first step toward the result of P.-G. Becker and W. Bergweiler [8] we are interested in(Theorem 3.1 of the present paper). After that, the interest for these equations wanes for almostsixty years, with a renewal of enthusiasm in the 80’s. Indeed, then, the sets of Fatou and Juliashare connexions with dynamical systems and fractals which are very prolific areas at that time.From the algebraic point of view, this surge of interest is visible in 1995 in the result of P.-G.Becker and W. Bergweiler [8], which completes the previous result of J. F. Ritt on the Schröder’sequation. The two authors also provide a partial result of Theorem 3.1 one year earlier in [7].Indeed, they give the list of all the algebraic solutions of the Böttcher’s equation ( B ). In this paper,we present their accomplished result [8], reproduced in Theorem 3.1. This statement provides thelist of all the differentially algebraic solutions of the Schröder’s, Böttcher’s and Abel’s equations.The proof of P.-G. Becker and W. Bergweiler relies on the results in [63, 7], a theorem on coherentfamilies developed by M. Boshernitzan and L. A. Rubel [11], and the theory of P. Fatou and G.Julia [25, 26, 27, 40]. 2 Schröder’s, Böttcher’s and Abel’s equations: their deepinteractions with the iteration theory through history
The Schröder’s equation appears in 1870 in a paper of the same author [69], in link with the stronginterest of the author in Newton’s method. This consists in finding approximations of real rootsof a real-valued function f ( x ). The idea is the following. Take a real number x and define thefollowing sequence for n ≥ x n +1 = x n − f ( x n ) f ( x n ) . (4)Graphically speaking, the number x n +1 is the abscissa of the intersection point between the tangentof the curve of f ( x ) at x = x n and the x -axis. Then, the general principle is that, if one takes x close enough to a root of f ( x ), then the sequence ( x n ) may converge to this root. As mentioned in[2], this problem seems to be one of the oldest processes of iteration in the history of mathematicsand we can find tracks of this in ancient Babylone or in the Arab world of the twelfth century.We shall point out that I. Newton did not present his method with the expression of the sequence(4) but with an equivalent approach based on algebra rather than calculus. An other equivalentformulation is then made by J. Raphson, again without the use of calculus. It is finally T. Simpsonwho introduces the closest procedure to (4), formally defined in this precise form by J. Fourier.The contribution of E. Schröder to the further development and generalization of Newton’smethod results from his idea of turning the discrete problem of the convergence of the sequence (4)into the iteration problem of the function: R ( x ) = x − f ( x ) f ( x ) . (5)With this point of view, a zero α of f ( x ) becomes a fixed point of R ( x ). This also allows theauthor to extend Newton’s method to the complex plane and to the search of complex fixed pointsof R ( x ), or complex zeros of f ( x ). We will be mainly interested in the case where R ( z ) is a rationalfraction with coefficients in C of degree at least two. Let R n ( z ) denote the n -th iterate of R ( z ).We remark that if we start with a point z and if R n ( z ) converges to a point α , then R ( α ) = α .That is, α is a fixed point of R ( z ). This explains why such points are important in the theory ofiteration. Now, by Taylor formula, in a neighbourhood of the fixed point α , the value | R ( z ) − α | can be approximated by | R ( α ) || z − α | . Hence, we guess that the behaviour of the sequence R n ( z )is not the same depending whether | R ( α ) | < | R ( α ) | > α (it is the fixed point theorem of E. Schröder and G. Koenigs mentionedbelow), and in the second, the only way for this sequence to converge to α is that R N ( z ) = α forsome N ∈ N . That is why we distinguish the following categories of fixed points α of R , accordingto the value of | R ( α ) | . We say that a fixed point α is:1. attracting if 0 < | R ( α ) | < super-attracting if | R ( α ) | = 0,3. repelling if | R ( α ) | > rationally indifferent if R ( α ) is a root of unity,3. irrationally indifferent if | R ( α ) | = 1 and if R ( α ) is not a root of unity.E. Schröder establishes (even if his proof is not rigorously explained) the following fixed pointtheorem in [69]: if R ( z ) is an analytic function in a neighbourhood of an attracting fixed point α of R , then, there exists a neighbourhood D of α such that:lim n → + ∞ R n ( z ) = α, ∀ z ∈ D. Note that G. Koenigs provides a complete proof of this theorem in [41]. This result can beobtained by applying the Taylor formula to R ( z ) near z = α . As α is attracting, we find theexistence of a neighbourhood D of α , and a real number (cid:15) such that 0 < (cid:15) < | R ( z ) − α | < (cid:15) | z − α | , ∀ z ∈ D. We deduce that R ( D ) ⊂ D . By induction on n , we find that R n ( D ) ⊂ D , for every integer n ≥ D .Now, let us assume that the rational fraction R ( z ) of the Newton’s method (5) satisfies theassumption of the fixed point theorem. Then, the aim of E. Schröder is to generalize Newton’smethod by finding rational fractions φ ( z ) distinct from R ( z ), such that φ ( α ) = α and such that φ n ( z ) converges to α when z is close enough to α , in order to improve the rate of convergenceof R n ( z ). To do so, it is important to understand the iterates of a rational fraction. But theseare in general difficult to compute. That is why E. Schröder thinks about finding these φ ( z ) suchthat their iterates are easy to compute, while keeping track of the initial rational fraction R ( z ).He solves this problem by the use of conjugation. We say that two rational fractions φ and R are conjugated if there exists an invertible function F ( z ) such that: R = F − φF, (6)where functional composition is multiplicatively written. Particularly interesting conjugations arethose with a Möbius transformation F , that is a rational fraction of the following form: F ( z ) = az + bcz + d , (7)where a, b, c, d ∈ C and ad − bc = 0. They are of degree one and stable under composition.Let us stress that the conjugation preserves the notions of fixed points and iteration. Indeed,if α is a fixed point of R ( z ), of one of the five kinds defined before (attracting, super-attracting,repelling, rationally indifferent or irrationally indifferent) then F ( α ) is a fixed point of φ ( z ) of thesame kind , and for every n ∈ N : R n = F − φ n F. The notion of fixed points is crucial in the study of equations ( S ), ( B ) and ( A ). If a rationalfraction R admits a fixed point α , we let S R,α , B R,α , A R,α denote respectively the Schröder’s,Böttcher’s and Abel’s equation ( S ) , ( B ) and ( A ) associated to the rational fraction R . The mentionof α means that we are interested in the behaviour of the iterations of R ( z ) in a neighbourhood ofthe fixed point α .Furthermore, the conjugation respects the solutions of Equations ( S ) , ( B ) and ( A ). Indeed, if R ( z ) is a rational fraction which admits a fixed point α , then we have the following:4. f is a solution of S R,α ⇔ gf is a solution of S gRg − ,g ( α ) . f is a solution of B R,α ⇔ gf is a solution of B gRg − ,g ( α ) . (8)3. f is a solution of A R,α ⇔ f g − is a solution of A gRg − ,g ( α ) . Let us go back to the approach of E. Schröder to conjugate the rational fraction of Newton’smethod to find an easier form. The choice of φ ( z ) = sz in (6), with a certain s ∈ C ∗ leads to theSchröder’s equation: F ( R ( z )) = sF ( z ) , ( S )in which the unknown is the function F ( z ).Note that this is slightly different from Equation ( S ), with the following link: if f is an invertiblesolution of Equation ( S ), then F = f − is a solution of Equation ( S ). E. Schröder also considersthe case of φ ( z ) = z + λ , for a fixed λ ∈ C , which leads to the Abel’s equation (Equation ( A ) is theparticular case of λ = 1): F ( R ( z )) = F ( z ) + λ. ( A )Let us note that the case φ ( z ) = z d , where d ≥ F ( R ( z )) = F ( z ) d . ( B )Likewise, if f is an invertible solution of Equation ( B ), then F = f − is a solution of Equation( B ).Moreover, E. Schröder is interested in the so-called analytic iteration problem , which he formu-lates in the following way in [70]. For a given analytic function φ ( z ), find a function Φ( w, z ) of twocomplex arguments, which is continuous (even analytic) in both variables and such that:Φ( w, z ) = Φ( w − , φ ( z )) (9)Φ(1 , z ) = φ ( z ) . Even if E. Schröder seems not to explicitly connect this problem to the resolution of theSchröder’s equation, it is worth pointing this link out here, as made in [2]. If there exists aninvertible analytic solution F to Equation ( S ), then, a solution of (9) with φ = R is given by:Φ( w, z ) = F − ( s w F ( z )) . (10)An other solution can be given based on an invertible analytic solution F of the Abel’s equation( A ) by: Φ( w, z ) = F − ( F ( z ) + wλ ) . (11)Finally, a solution can be given based on an invertible analytic solution F of the Böttcher’sequation ( B ) by: Φ( w, z ) = F − (cid:16) F ( z ) d w (cid:17) . (12)Thus, solving Equation ( S ) , ( B ) or ( A ) implies solving the analytic iteration problem for therational fraction R ( z ). In [41], G. Koenigs proves the existence of an analytic solution F of Equation5 S ) around an attracting fixed point α , and also (applying his previous reasoning to F − instead of F ) around a repelling fixed point α . Note that in each case, the solution is unique up to a constantmultiplier. Let us consider the attracting case. Recall that s = R ( α ). The proof of the authorconsists in showing that the following function is analytic in a neighbourhood D of α . F ( z ) = lim n → + ∞ R n ( z ) − αs n . Indeed, if we write F n ( z ) = R n ( z ) − αs n , (13)we have : F n ( R ( z )) = R n +1 ( z ) − αs n = sF n +1 ( z ) . Then, taking the limit when n tends to infinity, we obtain that F is solution of Equation ( S ).Now, to prove that F is analytic in a neighbourhood of α , G. Koenigs reduces the problem to theconvergence of a series of functions P f i ( z ). He then uses a result from G. Darboux which statesthat if the series P f i ( z ) and P f i ( z ) both converge uniformly in a disc, then P f i ( z ) converges toan analytic function on this disk. Let us reproduce here a shorter proof we can find in [5, page 49]for example, replacing G. Darboux result by a theorem which states that a locally uniform limit ofa sequence of analytic functions in an open set D is analytic on D . First, there exist σ ∈ R , suchthat s < σ <
1, a real number r > A ∈ R ∗ + , such that R is analytic in the disc D ( α, r ) centred at α and of radius r , and such that for all z ∈ D ( α, r ): | R ( z ) − α | < σ | z − α | , (14) | R ( z ) − α − s ( z − α ) | < A | z − α | . (15)This arises from the Taylor formula applied to R ( z ) near z = α and the fact that α is anattracting fixed point of R . We deduce from the first inequality above that R ( D ( α, r )) ⊂ D ( α, r ).Besides, even if this means reducing r we assume that α is the only solution to R ( z ) = α in D ( α, r ).Moreover, we see that for all z ∈ D ( α, r ): F n ( z ) = ( z − α ) n − Y k =0 R k +1 ( z ) − αs ( R k ( z ) − α ) . (16)The fact that R ( D ( α, r )) ⊂ D ( α, r ) implies by induction that R n ( D ( α, r )) ⊂ D ( α, r ), for everyinteger n ≥
1. Then, for all z ∈ D ( α, r ) we can apply (14) to R k − ( z ) and (15) to R k ( z ). Thisgives, for all z ∈ D ( α, r ) \ { α } : (cid:12)(cid:12)(cid:12)(cid:12) R k +1 ( z ) − αs ( R k ( z ) − α ) − (cid:12)(cid:12)(cid:12)(cid:12) < A | s | | R k ( z ) − α | =: u k ( z ) . But it appears that for all z ∈ D ( α, r ) \ { α } : u k +1 ( z ) u k ( z ) < σ < .
