Tame rational functions: Decompositions of iterates and orbit intersections

. Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A.x/ (cid:0) A.y/ D 0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A ı d ; d (cid:21) 1; into compositions of rational functions can be obtained from decompositions of a single iterate A ı N for N large enough.


Introduction
Let A be a rational function of degree at least 2 on the Riemann sphere. For a point z 1 2 CP 1 we denote by O A .z 1 / the A-forward orbit of z 1 , that is, the set ¹z 1 ; A.z 1 /; A ı2 .z 1 /; : : : º: In this paper, we address the following problem: given two rational func- for some k; l 1: Put another way, unless rational functions A and B have the same global dynamics, an orbit of A may intersect an orbit of B at most in finitely many places. In the particular case where A and B are polynomials, the problem under consideration was completely settled in [7,8], where it was shown that the above condition on orbits is equivalent to (1). An essential ingredient of the proof was a result of [32], concerning functional decompositions of iterates of polynomials, which can be described as follows. be a decomposition of an iterate A ıd of a rational function A into a composition of rational functions X and Y . We say that this decomposition is induced by a decomposition A ıd 0 D X 0 ı Y 0 , where d 0 < d; if there exist k 1 ; k 2 0 such that In general, decompositions of A ıd are not exhausted by decompositions induced by decompositions of smaller iterates. However, the main result of [32] states that if A is a polynomial of degree n 2 not conjugate to z n or to˙T n , where T n stands for the Chebyshev polynomial, then there exists an integer N 1 such that every decomposition of A ıd with d N is induced by a decomposition of A ıN . Moreover, the number N depends on n only.
It seems highly likely that the result of [7,8] about orbit intersections of polynomials remains true for all rational functions, while the result of [32] about decompositions of iterates of polynomials not conjugate to z n or to˙T n remains true for all non-special rational functions, where by a special function we mean a rational function A that is either a Lattès map or is conjugate to z˙n or˙T n . However, the approach of the papers [7,8,32] cannot be extended to the general case, since it crucially depends on results of the Ritt theory of functional decompositions of polynomials [27], some of which have no analogues in the rational case while others are known not to be true. The result of [32] was proved by a different method in [16]. Nevertheless, the method of [16] does not extend to rational functions either.
A partial generalization of the result of [32] to rational functions was obtained in [25]. Namely, it was shown that there exists a function with integer arguments N D N.n; l/ such that for every rational function A of degree n 2 decompositions (2) with deg X Ä l and d N are induced by decompositions of A ıN . Other related results in the rational case were obtained in [2,3]. Specifically, it was shown in [2] that decompositions of iterates of a rational function A correspond to equivalence classes of certain analytic spaces defined in dynamical terms. On the other hand, in [3], an analogue of the problem about orbits was considered for semigroups of rational functions, and the results obtained were formulated in terms of the amenability of the corresponding semigroups. Giving a new look at the problems considered, the papers [2,3], however, do not provide handy conditions on rational functions A and B under which the results of [7,8,32] remain true.
To formulate our results explicitly, we introduce the following definition. Let A be a rational function of degree at least 2. We say that A is tame if the algebraic curve A.x/ A.y/ D 0 has no factors of genus 0 or 1 distinct from the diagonal. Otherwise, we say that A is wild. By the Picard theorem, the condition that A is tame is equivalent to the condition that for any functions f and g meromorphic on C the equality implies that f Á g: The problem of describing tame rational functions appears in holo-morphic dynamics [10]. It is also closely related to the problem of describing rational functions sharing the measure of maximal entropy [23,31]. It is easy to see that every rational function of degree 2 is wild. Consequently, a tame rational function has degree at least 3. On the other hand, a generic rational function of degree at least 4 is tame. Specifically, a rational function of degree at least 4 is tame whenever it has only simple critical values [15]. A comprehensive classification of wild rational functions is not known. The most complete result in this direction, obtained in [1], is the classification of solutions of equation (3) under the assumption that A is a polynomial and f , g are rational functions. For an account of recent progress in the general case we refer the reader to [29].
Our first main result is a generalization of the result of [32] to tame rational functions. Our second main result is a similar generalization of the result of [7,8].
Theorem 1.2. Let A and B be tame rational functions such that an orbit of A has an infinite intersection with an orbit of B. Then A and B have a common iterate.
Our proof of Theorem 1.1 is based on the result of [25] about decompositions of iterates cited above and the following statement of independent interest, providing lower bounds for genera of irreducible components of algebraic curves of the form where A and B are rational functions.
unless B D A ı S for some rational function S , and C is the graph x S.y/ D 0: Since equality (2) implies that the curve C A;X has a factor of genus 0, it follows from Theorem 1.3 that if deg X is large enough, then X D A ı S for some S 2 C.z/, and further analysis combined with the result of [25] permits us to prove Theorem 1.1.
In turn, the proof of Theorem 1.2 goes as follows. First, using the theorem of Faltings, we conclude that if has a factor of genus 0 or 1. Then, using Theorem 1.3, we prove that each iterate of B is a compositional left factor of some iterate of A, where by a compositional left factor of a rational function f we mean any rational function g such that f D g ı h for some ratio- (i) Each iterate of B is a compositional left factor of some iterate of A.
(ii) Each iterate of B is a compositional right factor of some iterate of A.
(iii) The functions A and B have a common iterate.
In addition to Theorem 1.2, we prove two other results supporting the conjecture that existence of orbits with an infinite intersection is equivalent to (1). The first result states that for arbitrary rational functions A and B the existence of such orbits imposes strong restrictions on their degrees consistent with condition (1). Specifically, letting P.n/ denote the set of prime divisors of a natural number n, we prove the following statement. Theorem 1.5. Let A and B be rational functions of degree at least 2 such that an orbit of A has an infinite intersection with an orbit of B. Then P.deg A/ D P.deg B/: The second result states that special rational functions, which are the simplest examples of wild rational functions and for which Theorem 1.1 is not true, cannot serve as counterexamples to Theorem 1.2. Theorem 1.6. Let A and B be rational functions of degree at least 2 such that an orbit of A has an infinite intersection with an orbit of B. Assume that at least one of these functions is special. Then A and B have a common iterate.
Besides the above results, we give new proofs of the main results of [7,8,32], using instead of Ritt theory the results of [19,20] and the classification of commuting polynomials.
The rest of the paper is organized as follows. In the second section, we discuss tame and wild rational functions, and provide a sufficient condition for a rational function to be wild. In the third section, we prove Theorem 1.3. In the fourth section, we prove Theorems 1.1, 1.2, and 1.4. In the fifth section, we deduce Theorems 1.5 and 1.6 from the results of [20]. Specifically, we use a description of pairs of rational functions A and U such that for every d 1 the algebraic curve has a factor of genus 0 or 1. Finally, in the sixth section, we reconsider the polynomial case and give new proofs of the main results of [7,8,32].