6e conclude that the product of (16) converges uniformly in D ( α, r ) \ { α } . Moreover, for every n ∈ N , F n ( z ) admits the finite limit 0 when z tends to α . By (see [68, 7.11 Theorem]), the sequence( F n ( z )) of analytic functions over D ( α, r ) converges uniformly on D ( α, r ). We deduce that F isanalytic in D ( α, r ). Notice that for every integer n ≥
1, ( R n ) ( α ) = s n . Furthermore, the uniformconvergence of ( F n ( z )) n near α makes the operations of derivation and limit commute in (13). Wededuce that F ( α ) = 1. Thus F is locally invertible. This result of G. Koenigs strengthen theconnection between the Schröder’s equation and the theory of iteration.For his part, P. Fatou points out the connections (10) and (11) between Schröder’s and Abel’sequations and the analytic iteration problem. Indeed, in [25], P. Fatou remarks that the solution ofthe Schröder’s equation given by G. Koenigs solves this problem. Then, P. Fatou studies the Abel’sequation ( A ) in the case of a rationally indifferent fixed point α . Inspired by a previous result fromL. Leau [43, 44], P. Fatou proves in [25] the existence of particular domains (that is open connectedsets) L , . . . , L k such that, for all j ∈ { , . . . , k } , α belongs to the boundary of L j , R ( L j ) ⊂ L j and the restriction to L j of R n converges uniformly to α on L j , when n tends to infinity. Suchdomains are called the petals of R and the result is called the Leau-Fatou Flower Theorem . Notethat a similar result is proved by G. Julia in [40]. In each petal L j , P. Fatou proves the existenceof an analytic solution of the Abel’s equation ( A ), and mentions again that this provides a solutionto the analytic iteration problem. Let us precise here that Equation ( A ) is introduced by N. H.Abel in [1]. The author remarks that it can be turned into a difference equation. Indeed, if f is aninvertible solution of Equation ( A ), then F = f − is solution of: F ( z + 1) = R ( F ( z )) . ( ˜ A )Now, let us consider the Böttcher’s equation ( B ), in the case of a super-attracting fixed point α . According to J. F. Ritt [61], the existence of an analytic solution in the neighbourhood of α is due to L. Böttcher in [12, 13, 14]. J. F. Ritt provides a short proof of this result in [61]. Letus sketch it here. Even if it means considering, instead of R ( z ), the conjugation of R ( z ) with anappropriate Möbius transformation (7), we may assume that α = 0. Now, let R ( z ) = + ∞ X n = d a n z n , (17)where d ≥
2. Considering a / ( d − d R ( z/a / ( d − d ), we may assume that a d = 1. We know that thereexists a disc D around zero in which the only zero of R ( z ) in D is the origin. Even if this meansreducing the radius of D , we assume that R ( D ) ⊂ D . This can be deduced from the Taylor formulaand the fact that 0 is a super-attracting fixed point of R ( z ). By induction, we find that R n ( D ) ⊂ D ,for every integer n ≥
1, and that the origin is the only zero of R n ( z ) in D . Moreover, the origin isa zero of R ( z ) of order d . Hence, by induction, the origin is a zero of R n ( z ) of order d n , for everyinteger n ≥
1. Then, for every integer n ≥ R n ( z ) z d n = 0 , ∀ n ∈ N , ∀ z ∈ D. (18)Hence, for every integer n ≥
1, there exists a d n -th root of (18) on D , that is an analytic function g n ( z ) over D such that g n ( z ) d n = R n ( z ) z d n , ∀ z ∈ D. h n ( z ) = zg n ( z ), which is analytic over D , we get: h n ( z ) d n = R n ( z ) , ∀ n ≥ , ∀ z ∈ D. We can then write: h n ( z ) = [ R n ( z )] /d n .Now, we remark that: h n +1 ( z ) = z n Y i =0 (cid:2) g (cid:0) R i ( z ) (cid:1)(cid:3) /d i , (19)Some calculation (see the details in [25, page 188]) prove that the product (19) converges uni-formly on D . We deduce that h n ( z ) converges uniformly on D to an analytic function F ( z ) over D . Finally, we have: [ R n ( R ( z ))] /d n = (cid:2) R n +1 ( z ) (cid:3) /d n = h(cid:0) R n +1 ( z ) (cid:1) /d n +1 i d . If we take the limit when n tends to infinity, we obtain that F is solution of ( B ), which concludesthe proof.Let us return to the interest of E. Schröder for the analytic iteration problem. As said before,the author seems not to relate this question to solutions of the Schröder’s or Abel’s equation in themanner of (10) and (11). But the author has another connection in mind. He links the existenceof a solution to the analytic iteration problem (9) to the one of a continuous curve which containsthe iterates of φ ( z ). This consideration of invariant structures of the complex plane with respectto the iteration is at the heart of the theory developed by P. Fatou and G. Julia. However, thestudy of E. Schröder and G. Koenigs are limited to a neighbourhood of a fixed point. One of themain innovations of the work of P. Fatou and G. Julia is their idea and tools to investigate thewhole complex plane, and in fact the compactification ˆ C = C ∪ {∞} of C , dividing it into zonesdepending on the behaviour of the sequence of iterates of a rational fraction. Let us note that ˆ C makes the study of rational fractions a central topic, as they are the only analytic functions overˆ C . In order to present the main discoveries of P. Fatou and G. Julia, let us first introduce somenotations and definitions. As before, and for now on, we let R ( z ) denote a rational fraction ofdegree at least two, and we let R n ( z ) denote the n -th iterate of R ( z ).Quite natural questions arise when considering a point z close to a fixed point α and thesequence of iterates z n = R n ( z ). From a local point of view, we can wonder if there exists aneighbourhood of α in which ( z n ) converges. From a global point of view, we can study thebehaviour of this sequence outside such a neighbourhood, and on its boundary. The work of P.Fatou and G. Julia is about the second point. We may see this as the study of the impact of thepoint z and its neighbourhood over the convergence of the sequence ( z n ). To translate this impact,P. Fatou [25, 26] involves the theory of normal families developed by P. Montel [52]. We say thata family F of analytic functions defined on a domain D of ˆ C is normal over D if from everyinfinite subsequence of F , we can extract a sub-sequence of F which converges uniformly locallyon D (that is in every compact set of D ). The link with our subject is given by the applicationof the Arzela-Ascoli theorem to F = { R n } n . Indeed, this states the equivalence for a family F ofcontinuous functions defined on a domain D of ˆ C to be normal over D or equicontinuous over D .But, by definition, { R n } n is equicontinuous over D if for every z ∈ D and every (cid:15) >
0, there8xists δ > n ∈ N and every z ∈ D : | z − z | < δ = ⇒ | R n ( z ) − R n ( z ) | < (cid:15). Thus, the notion of equicontinuity exactly transcribes the fact that the behaviour of the sequence( z n ) depends on z . In order to understand the boundary of this property, P. Fatou defines andstudies the set of all the points of ˆ C for which the family { R n } n is not normal (that is there exists noneighbourhood of these points in which the family is normal). He denotes F this set, for Frontière (french word for boundary ). This set is nowadays written as J ( R ) (or J ), for the Julia set associatedto R , and it is its complement ˆ C \ J ( R ) that is denoted as F ( R ) (or F ), this time because of theinitial letter of Fatou, and called the Fatou set associated with R . At the same time, G. Julia definesthe set E consisting of all the repelling fixed points of all the iterates R n , n ∈ N , and studies thederived set E composed by all the accumulation points of E , which coincides with J (see Theorem3.9 below). The study of the set J can provide information on the solutions of functional equations.Let us illustrate this fact with the analytic extension of a solution of the Schröder’s equation ( S )in a neighbourhood of an attracting fixed point α , following a method introduced by P. Fatou [27].Recall that s = R ( α ). According to G. Koenigs, there exists a neighbourhood D α of α and ananalytic function F on D α such that for all z ∈ D α : F ( R ( z )) = sF ( z ) . Even if this means reducing D α , we assume that R ( D α ) ⊂ D α . This comes from the Taylorformula and the fact that α is attracting. Let us consider ˜ D = R − ( D α ). The fact that R ( D α ) ⊂ D α implies that D α ⊂ ˜ D . Then, let us define for all ˜ z ∈ ˜ D :˜ F (˜ z ) = 1 s F ( R (˜ z )) . We see that ˜ F ( z ) = F ( z ) , ∀ z ∈ D α . Moreover, as R (˜ z ) ∈ D α , for all z ∈ ˜ D , we have:˜ F ( R (˜ z )) = F ( R (˜ z )) , ∀ z ∈ ˜ D = s ˜ F (˜ z ) , ∀ z ∈ ˜ D. Hence, ˜ F extends analytically F on ˜ D and remains a solution of the Schröder’s equation over ˜ D .Iterating the process, we can extend F to an analytic function G over the domain of attraction D of α (that is the set of all the elements z such that there exists an integer N such that R N ( z ) ∈ D α ),such that G remains a solution of the Schröder’s equation over D . Thus, the understanding ofproperties of R ( z ), namely the nature of its fixed points, can provide information about a solution F ( z ) of the Schröder’s equation ( S ).However, Equation ( S ), the other form of the Schröder’s equation, also provides a connectionwith the theory of P. Fatou and G. Julia, maybe even better in some cases. Indeed, when α is arepelling fixed point of R , H. Poincaré [59] proves that the solution f of the Schröder’s equation ( S )(also called Poincaré’s equation in this form) can be extended as a meromorphic function over C .Hence, this allows a global study of the structure of the Julia set J . As explained in [6, Theorem6.3.2], a computation of the coefficients of the Taylor series expansion of a formal solution f ( z ) of9 S ) implies that this series has a positive radius of convergence. Hence, there exists r > f ( z ) is analytic over the disc D ( α, r ) centred at α and of radius r >
0. Then, one can extend f ( z )by induction as follows. The analyticity of f ( z ) over D ( α, r ) implies that R ( f ( z )) is meromorphicover D ( α, r ). Hence, by Equation ( S ), so is f ( sz ). Therefore, f ( z ) is meromorphic over D ( α, sr ).By induction, we find that f ( z ) is meromorphic over D ( α, s n r ), for every n ∈ N . As | s | >
1, weobtain that f ( z ) is meromorphic over C .The Schröder’s equation is also studied from an algebraic point of view. Indeed, J. F. Rittestablishes in [63] the list of all the differentially algebraic solutions of this equation, when | s | > R ( z ). This is Theorem 3.1 of Section 3.2. During the 30’s, the interest for functional equations ( S ), ( B ) and ( A ) and the theory of iterationis less vigorous. Nonetheless, let us note the work of C. L. Siegel [72] on the existence of a solutionto Equation ( S ) for some indifferent fixed points, and the one of H. Brolin [15] on the structure ofthe Julia set. The enthusiasm for this subject rises again sixty years later, during the 80’s. Thisis mainly due to the connections the theory of P. Fatou and G. Julia shares with the active areaof dynamical systems and fractals, and the possibility to make computational experiences. Indeed,as said before, the iteration of a rational function R ( z ) gives rise to a dynamical system, whichdivides ˆ C into different areas, depending on the concordance or disparity of the local behaviourof the sequence { R n ( z ) } n around a point z = z . The concordance is formalized by the notionof equicontinuous and normal families. The set of points with this local concordance is the Fatouset F ( R ) and its complement in ˆ C is the Julia set J ( R ). Let us assume that R ( z ) is of degreeat least two. Note that F ( R ) is open in ˆ C and J ( R ) is a closed compact subset of ˆ C . Let usmention that J ( R ) is always non-empty [25] and perfect, that is, closed and without any isolatedpoint [26, 40]. Moreover, Julia sets provide lots of examples of fractals. These objects are definedby B. Mandelbrot in the 90’s [51]. We refer to [21, 18, 58] for more details about what follows.The story of fractals actually goes back to the works of B. Riemann and K. Weierstrass and theirdiscoveries of continuous functions with no derivative, at any point. This created a lot of confusionin the mathematical community which had trouble to apprehend such strange objects (see moredetails in [2, page 88])! This discomfort was even increased with the work of G. Cantor and hisperfect, totally disconnected (that is with all connected components reduced to a point) sets, whichquestioned the notion of dimension of his time. To deal with the complexity of such objects, theclassical topological dimension is replaced by the Hausdorff dimension, which may be a non integralnumber. Intuitively (see for example [58]), this notion measures the growth of the number of setsof diameter (cid:15) needed to cover the concerned set, when (cid:15) tends to zero. The Hausdorff dimension isalways greater than or equal to the topological one. B. Mandelbrot defines the fractals as sets forwhich the Hausdorff dimension is strictly greater than the topological one. For example, the triadicCantor set has topological dimension 0 and Hausdorff dimension log(2) / log(3) : it is a fractal. Assaid before, lots of Julia sets provide examples of fractals. Julia sets still fuel the current research,with different points of view, namely, investigations on their Hausdorff dimension [45, 74], theirLebesgue measure [16], or their computational complexity [24]. Apart from the structure of theJulia set J ( R ), there is the question of its variation when the coefficients of R ( z ) depend on aparameter. This question is raised by P. Fatou in [26]. In the case of polynomials of degree two, ofthe form R c ( z ) = z + c , where c ∈ C , there exists a classification of the Julia sets J ( R c ). Note thateach polynomial of degree two is conjugated to a polynomial of such a form. This classification is10ncoded by the Mandelbrot set, defined as the set of the complex numbers c ∈ C for which J ( R c )is connected. Indeed, the dynamic of R c ( z ) changes as c moves from a cardioid to a disc of theMandelbrot set. For example (see [6, paragraph 1.6]), let us focus on the cardioid C , and the disc D , where C = u ( D (0 , / , u ( z ) = z − z , and D = D ( − , / C and the disc D are part of the Mandelbrot set (see the figure below). When c ∈ C , the rational fraction R c admitsa unique attracting fixed point α and a pair ( u, v ) such that R c ( u ) = v, R c ( v ) = u , and u, v arerepelling fixed points of R c . Then, when c enters inside D , the point α becomes a repelling fixedpoint of R c and the points u, v become attracting fixed point of R c . As an illustration, the figurebelow represents the Mandelbrot set and the form of the Julia sets J ( R c ) attached to R c for someof the points c inside and outside the Mandelbrot set. Note that the Mandelbrot set appears to beconnected itself [22].Figure 1: The Mandelbrot set (in grey) and Julia sets J ( R c ) (at the end of the blue lines) for somepoints c ∈ C inside and outside the Mandelbrot set. This image is taken from [19].Besides, the Mandelbrot set M gives rise to the following interesting transcendental result [57].A certain conformal map Φ( z ), constructed by A. Douady and J. H. Hubbard [22], defined on thecomplement of M , admits transcendental values Φ( α ) at each algebraic number α of the complement11f M . A link with the Böttcher’s equation is that Φ( c ) = f c ( c ), for every c in the complement of M , where f c satisfies Equation ( B ) for d = 2 and R ( z ) = R c ( z ). In the same area, the algebraicaspects of the solutions of Equations ( S ) , ( B ) and ( A ) are also investigated by many authors, alongwith other kinds of functional equations. The first main result of this type is due to 0. Hölder in1887. Indeed, the author proves that the Euler’s Gamma function defined for every z ∈ C suchthat Re( z ) >
0, by: Γ( z ) = Z + ∞ t z e − t dtt , is hypertranscendental over C ( z ). As said in [66], the proof of O. Hölder is based on the followingfunctional equation: Γ( z + 1) = z Γ( z ) . This is a non-autonomous version of the Abel’s difference equation ( ˜ A ), of the form: G ( z + 1) = R ( z, G ( z )) , where R ( z, X ) is this time a complex rational fraction of two variables. In [67, Problem 69], L.A. Rubel proposes the study of such a generalised functional equation for the Schröder’s equa-tion. This gives rise to further studies. For example, K. Ishizaki [39] considers the case where R ( z, X ) = a ( z ) X + b ( z ), with a ( z ) , b ( z ) rational fractions. The author proves that every transcen-dental meromorphic solution of the associated generalised Schöder’s equation: G ( sz ) = a ( z ) G ( z ) + b ( z ) , with | s | / ∈ { , } , is hypertranscendental.Concerning the generalised equation of ( B ), that is: G ( z d ) = R ( z, G ( z )) , (20)this is called a d -Mahler equation and was introduced by K. Mahler in 1929 in [47, 48, 50]. Asolution G of (20) is called a d -Mahler function. Ke. Nishioka [56] proves that a Mahler functionis transcendental if and only if it is not rational (see also [57]). The same author provides in [55] asufficient condition for the hypertranscendence of Mahler functions of order one. Before this time,K. Mahler proved that the d -Mahler function P + ∞ n =0 z d n is hypertranscendental [49]. This result isgeneralised in [46] with the study by J. H. Loxton and A. J. Van der Poorten of inhomogeneouslinear Mahler systems of order one. Such functional results are motivated by a theorem of K. Mahler[47, 48, 50] which establishes, under some assumptions, the equivalence between the transcendenceof a Mahler function f ( z ) and the one of its value f ( α ) at a non-zero algebraic number α . Thus,results on functional algebraic independence turns into results on algebraic independence of values.More results are obtained using an adapted Galois theory. For short, this theory (see [60] for lineardifference equations) associates to a system of linear functional equations an algebraic group, calledthe Galois group , which encodes the algebraic relations between the solutions of the system. Ifthe Galois group is big enough , the solutions are algebraically independent. This approach allowsK. Nguyen [54] to recover the hypertranscendental result of Ke. Nishioka mentioned above. Toillustrate further use of Galois theory for linear Mahler functions, let us mention the result in [23]which provides sufficient conditions for a Mahler function to be hypertranscendental, and the workof J. Roques [64]. Note that these kinds of functional results are still open questions for linear Mahler12quations over a function field of positive characteristic (see for example [30, 31, 29]). Similarly,the Siegel-Shidlovskii theorem [71], which is a kind of analogue of the Mahler’s theorem for certainsolutions of linear differential equations over C ( z ) called E -functions, motivates the study of thealgebraic independence of solutions of such equations. As for Mahler functions, there is a dichotomybetween rational and transcendental E -functions. Moreover, the Hrushovski-Feng algorithm [38, 28]computes the Galois group of systems of linear differential equations. More generally, the study ofhypertranscendence or algebraic independence of solutions of different types of functional equationsis currently a very dynamic area (see for example [20, 36]). This frequently involves the developmentand use of Galois theories adapted to the different settings. As an illustration, let us mention [37]for linear difference equations, and the work of C. Hardouin for q -difference systems [34], whichgeneralises the work of K. Ishizaki mentioned above, and the development of a general tannakianGalois theory, which in particular apply for τ -difference systems in positive characteristic [35],developed by G. Anderson, W. D. Brownawell and M. Papanikolas [3].Let us go back to Equations ( S ) , ( B ) and ( A ). A fruitful link between algebraic properties ofsolutions of these equations and the iteration theory is given in 1986 by a result of M. Boshernitzanand L. A. Rubel in [11]. This states the equivalence for a solution of Equation ( S ) , ( B ) or ( A ) tobe differentially algebraic and the family { R n ( z ) } n to be coherent . The latter means that all therational fractions R n ( z ) satisfy a same algebraic differential equation (1). Note that L. A. Rubelasked in [67, Problem 27] if there exists a transcendental entire function whose iterates form acoherent family. This is answered negatively by W. Bergweiler in [9].The theorem of M. Boshernitzan and L. A. Rubel, is one of the ingredients of the proof of theresult of P.-G. Becker and W. Bergweiler [8] we are interested in. This statement, reproduced hereas Theorem 3.1, explicitly lists all the differentially algebraic solutions of Equations ( S ) , ( B ) and( A ). Let us note that partial results have already been found. We mentioned earlier the resultof J. F. Ritt [63]. But we can also indicate the work of F. W. Carroll [17]. The author considersthe case where R ( z ) is a finite Blaschke product with an attracting fixed point (see for examplethe survey [32] for the definition and more information about such products). Then, he guaranteesthat a solution of the Schröder’s equation ( S ) is hypertranscendental. Furthermore, P. Borweinexamines the case where d = 2, and R ( z ) = z + c , with c >
0, in the Böttcher’s equation ( B ).Then, the author states that every solution of this equation is hypertranscendental. Moreover, P.Borwein points out that his proof, as well as the one of J. F. Ritt in [63], shares analogies withthe proof of the hypertranscendence of the Gamma function by O. Hölder. Finally, P.-G. Beckerand W. Bergweiler themselves previously obtained in [7] the list of all the algebraic solutions of theBöttcher’s equation ( B ) when R ( z ) is conjugated to a polynomial. In fact, this list concerns thesolutions of the more general following equation: f ( p ( z )) = q ( f ( z )) , (21)where p ( z ) , q ( z ) are polynomials of the same degree d ≥
2, and the attracting fixed point is ∞ .In the same paper [7], the authors conjecture that the transcendental solutions of Equation(21) are in fact hypertranscendental. For the Böttcher’s equation ( B ), Theorem 3.1, their furtherresult, implies that this conjecture is satisfied. To conclude, let us mention some recent works. K.D. Nguyen [53] studies systems of n Böttcher’s equations ( B ) for polynomials R ( z ) , . . . , R n ( z ).The author proves a result that links the algebraic independence of some transformations of thesolutions f R i associated to each Böttcher’s equation to the conjugacy of some iterates of somepolynomials among R ( z ) , . . . , R n ( z ). Finally, M. Aschenbrenner and W. Bergweiler prove in [4]13he hypertranscendence over C ( z ) of the iterative logarithm itlog( R ) of a non-linear rational orentire function R with a rationally indifferent fixed point. The function itlog( R ) is the uniqueformal power series solution f of the equation: f ( R ( z )) = R ( z ) f ( z ) , (22)The proof of the authors is similar to the one of Theorem 3.1 we present later. Note that theauthors also prove that in the case where R ( z ) is a non-linear entire function, the function itlog( R )is even hypertranscendental over the ring of entire functions. Note that Equation (22) is useful tostudy the iteration of R ( z ) not only in the petals of R ( z ), given by the Leau-Fatou Flower theoremmentioned above, but in a neighbourhood of the concerned fixed point. Before presenting the statement of P.-G. Becker and W. Bergweiler we are interesting in, let usprecisely define its setting. Let R ( z ) denote a rational fraction with coefficients in C and of degreeat least two. Even if this means replacing R ( z ) by a conjugate with the appropriate Möbiustransformation, we may assume that 0 is a fixed point of R ( z ). Let us write s = R (0). Theframework considered is the following (see [25, Chapitre II]):1. If 0 is an attracting, repelling or irrationally indifferent fixed point of R ( z ), we consider theSchröder’s equation ( S ). The Schröder’s equation admits a unique solution of the form f ( z ) = + ∞ X n =1 a n z n , with a = 1 . (23)If f converges in a neighbourhood of 0, we say that f is a Schröder function. In the case ofan attracting or repelling fixed point, the solution is always convergent, as seen before. Butin the case of an irrationally indifferent fixed point, there always exists a formal solution, butnot necessarily convergent [72]. The question of the convergence of this formal solution is adynamic area of research [75, 33].2. If 0 is a super-attracting fixed point of R ( z ), there exists an integer d ≥ R ( z ) = + ∞ X n = d b n z n , b d = 0 . (24)Then, we consider the Böttcher’s equation ( B ). For every a ∈ C such that a − d = b d , theBöttcher’s equation admits a unique solution of the form f ( z ) = + ∞ X n =1 a n z n . (25)The function f converges in a neighbourhood of 0, and we say that f is a Böttcher function.14. If 0 is a rationally indifferent fixed point of R ( z ), even if this means replacing R ( z ) by aniterate R k ( z ), we may assume that s = 1. Indeed, if s k = 1, we have ( R k ) (0) = s k = 1. Letus write: R ( z ) = z + + ∞ X n = d b n z n , where d ≥ , b d = 0 . (26)Then, we consider the Abel’s equation ( A ). In each petal given by the Leau-Fatou Flowertheorem mentioned earlier, the Abel’s equation admits an analytic solution f ( z ). We say that f is an Abel function.Let us introduce some vocabulary. Let us remind that we say that two analytic functions S and S are conjugated to each other via an invertible analytic function g if S = g − S g , wherefunctional composition is denoted multiplicatively. If we say that S and S are conjugated , withno precision about g , we mean that g is a Möbius transformation (7).We are now able to present the result of P.-G. Becker and W. Bergweiler, as it is written in [8].We give several explanations and comments about Theorem 3.1 directly after its statement. Theorem 3.1 (P.-G. Becker and W. Bergweiler )
Let f ( z ) be a Schröder, Böttcher or Abel function. Let us assume that f is differentially algebraic.Then, we have the following.1. If f is a Schröder function, then is a repelling fixed point of R ( z ) and f is a Möbiustransformation of a function of one of the following forms:(a) exp( αz r ) . In this case, R ( z ) is conjugated to z d or z − d .(b) cos( αz r + β ) . In this case, R ( z ) is conjugated to T d or − T d , where T d is the d -thTchebychev polynomial.(c) ℘ ( αz r + β ) , ℘ ( αz r + β ) , ℘ ( αz r + β ) , ℘ ( αz r + β ) , where ℘ denotes the Weierstrassfunction,where the constant r is a rational number such that the concerned functions are meromorphicover C , α is a non-zero complex number and β is a fraction of a period of the concernedfunction.2. If f is a Böttcher function, then, f is a Möbius transformation or is a Möbius transformationof a function of one of the following forms:(a) ρz , where ρ d − = 1 . In this case, R ( z ) is conjugated to z d .(b) ρz + ρz , where ρ d − = 1 . In this case, R ( z ) is conjugated to T d .(c) ρz + ρz , where ρ d − = − . In this case, R ( z ) is conjugated to − T d .3. The function f is not a Abel function.In particular, Abel functions are always hypertranscendental. S ) extends to a meromorphic function on thecomplex plane, as seen at the end of Section 2.1. Remark that, even if the function z → z r is notmeromorphic on the complex plane when r is not an integer, this case may happen. There is anexample below, with the meromorphic function cos( √ z ) − r = 1 / R ( z ) involved in this statement is not necessarily 0. We clarify this point below.1. If f is a Schröder function(a) The rational fractions R ( z ) = z d or R ( z ) = z − d are considered at the repelling fixedpoint z = 1. To move it at the origin, we have to conjugate R ( z ) with L ( z ) = z − R ( z ) = ( z + 1) d −
1, or ˜ R ( z ) = ( z + 1) − d −
1, and associated solutionsof equation S ˜ R, are of the form exp( αz r ) −
1, where α, r are as in the statement ofTheorem 3.1. In particular, ˜ f ( z ) = exp( z ) − f (0) = 0 and ˜ f (0) = 1.(b) The rational fractions R ( z ) = T d or R ( z ) = − T d are considered at the repelling fixedpoint z = 1. To move it at the origin, we have to conjugate R ( z ) with L ( z ) = z − R ( z ) = T d ( z + 1) −
1, or ˜ R ( z ) = T − d ( z + 1) −
1, and associated solutionsof equation S ˜ R, are of the form cos( αz r + β ) −
1, where α, β, r are as in the statementof Theorem 3.1. In particular, ˜ f ( z ) = cos( √ z ) − f (0) = 0 and ˜ f (0) = 1.2. If f is a Böttcher function,(a) The rational fraction R ( z ) = z d is considered at the super-attracting fixed point z = 0.(b) The rational fraction R ( z ) = T d ( z ) is considered at the super-attracting fixed point z = ∞ . To move it at the origin, we have to conjugate R ( z ) with L ( z ) = 1 /z . Thenwe have: ˜ R ( z ) = 1 / ( T d (1 /z )), and associated solutions of equation B ˜ R, are of the form1 / ( ρz + ρz ), where ρ d − = 1.(c) The rational fraction R ( z ) = T − d ( z ) is considered at the super-attracting fixed point z = ∞ . To move it at the origin, we have to conjugate R ( z ) with L ( z ) = 1 /z . Then wehave: ˜ R ( z ) = 1 / ( T − d (1 /z )), and associated solutions of equation B ˜ R, are of the form1 / ( ρz + ρz ), where ρ d − = − Weierstrass function ℘ is a meromorphic functionover C , attached to a lattice Λ ⊂ C , that is a discrete subgroup of C that contains an R -basis of C , and defined by: ℘ ( z ) := ℘ Λ ( z ) = 1 z + X w ∈ Λ ,w =0 (cid:18) z − w ) − w (cid:19) . The function ℘ Λ is periodic with respect to Λ, that is: ℘ Λ ( z + ω ) = ℘ Λ ( z ) , ∀ z ∈ C , ∀ ω ∈ Λ . (27)16esides, the function ℘ Λ satisfies the following algebraic differential equation: ℘ = 4 ℘ − g (Λ) ℘ Λ − g (Λ) , (28)where g (Λ) , g (Λ) ∈ C depend on Λ and satisfy g (Λ) − g (Λ) = 0. For more details, see forexample [73].Finally, Tchebytchev polynomials are defined by induction with: T ( X ) = 1 , T ( X ) = X, T n +2 ( X ) = 2 XT n +1 ( X ) − T n ( X ) , ∀ n ≥ . Let us note that for every integer n ≥ x ∈ [ − , T n (cos( x )) = cos( nx ). ( S ) and ( B ) As mentioned earlier, J. F. Ritt gives in 1926 the list of all the differentially algebraic Schröderfunctions when 0 is a repelling fixed point of R ( z ) [63]. These are exactly the functions listed inthe first point of Theorem 3.1. Thus, when the Schröder’s equation admits a differentially algebraicsolution, then, the considered fixed point of the associated rational fraction is always repelling. Theorem 3.2 (J. F. Ritt )
Let us assume that is a repelling fixed point of a rational fraction R ( z ) of degree at least two. Let f be a solution of the associated Schröder’s equation ( S ) . If f is differentially algebraic, then, f isin the list of the first point of Theorem 3.1. The proof of this theorem is based on the theories of differentiation and elimination. Thesetechniques allow the author to prove that a solution of Equation ( S ) is (after an appropriate changeof variables) a solution of a Schwarzian differential equation of the following form: g (3) g − / g (2) ) = A ( g ) g (4) , (29)where A is a rational fraction.The author then uses a previous classification of his own [62] for differentially algebraic solutionsof (29) to deduce that they are of the desired forms. As noted earlier, P. Borwein in [10] points outthat the techniques of this proof share analogies with the proof of the hypertranscendence of theGamma function by O. Hölder. One year before proving Theorem 3.1, P.-G. Becker and W. Bergweiler [7] provided the list of all thealgebraic Böttcher functions, when R ( z ) is conjugated to a polynomial. It is exactly the functionslisted in the second point of Theorem 3.1. Thus, the Böttcher functions are transcendental if andonly if they are hypertranscendental. This was a part of a conjecture of the two authors in [7]. Letus state this previous result in the case of Böttcher functions. As said before, this applies to themore general equations (21). 17 heorem 3.3 (P.-G. Becker and W. Bergweiler ) Let R ( z ) be a polynomial of degree at least two and let f be a solution of the associated Böttcher’sequation ( B ) . Then, f is algebraic if and only if f is in the list of the second point of Theorem 3.1.In particular, all the algebraic Böttcher functions are rational. The proof of Theorem 3.3 is based on the analysis of the finite singularities (algebraic branchpoints) of the Böttcher function. Indeed, the authors prove that if f is an algebraic solution of( B ), different from a Möbius transformation, then its local inverse f − has exactly two such finitesingularities. This result, and [6, Theorem 4.1.2] about exceptional points of a rational fraction (see[6, Definition 4.1.1]), allow them to find the corresponding forms of R ( z ). Then, the first remarksof the paper [7] provide the Böttcher functions when R ( z ) ∈ { z d , T d , − T d } . Notice that a Möbiustransformation of an algebraic function remains algebraic. Remind that a formal power series f ( z ) is differentially algebraic if there exists a non-zero poly-nomial P ( z, X , . . . , X n ) such that f satisfies (1). Examples of such functions are given by poly-nomials, algebraic functions, Bessel functions and classical functions as the exponential, logarithm,cosinus or sinus for example. As said before, the first example of hypertranscendental function doesnot appear before 1887, with the Euler’s Gamma function given by O. Hölder. But most entirefunctions, or analytic functions over a domain of C , are hypertranscendental [66, Theorem 5].In [67, Problem 26”], L. A. Rubel asks the following question: are there any boundary on thegrowth of entire differentially algebraic functions ? More precisely, given an entire differentiallyalgebraic function satisfying an n -order equation (1), do there exist constants A, α such that: | f ( z ) | ≤ A exp n ( | z | α )? (30)L. A. Rubel indicates that a strategy to answer the question negatively should be to construct afunction f big enough (compared to the exponential) such that all its iterates f k ( z ) satisfy the samealgebraic differential equation. This is the notion of coherent family studied by M. Boshernitzanand L. A. Rubel in [11]. Indeed, recall that a family of functions is said to be coherent if allits elements satisfy the same algebraic differential equation (1). As an example, let us considerthe family { cz n , c ∈ C , n ∈ N } . This family is coherent because each of its elements satisfies thefollowing algebraic differential equation: zf (2) ( z ) f ( z ) + f ( z ) f ( z ) − zf ( z ) = 0 . (31)However, the family of all polynomials with rational coefficients is proved not to be coherentin [11]. Before stating the main result of M. Boshernitzan and L. A. Rubel, let us mention twoimportant results concerning coherent families and sketch their proof. Theorem 3.4
Let f be an analytic differentially algebraic function. Then, f satisfies an autonomous algebraicdifferential equation. This means that there exist a non-zero polynomial Q ( X , . . . , X n ) , with coef-ficients in C , independent of z , such that: Q ( f ( z ) , f ( z ) , . . . , f ( n ) ( z )) = 0 . (32)18 roof. We can find a proof of this well-known result in [11], and a more detailed reasoning in [65].Let us gather the explanations here. Let P ( z, X , . . . , X n ) be a non-zero polynomial such that f ( z ) satisfies Equation (1). Without any loss of generality, we can assume that P is irreduciblein C [ z, X , . . . , X n ]. The goal is then to remove the variable z from the equation. The notion ofthe resultant of two polynomials will solve the problem. Let us define the following operator onpolynomials S ( z, X , . . . , X n ) ∈ C [ z, X , . . . , X n ], over C [ z, X , . . . , X n , X n +1 ]: D : S ( z, X , . . . , X n ) DS ( z, X , . . . , X n +1 ) = dSdz ( z, X , . . . , X n )+ n X k =0 dSdX k ( z, X , . . . , X n ) X k +1 . The advantage of this definition is that for all S ∈ C [ z, X , . . . , X n ] we have: DS ( z, f ( z ) , . . . , f ( n +1) ( z )) = ddz [ S ( z, f ( z ) , . . . , f ( n ) ( z ))]Hence, DP ( z, f ( z ) , . . . , f ( n +1) ( z )) = 0. Hence, if R is the resultant of the polynomials P and DP with respect to the variable z , properties of the resultant guarantee that R ∈ C [ X , . . . , X n +1 ]and that there exist A, B ∈ C [ z, X , . . . , X n +1 ] such that: R = AP + B ( DP ) . (33)Now, if we specialize (33) at ( z, f ( z ) , . . . , f ( n +1) ( z )), we obtain: R ( f ( z ) , . . . , f ( n +1) ( z )) = 0 . (34)This provides an autonomous algebraic differential equation for f if we prove that R is a non-zero polynomial. To do so, let us assume by contradiction that R = 0. Then, P and DP admita non-constant common factor. As P is irreducible, P | DP in C [ z, X , . . . , X n +1 ]. Let T ∈ C [ z, X , . . . , X n +1 ] such that:( DP )( z, X , . . . , X n +1 ) = P ( z, X , . . . , X n ) T ( z, X , . . . , X n +1 ) . Then, for all U ( z ) ∈ C [ z ], we have:( DP )( z, U ( z ) , . . . , U ( n +1) ( z )) = P ( z, U ( z ) , . . . , U ( n ) ( z )) T ( z, U ( z ) , . . . , U ( n +1) ( z )) . (35)Let U ( z ) ∈ C [ z ]. Let us write Q ( z, U ( z ) , . . . , U ( n +1) ( z )) = ˜ Q ( z ), for all polynomial Q ∈ C [ z, X , . . . , X n +1 ].Then, by (35) we have: ˜ P ( z ) = ˜( DP )( z ) = ˜ P ( z ) ˜ T ( z ) . (36)This provides: ˜ P ( z ) ∈ C . (37)Now, an argument from linear algebra allows us to conclude that P ∈ C , which is a contradiction.More precisely, if x , x ∈ C , there is a surjective morphism of C -vector spaces: φ x ,x : C n +2 [ z ] −→ C n +2 U ( z ) U ( x ): U ( n ) ( x ) , U ( x ): U ( n ) ( x ) , (38)19here C n +2 [ z ] denotes the C -vector space of complex polynomials of degree less that or equal to2 n + 2. Then, for all x ∈ C and for all ( z , . . . , z n ) ∈ C n +1 , there exists U ( z ) ∈ C n +2 [ z ] suchthat ( U ( x ) , . . . , U ( n ) ( x )) = ( z , . . . , z n ) and ( U (0) , . . . , U ( n ) (0)) = (0 , . . . , x , U ( x ) , . . . , U ( n ) ( x )) and at (0 , U (0) , . . . , U ( n ) (0)) implies that P ( x , z , . . . , z n ) = P (0 , . . . , , ∀ ( x , z , . . . , z n ) ∈ C n +2 . This implies that P = P (0 , . . . , ∈ C and yields a contradiction. Theorem 3.4 is proved.Another useful result about coherent families is that they are stable with respect to manyoperations. This is the following statement, proved by M. Boshernitzan and L. A. Rubel in [11]. Theorem 3.5 (M. Boshernitzan and L. A. Rubel)
Let f, g be two analytic differentially algebraic functions. Let
P, Q be non-zero differential algebraicpolynomials providing an autonomous differential algebraic relation for f and g respectively. Let usconsider the functions : f + g, f − g, f × g, f /g, f g, f G, f g , (39) where G is a primitive of g , and the composition of applications is denoted multiplicatively.Then, for every function h in this list, there exists a complex autonomous polynomial T ( X , . . . , X n ) , which depends only on P and Q ( and not on f nor g ) such that: T ( h ( z ) , h ( z ) , . . . , h ( n ) ( z )) = 0 . (40) In particular , coherent families are stable under the operations in (39) . In other words, if F and G are two coherent families, the family { f ∗ g | f ∈ F , g ∈ G } is a coherent family, where ∗ is a fixed operation of the set { + , − , × , ÷} , or the operation of composition. And the two families { f G | f ∈ F , G ∈ G } , { f g | f ∈ F , g ∈ G } are coherent families.Proof. In order to explain the reasoning, we only sketch the proof for h = f + g (it is similar in theother cases) and the case where P, Q are of order 1 (to reduce the notations). In other words, wehave the equations: P ( f ( z ) , f ( z )) = 0 and Q ( g ( z ) , g ( z )) = 0, with P, Q ∈ C [ X , X ] autonomous.As P is autonomous, if we derive the first equation with respect to the variable z , we obtain: f ( z ) S ( P )( f ) + ˜ P ( f ( z ) , f ( z )) = 0 , (41)where ˜ P ∈ C [ X , X ], and S ( P ) is the separant of the polynomial P . For a polynomial U ( X , . . . , X n )in C [ X , . . . , X n ], S ( U ) is defined by: S ( U )( X , . . . , X n ) = dUdX n ( X , . . . , X n ) , where the derivation is made with respect to the biggest variable appearing in U . For every analyticfunction φ , we let S ( U )( φ ) = S ( U )( φ ( z ) , φ ( z ) , . . . , φ ( n ) ( z )). Similarly, we have: g ( z ) S ( Q )( g ) + ˜ Q ( g ( z ) , g ( z )) = 0 , (42)where ˜ Q ∈ C [ X , X ].Let us first assume that S ( P )( f ) = 0 and S ( Q )( g ) = 0. Then, by (41) and (42), there exists anon-zero rational fraction R ∈ C ( Z , . . . , Z ) such that: h (2) = R ( f, f , g, g ) . R only depends on P and Q . But, g = h − f and g = h − f . Thus, there exists anon-zero rational fraction H ∈ C [ X , . . . , X ]. Such that: h (2) ( z ) = H ( f ( z ) , f ( z ) , h ( z ) , h ( z )) . Note that H only depends on P and Q . We deduce the existence of non-zero rational fractions H , H ∈ C ( X , . . . , X ) such that: h (3) ( z ) = H ( f ( z ) , f ( z ) , h ( z ) , h ( z )) .h (4) ( z ) = H ( f ( z ) , f ( z ) , h ( z ) , h ( z )) . Note that H , H only depend on P and Q .Now, let us consider for every i ∈ { , , } , H i ( X , . . . , X ) as H i ( X , . . . , X ) ∈ K := [ C ( X , X )]( X , X ) . The transcendence degree of K over L := C ( X , X ) is equal to 2. Hence, there exists a non-zero monic polynomial S ( X , X )( Y , Y , Y ) ∈ L [ Y , Y , Y ], where the Y i ’s are formal variables for i ∈ { , , } , such that: S ( X , X )( H ( X , . . . , X ) , H ( X , . . . , X ) , H ( X , . . . , X )) = 0 . (43)Note that the coefficients of S are in L . Hence, S only depends on the H i ( X , . . . , X ), that is,only on P and Q .Now, if we let X = f ( z ) , X = f ( z ) , X = h ( z ) , X = h ( z ) in (43), we get: S ( h ( z ) , h ( z ))( h (2) ( z ) , h (3) ( z ) , h (4) ( z )) = 0 . Now, if we formally replace h ( j ) ( z ) by a variable X j , for every j ∈ { , . . . , } , we find a polynomial T ∈ C [ X , . . . , X ] such that T ( h ( z ) , h ( z ) , h (2) ( z ) , h (3) ( z ) , h (4) ( z )) = 0 . As, S ∈ L [ Y , Y , Y ] is monic with respect to the variables Y , Y , Y , T is a non-zero polynomial.Finally, as S only depend on P and Q , the same is true for T . Hence, T provides a non-zeroautonomous differential algebraic relation for h ( z ), which only depends on P and Q .Then, it remains to treat the case where S ( P )( f ) = 0, or S ( Q )( g ) = 0.First, let us notice that there exist integers k, l ≥ S k ( P ) and S l ( Q ) arenon-zero constants. Indeed, the separant of a polynomial strictly reduces its total degree. Hence,as polynomials, S k ( P ) , S l ( Q ) = 0. Thus, there exist k ≤ k − l ≤ l − S k ( P )( f ) = 0 and S k +1 ( P )( f ) = 0 (44) S l ( Q )( g ) = 0 and S l +1 ( Q )( g ) = 0 . Note that for all k ≤ k and l ≤ l , S k ( P ) , S l ( Q ) = 0, as polynomials. Besides, we notice that k, l only depend on P and Q , but k , l depend on f and g . The idea is then to apply the first part of21he proof to S k ( P ) , S l ( Q ), instead of P, Q respectively, but we need to get rid of the dependenceon f and g .To do so, let us consider all the integers k ≤ k − l ≤ l −
1. As S k ( P ) and S l ( Q ) are non-zero polynomials, we can consider two formal variables ˜ f k , ˜ g k and assume that they formally satisfy(44), for k , l . We deduce from the first part of the proof that there exists a differential algebraicrelation satisfied by the formal variable h k ,l = ˜ f k + ˜ g l , which only depends on S k ( P ) , S l ( Q ).Let us note T k ,l the associated non-zero autonomous differential algebraic polynomial, which onlydepends on P, Q and k , l . Moreover, for every functions u, v which satisfy (44) for some k , l , wehave: T k ,l ( u + v ) = 0 . Then, let us note: T = Y k ≤ k ; l ≤ l T k ,l . Then, T is a non-zero autonomous differential algebraic polynomial which only depends on P and Q . Moreover, T ( f + g ) = 0 and this concludes the proof.Let us notice that if f is an invertible analytic function which is differentially algebraic, then f − is also differentially algebraic. Indeed, for all n ∈ N , (cid:0) f − (cid:1) ( n ) ( z ) is a rational fraction of f ( w ) , f ( w ) , . . . , f ( n ) ( w ), where w = f − ( z ). But f is differentially algebraic. Therefore, the family { f ( n ) ( w ) } n has a finite transcendence degree over C ( z ), and so has the family n(cid:0) f − (cid:1) ( n ) ( z ) o n .We are finally able to state the main result of M. Boshernitzan and L.A. Rubel [11], which linkscoherent families to the algebraic properties of Schröder, Böttcher and Abel functions. Theorem 3.6 (M. Boshernitzan and L.A. Rubel)
Let f be a Schröder, Böttcher or Abel function. Let R ( z ) be the associated rational fraction of degreeat least two. Then, f is differentially algebraic if and only if the family { R n ( z ) } n ∈ N is coherent. The proof of Theorem 3.1 only uses the direct implication of Theorem 3.6, that is why we willonly reproduce this part of the proof here.
Proof of the direct implication of Theorem 3.6.
Let us assume that f is differentially algebraic. Thegoal is to prove that the family { R n ( z ) } n is coherent. First, let us assume that f is a Schröderfunction. Then, g = f − is differentially algebraic and satisfy Equation ( S ). Thus, we have: R = g − ( sz ) g. (45)Then, for all n ∈ N : R n = g − ( s n z ) g. We have seen that the family { s n z } n is coherent (see (31)). Hence, by Theorem 3.5, { R n } n iscoherent.If f is a Böttcher function, then, g = f − is differentially algebraic and satisfy equation ( B ).Similarly, we obtain R n = g − ( z d n ) g.
22e have seen that the family { z d n } n is coherent (see (31)). Hence, by Theorem 3.5, { R n } n iscoherent.Finally, if f is an Abel function, we find: R n = f − ( z + n ) f. (46)But the family { z + n } n is coherent because { n } n is (see (31)) and the coherence is stable underaddition by Theorem 3.5. Hence, again by Theorem 3.5, { R n } n is coherent. The ingredients from the theory of P. Fatou and G. Julia needed in the proof of Theorem 3.1 arestated as Theorems 3.8 and 3.9 below. Let R ( z ) be a rational fraction of degree at least two. Let F ( R ) denote the Fatou set of R . Recall that it is the open set of all the elements z ∈ ˆ C for whichthe family { R n ( z ) } n is normal in a neighbourhood of z . A fundamental result about normalfamilies is the following normality criterion from P. Montel [52]. Theorem 3.7 (P. Montel)
Let D be a domain in ˆ C . Let Ω = ˆ C \ { , , ∞} . Then, the family F = { f : D −→ Ω | f is analytic over D } is normal over D . Among other things, this allows the author to prove various theorems from É. Picard. One ofthem states that an analytic function on a punctured neighbourhood of the origin, which admitsan essential singularity at 0, takes all the values of ˆ C , except at most two.Now, let J ( R ) = ˆ C \ F ( R ) denote the Julia set of R . Remind that J ( R ) is a closed compactsubset of ˆ C , which is non empty and perfect. The Fatou set, however, can be empty, as we willsee in Theorem 3.8. Let us note that the Fatou and Julia sets are compatible with the notions ofiteration and conjugation via a Möbius transformation g . Indeed, we can prove that: F ( R n ) = F ( R ) , J ( R n ) = J ( R ) , ∀ n ∈ N . (47)Moreover, if g is a Möbius transformation and S = gRg − , we have: F ( S ) = g ( F ( R )) , J ( S ) = g ( J ( R )) . (48)Now, let us discuss the two results used in the proof of Theorem 3.1. The first one deals withthe cases where the Fatou set is empty or not. Theorem 3.8
Let R ( z ) be a rational fraction of degree at least two.1. Assume that R ( z ) admits a non repelling fixed point α . If α is irrationally indifferent, as-sume further that the associated Schröder equation ( S ) admits a convergent solution in aneighbourhood of α . Then F ( R ) = ∅ . . Assume that R ( z ) is the rational fraction of the Schröder’s equation associated with one ofthe functions ℘ ( αz r + β ) , ℘ ( αz r + β ) , ℘ ( αz r + β ) , or ℘ ( αz r + β ) which appears in the firstpoint of Theorem 3.1. Then F ( R ) = ∅ . The second result used by P.-G. Becker and W. Bergweiler to prove Theorem 3.1 is the following.
Theorem 3.9
Let R ( z ) be a rational fraction of degree d at least two. The set of all the repelling fixed points ofall the iterates R n ( z ) , n ∈ N , is dense in J ( R ) . Let us sketch the proof of Theorem 3.8 (see details in [6]).
Proof of Theorem 3.8.
First, we can prove that an attracting or super-attracting fixed point of R ( z ) belongs to F ( R ). Indeed, based on the Taylor development of R , there exists σ < D of the fixed point α such that: | R ( z ) − R ( α ) | ≤ σ | z − α | , ∀ z ∈ D if α is attracting; and even: | R ( z ) − R ( α ) | ≤ σ | z − α | , ∀ z ∈ D, if α is super-attracting. Secondly, if α is rationally indifferent, α ∈ J ( R ). This is proved by P. Fatou[26] and G. Julia [40] (see also [6, Theorem 6.5.1]). But the Leau-Fatou Flower theorem implies theexistence of domains L , . . . , L k called the petals of R such that, for all j ∈ { , . . . , k } , α belongs tothe boundary of L j , R ( L j ) ⊂ L j and the restriction to L j of R n converges uniformly to α on L j ,when n tends to infinity. This latter property shows that each petal is included in F ( R ). Hence, F ( R ) is not empty.Finally, if α is an irrationally indifferent fixed point of R ( z ), we can use a theorem stated in [6]which establishes the following equivalence, valid for an indifferent fixed point of R ( z ): R ( z ) is linearizable in a neighbourhood of α ⇔ α ∈ F ( R ) . (49)We say that R ( z ) is linearizable in a neighbourhood D of α if there exists an invertible analyticfunction g over D such that R is locally conjugated via g to a function of the form: h ( z ) = α + ( z − α ) h ( α ) . But this is precisely the case for R , which is conjugated, via the solution of the associated Schröder’sequation ( S ), to h ( z ) = sz . Then α ∈ F ( R ).Besides, for the second part of the theorem, we only sketch the proof for the Weierstrass function ℘ ( z ), following [6, p. 74]. Recall that this function is periodic , that is, satisfies (27). Moreover,this function satisfies the following Schröder’s equation: ℘ (2 z ) = R ( ℘ ( z )) , (50)for a certain rational fraction R ( z ). This is the duplication formula for elliptic curves (see forexample [73, page 54, page 170] and [42, Chapter 1]). Now, let D be a disc in C and let U = ℘ − ( D ).Let φ ( z ) = 2 z . Then, φ n ( U ) = 2 n U . Hence, for N big enough, 2 N U will contain a period24arallelogram of the lattice Λ associated with ℘ . This means that the values that ℘ will take in2 N U are exactly the one it takes on C . But it is known, as a consequence of the open mappingtheorem, that ℘ ( C ) = ˆ C . Hence, using (50), we obtain: R N ( D ) = R N ( ℘ ( U )) = ℘ (2 N U ) = ˆ C. This implies that { R n } n is not equicontinuous over D . Indeed, the local behaviour of the iteratesdoes not respect the proximity of antecedent points, because D is sent onto the whole Riemannsphere by R N . As D is arbitrary, we conclude that { R n } n is not equicontinuous over any opensubset of C . This implies that J ( R ) = ˆ C .Note that, as quickly mentioned in Section 2.2, the case of an irrationally indifferent fixed point α was proved by C. L. Siegel in 1942 for particular cases of α , called diophantine fixed points.Now, let us reproduce the proof of Theorem 3.9, as detailed in [6, Theorem 6.9.2]. First let usrecall the following facts. Let R ( z ) be a rational fraction of degree d ≥
2. Then, R ( z ) is a d -foldmap of ˆ C onto itself. That is, for all w ∈ ˆ C , the equation R ( z ) = w admits exactly d solutionsin ˆ C , when counting multiplicities. Now, we say that w is a critical value of R ( z ), if there exists z ∈ ˆ C such that R ( z ) = w and if there is no neighbourhood of z in which R ( z ) is injective. Butthere are only finitely many critical values of R ( z ) in ˆ C . Indeed, for an element x ∈ ˆ C , there existsa neighbourhood of x in which R ( z ) is injective if R ( z ) has neither a zero nor a pole at x . When w is not a critical value of R ( z ), there exists exactly d pairwise distinct elements z i ∈ ˆ C , i = 1 , . . . , d such that R ( z i ) = w , for every i ∈ { , . . . , d } . Proof of Theorem 3.9.
The first part of the proof consists in showing that J ( R ) is contained in thetopological closure of the set of all the fixed points of all the R n ( z ). Then, by [6, Theorem 9.6.1],the set of all the non repelling fixed points of all the R n is finite. We deduce that J ( R ) is containedin the topological closure of the set P of all the repelling fixed points of all the R n ( z ). Finally, as,for every integer n , each repelling fixed point of R n ( z ) is contained in the closed set J ( R n ) = J ( R ),we have that J ( R ) is the topological closure of P . This gives the result of Theorem 3.9.It thus remains to show that J ( R ) is contained in the topological closure of the set of all thefixed points of all the R n ( z ). It suffices to consider an open set N of ˆ C such that N ∩ J = ∅ , andprove that N contains a fixed point of one of the R n ( z ). Let w ∈ N ∩ J = ∅ . We may assumethat w is not a critical value of R . Indeed, the number of such values is finite and we may thusfind a non-critical value of R in a neighbourhood of w in N ∩ J . Then, R − ( w ) contains at leastfour distinct points w j , j = 1 , . . . ,
4. Indeed, the fact that d ≥ R ismore than or equal to four. At least three of these points, say w , w , w are distinct from w . Then,we construct three neighbourhoods N i of w i , i = 1 , ,
3, whose topological closures are pairwisedisjoint, and a neighbourhood N ⊂ N of w , disjoint from every N i and such that R : N i −→ N is a homeomorphism, with reciprocal S i , for every i ∈ { , . . . , } . But { R n ( z ) } n is not a normalfamily on N , because w ∈ N ∩ J ( R ). Then [6, Theorem 3.3.6] (which is a corollary of Theorem3.7) gives the existence of z ∈ N , n ≥ i ∈ { , . . . , } such that: R n ( z ) = S i ( z ) . We deduce that R n ( z ) = R ( S i ( z )) = z . Hence z is a fixed point of an iterate of R ( z ),contained in N .Note that P. Fatou [26] and G. Julia [40] proved the finiteness of the sets of attracting andrationally indifferent fixed points of all the R n ( z ).25 .5 The proof of Theorem 3.1 In this section, we reproduce the proof of Theorem 3.1 established in [8] by P.-G. Becker and W.Bergweiler . We give detailed explanations about the way that the three ingredients exposed earliermerge. This is a beautiful illustration of the power of the interactions between distinct domainsof mathematics. As mentioned before, the statement of Theorem 3.1 belongs to the theory ofhypertranscendence of solutions of functional equations. The proof of Theorem 3.1 uses previousresults of hypertranscendence and coherent families, along with the theory of iteration of a rationalfraction and analytic properties of the sets of Fatou and Julia.
Proof of Theorem 3.1.
We will use the notations of (8) and the framework of Section 3.1. Let R ( z )be a rational fraction of degree at least two, which admits 0 as a fixed point. According to thenature of this fixed point, we let f be either a Schröder solution of S R, (recall that we assumethat this equation admits a convergent solution), a Böttcher solution of B R, , or a Abel solution of A R, . Let us assume that f is differentially algebraic. Our goal is to prove that f cannot be a Abelfunction and that f is in the list 1 or 2 of the statement of Theorem 3.1, depending on whether f is a Schröder or Böttcher function.Let us immediately remark that Theorem 3.6 guarantees that the family { R n ( z ) } n is coherent.To begin with, if 0 is repelling (that is, f is a Schröder function), we can apply Theorem 3.2 andget that f is in the list of the first point of Theorem 3.1.Now, let us deal with the case where 0 is not repelling. The goal is to reduce this case tothe repelling one. Let us first notice that Theorem 3.8 implies that F ( R ) = ∅ . Moreover, byTheorem 3.9, the set of all the repelling fixed points of all the iterates R n ( z ) are dense in J ( R ).As J ( R ) is always a non-empty set [25], we deduce that there exist w ∈ ˆ C and k ∈ N such that w is a repelling fixed point of R k ( z ). We may then consider a Schröder solution φ of S R k ,w . But { (cid:0) R k (cid:1) n } n is coherent, as a subfamily of the coherent family { R n ( z ) } n . Hence, by Theorem 3.6, φ is differentially algebraic.Then, Theorem 3.2, implies that there exists a Möbius transformation g ( z ) such that g − R k g is one of the rational fractions which appear in the first point of Theorem 3.1 (with d k instead of d ). Let us note S ( z ) = g − R k g . By (47) and (48), the fact that F ( R ) = ∅ implies that F ( S ) = ∅ .Hence, by Theorem 3.8, S ( z ) ∈ { z d k , z − d k , T d k , − T d k } . Besides, as 0 is a non repelling fixedpoint of R ( z ), as noticed in Section 3.4, g − (0) is a non repelling fixed point of S ( z ). But z − d k does not admit any such fixed points, and all the non repelling fixed points of z d k , T d k or − T d k aresuper-attracting (a computation shows that a fixed point of a Tchebychev polynomial distinct from ∞ is repelling and that ∞ is a super-attracting fixed point). Hence, S ( z ) ∈ { z d k , T d k , − T d k } , and g − (0) is a super-attracting fixed point of one of these three polynomials. Note that, by Section3.4 again, this implies that 0 is a super-attracting fixed point of R k ( z ) and R ( z ).We deduce that f is a solution of B R, . Iterating this equation, we see that f is a solution of B R k , . Now, by (8), g − f is a solution ψ of B S,g − (0) . But the fact that S ( z ) ∈ { z d k , T d k , − T d k } guarantees, via Theorem 3.3 that ψ is algebraic and is a Möbius transformation or a Möbiustransformation of one of the functions of the list of the second point of Theorem 3.1. As f = gψ ,we deduce that f is a Möbius transformation or a Möbius transformation (composition with theMöbius transformation g ) of a function of the list of the second point of Theorem 3.1.26ence, f ( z ) is solution of B M, , where M ( z ) is of the form of the rational fractions appearingin the second point of Theorem 3.1. As f ( z ) is solution of B R, and B M, , we have R = M and R ( z ) is of the form of the rational fractions appearing in the second point of Theorem 3.1.To conclude, we have proved that f is either a Schröder or Böttcher function (thus f is not aAbel function). Moreover, we have proved that, when f is a Schröder function, f is of the formdescribed in the first point of Theorem 3.1, and when f is a Böttcher function, f is of the formdescribed in the second point of Theorem 3.1. Theorem 3.1 is thus proved. Acknowledgement.
The author would like to warmly thank Alin Bostan, Lucia Di Vizio, andKilian Raschel for their enthusiastic proposal to write this paper. She would also like to thank thedetailed reports of the reviewers which allowed her to improve the fluidity, clarity and precision ofthis survey. Finally, she would like to thank Lucia Di Vizio for her support and her reviews andcorrections of this paper.
References [1]
N. H. Abel – œuvres complètes de Niels Henrik Abel. Tome II , Imprimerie de Grøndahl &Son, Christiania; distributed by the Norwegian Mathematical Society, Oslo, 1981, Contenantles mémoirs posthumes d’Abel. [Containing the posthumous memoirs of Abel], Edited and withnotes by L. Sylow and S. Lie.[2] D. S. Alexander – A history of complex dynamics-From Schröder to Fatou and Julia , Aspectsof Mathematics, E24, Friedr. Vieweg & Sohn, Braunschweig, 1994.[3]
G. W. Anderson, W. D. Brownawell and
M. A. Papanikolas – “Determination of thealgebraic relations among special Γ-values in positive characteristic”,
Ann. of Math. (2) (2004), 237–313.[4]
M. Aschenbrenner and
W. Bergweiler – “Julia’s equation and differential transcen-dence”,
Illinois J. Math. (2015), no. 2, 277–294.[5] M. Audin – Fatou, Julia, Montel, le grand prix des sciences mathématiques de 1918, et après ,Springer-Verlag, Berlin, 2009.[6]
A. F. Beardon – Iteration of rational functions-Complex analytic dynamical systems , Grad-uate Texts in Mathematics , Springer-Verlag, New York, 1991.[7]
P.-G. Becker and
W. Bergweiler – “Transcendency of local conjugacies in complex dy-namics and transcendency of their values”,
Manuscripta Math. (1993), no. 3-4, 329–337.[8] — , “Hypertranscendency of conjugacies in complex dynamics”, Math. Ann. (1995), no. 3,463–468.[9]
W. Bergweiler – “Solution of a problem of Rubel concerning iteration and algebraic differ-ential equations”,
Indiana Univ. Math. J. (1995), no. 1, 257–267.[10] P. Borwein – “Hypertranscendence of the functional equation g ( x ) = [ g ( x )] + cx ”, Proc.Amer. Math. Soc. (1989), no. 1, 215–221.2711]
M. Boshernitzan and
L. A. Rubel – “Coherent families of polynomials”,
Analysis (1986),no. 4, 339–389.[12] L. Böttcher – “Principaux résultats de convergence des itérées et applications à l’analyse(en russe)”,
Izv. Kazan (1904), 1–37.[13] — , “Principaux résultats de convergence des itérées et applications à l’analyse (en russe)”, Izv. Kazan (1904), 155–200.[14] — , “Principaux résultats de convergence des itérées et applications à l’analyse (en russe)”, Izv. Kazan (1904), 201–234.[15] H. Brolin – “Invariant sets under iteration of rational functions”,
Ark. Mat. (1965), 103–144(1965).[16] X. Buff and
A. Chéritat – “Ensembles de Julia quadratiques de mesure de Lebesgue stricte-ment positive”,
C. R. Math. Acad. Sci. Paris (2005), no. 11, 669–674.[17]
F. W. Carroll – “Transcendental transcendence of solutions of Schröder’s equation associ-ated with finite Blaschke products”,
Michigan Math. J. (1985), no. 1, 47–57.[18] J.-L. Chabert – “Un demi-siècle de fractales: 1870–1920”,
Historia Math. (1990), no. 4,339–365.[19] A. Chéritat – “L’ensemble de mandelbrot”,
Images des mathématiques, CNRS (Novembre2010).[20]
L. Di Vizio – “Approche galoisienne de la transcendance différentielle”, Transcendance etirrationalité, SMF Journ. Annu. , Soc. Math. France, Paris, 2012, 1–20.[21]
A. Douady – “Systèmes dynamiques holomorphes”, Bourbaki seminar, Vol. 1982/83,Astérisque , Soc. Math. France, Paris, 1983, 39–63.[22]
A. Douady and
J. H. Hubbard – “Itération des polynômes quadratiques complexes”,
C. R.Acad. Sci. Paris Sér. I Math. (1982), no. 3, 123–126.[23]
T. Dreyfus, C. Hardouin and
J. Roques – “Hypertranscendence of solutions of Mahlerequations”,
J. Eur. Math. Soc. (2018), 2209–2238.[24] A. Dudko and
M. Yampolskyy – “On computational complexity of cremer Julia sets.”,(2019), arxiv:1907.11047v3[math.DS].[25]
P. Fatou – “Sur les équations fonctionnelles”,
Bull. Soc. Math. France (1919), 161–271.[26] — , “Sur les équations fonctionnelles”, Bull. Soc. Math. France (1920), 33–94.[27] — , “Sur les équations fonctionnelles”, Bull. Soc. Math. France (1920), 208–314.[28] R. Feng – “Hrushovski’s algorithm for computing the Galois group of a linear differentialequation”,
Adv. in Appl. Math. (2015), 1–37.[29] G. Fernandes – “Méthode de Mahler en caractéristique non nulle (thèse)”, 〈tel-02386667〉 ,https://tel.archives-ouvertes.fr/tel-02386667/document.2830] — , “Méthode de Mahler en caractéristique non nulle: un analogue du théorème de Ku.Nishioka”,
Ann. Inst. Fourier (Grenoble) (2018), no. 6, 2553–2580.[31] — , “Regular extensions and algebraic relations between values of Mahler functions in positivecharacteristic”, Trans. Amer. Math. Soc. (2019), no. 10, 7111–7140.[32]
S. R. Garcia, J. Mashreghi and
W. T. Ross – “Finite Blaschke products: a survey”,
Mathand Computer Science Faculty Publications (2018).[33]
L. Geyer – “Linearizability of saturated polynomials”,
Indiana Univ. Math. J. (2019),no. 5, 1551–1578.[34] C. Hardouin – “Hypertranscendance des systèmes aux différences diagonaux”,
Compos. Math. (2008), no. 3, 565–581.[35] — , “Unipotent radicals of Tannakian Galois groups in positive characteristic”, Arithmeticand Galois theories of differential equations, Sémin. Congr. , Soc. Math. France, Paris,2011, 223–239.[36] — , “Galoisian approach to differential transcendence”, Galois theories of linear differenceequations: an introduction, Math. Surveys Monogr. , Amer. Math. Soc., Providence, RI,2016, 43–102.[37] C. Hardouin and
M. F. Singer – “Differential Galois theory of linear difference equations”,
Math. Ann. (2008), no. 2, 333–377.[38]
E. Hrushovski – “Computing the Galois group of a linear differential equation”, DifferentialGalois theory, Banach Center Publ. , Polish Acad. Sci. Inst. Math., Warsaw, 2002, 97–138.[39] K. Ishizaki – “Hypertranscendency of meromorphic solutions of a linear functional equation”,
Aequationes Math. (1998), no. 3, 271–283.[40] G. Julia – “Mémoire sur l’itération des fonctions rationnelles (8)”,
J. Math. Pures Appl. (1918), 47–246.[41] G. Koenigs – “Recherches sur les intégrales de certaines équations fonctionnelles”,
Ann. Sci.École Norm. Sup. (3) (1884), 3–41.[42] S. Lang – Elliptic curves: Diophantine analysis , Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences] , Springer-Verlag, Berlin-New York, 1978.[43]
L. Leau – “Étude sur les équations fonctionnelles à une ou à plusieurs variables”,
Ann. Fac.Sci. Toulouse Sci. Math. Sci. Phys. (1897), no. 2, E1–E24.[44] — , “Étude sur les équations fonctionnelles à une ou à plusieurs variables”, Ann. Fac. Sci.Toulouse Sci. Math. Sci. Phys. (1897), no. 3, E25–E110.[45] G. Levin and
M. Zinsmeister – “On the Hausdorff dimension of Julia sets of some realpolynomials”,
Proc. Amer. Math. Soc. (2013), no. 10, 3565–3572.2946]
J. H. Loxton and
A. J. Van der Poorten – “A class of hypertranscendental functions”,
Aequationes Math. (1977), no. 1-2, 93–106.[47] K. Mahler – “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgle-ichungen”,
Math. Ann. (1929), no. 1, 342–366.[48] — , “Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen”,
Math. Z. (1930), no. 1, 545–585.[49] — , “Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen”, Math. Z. (1930), no. 1, 545–585.[50] — , “Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punkt-folgen”, Math. Ann. (1930), no. 1, 573–587.[51]
B. Mandelbrot – Les objets fractals , Flammarion, Editeur, Paris, 1975, Forme, hasard etdimension, Nouvelle Bibliothèque Scientifique.[52]
P. Montel – “Sur les familles de fonctions analytiques qui admettent des valeurs exception-nelles dans un domaine”,
Ann. Sci. École Norm. Sup. (3) (1912), 487–535.[53] K. D. Nguyen – “Algebraic independence of local conjugacies and related questions in poly-nomial dynamics”,
Proc. Amer. Math. Soc. (2015), no. 4, 1491–1499.[54]
P. Nguyen – “Hypertranscedance de fonctions de Mahler du premier ordre”,
C. R. Math.Acad. Sci. Paris (2011), no. 17-18, 943–946.[55]
K. Nishioka – “A note on differentially algebraic solutions of first order linear differenceequations”,
Aequationes Math. (1984), no. 1-2, 32–48.[56] — , “Algebraic function solutions of a certain class of functional equations”, Arch. Math.(Basel) (1985), no. 4, 330–335.[57] K. Nishioka – Mahler functions and transcendence , Lecture Notes in Mathematics ,Springer-Verlag, Berlin, 1996.[58]
H.-O. Peitgen and
P. H. Richter – The beauty of fractals-Images of complex dynamicalsystems , Springer-Verlag, Berlin, 1986.[59]
H. Poincaré – “Sur une nouvelle classe de transcendantes uniformes”,
Journ. de Math. (4) (1890), 313–365.[60] M. van der Put and
M. F. Singer – Galois theory of difference equations , Lecture Notesin Mathematics , 1997.[61]
J. F. Ritt – “On the iteration of rational functions”,
Trans. Amer. Math. Soc. (1920),no. 3, 348–356.[62] — , “Periodic functions with a multiplication theorem”, Trans. Amer. Math. Soc. (1922),no. 1, 16–25.[63] — , “Transcendental transcendency of certain functions of Poincaré”, Math. Ann. (1926),no. 1, 671–682. 3064] J. Roques – “On the algebraic relations between Mahler functions”,
Trans. Amer. Math. Soc. (2018), 321–355.[65]
L. A. Rubel – “Generalized solutions of autonomous algebraic differential equations”,
Canad.Math. Bull. (1986), no. 3, 372–374.[66] — , “A survey of transcendentally transcendental functions”, Amer. Math. Monthly (1989),no. 9, 777–788.[67] — , “Some research problems about algebraic differential equations. II”, Illinois J. Math. (1992), no. 4, 659–680.[68] W. Rudin – Principles of mathematical analysis , third éd., McGraw-Hill Book Co., NewYork-Auckland-Düsseldorf, 1976, International Series in Pure and Applied Mathematics.[69]
E. Schröder – “Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen”,
Math.Ann. (1870), no. 2, 317–365.[70] E. Schröder – “Ueber iterirte Functionen”,
Math. Ann. (1870), no. 2, 296–322.[71] A. B. Shidlovskii – Transcendental numbers , De Gruyter Studies in Mathematics , Walterde Gruyter & Co., Berlin, 1989, Translated from the Russian by Neal Koblitz, With a forewordby W. Dale Brownawell.[72] C. L. Siegel – “Iteration of analytic functions”,
Ann. of Math. (2) (1942), 607–612.[73] J. H. Silverman – The arithmetic of elliptic curves , Graduate Texts in Mathematics ,Springer, Dordrecht, 2009.[74]
F. Yang – “Cantor Julia sets with hausdorff dimension two”, (2018),arxiv:1802.01063v1[math.DS].[75]