Tameness and normalization
Let f W S ! CP 1 be a holomorphic function on a compact Riemann surface S . Let us recall that the normalization of f is defined as a holomorphic function of the lowest possible degree between compact Riemann surfaces z f W z S f ! CP 1 such that z f is a Galois covering and for some holomorphic map h W z S f ! S . Equivalently, z f can be defined as a Galois covering z f W z S f ! CP 1 of the form (7) such that deg z f D jMon.f /j; (8) where Mon.f / is the monodromy group of f (see e.g. [9, Proposition 2.72]). We will denote by †.f / the subgroup of Aut.S / consisting of automorphisms such that f ı D f: Let A be a rational function of degree at least 2. Assume that there exist a compact Riemann surface S of genus 0 or 1, a holomorphic function U W S ! CP 1 ; and a Galois covering ‰ W S ! CP 1 such that A ı U is a rational function in ‰, but U is not a rational function in ‰. Then A is wild.
Proof. Since the assumptions of the theorem imply that for every˛2 †.‰/, to prove that the algebraic curve has a factor of genus 0 or 1, it is enough to show that there exists˛2 †.‰/ such that U ı˛6 Á U: Assume to the contrary that U ı˛Á U for any˛2 †.‰/: Since the equality ‰.x/ D ‰.y/ holds for x; y 2 S if and only if y D .x/ for some 2 †.‰/; in this case the algebraic function S D U ı ‰ 1 is single-valued and therefore rational. Thus, U D S ı ‰, in contradiction with the assumption.
(2) X is not a rational function of Y , Then A is wild.
On the other hand, X ı H is not a rational function of z Y for otherwise X would be a rational function of Y . Thus, the assumptions of Theorem 2.1 are satisfied for S D z S Y ; U D X ı H , and ‰ D z Y : Let f W R 1 ! R 2 be a holomorphic map between Riemann surfaces. We say that a holomorphic map h W R 1 ! R 0 is a compositional right factor of f if f D g ı h for some holomorphic map g W R 0 ! R 2 . Compositional left factors are defined similarly.
Corollary 2.4. Every rational function A that has a compositional right factor Y of degree at least 2 with g. z S Y / Ä 1 is wild. In particular, a rational function A of degree at least 2 is wild whenever g. z S A / Ä 1.
Proof. Let B be a rational function such that A D B ı Y . Then the assumptions of Corollary 2.3 are satisfied for B; Y; and X D z: Notice that rational functions A with g. z S A / D 0 can be listed explicitly as compositional left factors of rational Galois coverings. On the other hand, functions with g. z S A / D 1 admit a simple geometric description (see [18]).
Corollary 2.5. Any special rational function is wild.
Proof. The function z˙n itself is a Galois covering. On the other hand,˙T n is a compositional left factor of the Galois covering z n C 1 z n , implying that g. z S˙T n / D 0. Finally, every Lattès map A satisfies g. z S A / Ä 1 (see [18]).
For a holomorphic function f W S ! CP 1 the condition g. z S f / Ä 1 can be expressed merely in terms of the ramification of f . The easiest way to formulate the corresponding criterion is to use the notion of Riemann surface orbifold (see e.g. [20, Section 2.1] for basic definitions). Specifically, with each holomorphic function f W S ! CP 1 one can associate in a natural way two orbifolds O f 1 D .S; f 1 / and O f 2 D .CP 1 ; f 2 /, setting f 2 .z/ equal to the least common multiple of the local degrees of f at the points of the preimage f 1 ¹zº, and In these terms, the following statement holds.
where l; m are coprime and l C m 3; found in [1]. It was shown in [1] that the corresponding curve C A l;m defined by (9) is irreducible and has the rational parametrization z 7 ! .X.z/; Z.z//; where Moreover, A l;m is an indecomposable rational function, that is, A l;m has no decompositions into a composition of rational functions of degree at least 2. Thus, any compositional right factor of A l;m of degree at least 2 has the form ı A l;m for some 2 Aut.CP 1 /: On the other hand, it is easy to see that if l C m > 4, then .O is .l C m; lcm.l; m/; 2/: Thus, for l C m > 4, we have Let us notice however that although the family A l;m for l C m > 4 does not satisfy the assumption of Corollary 2.4, it does satisfy the assumptions of Theorem 2.1. Indeed, one can check that Z D X ı 1 z ; implying that the function A l;m ı X D A l;m ı Z is invariant with respect to the transformation z 7 ! 1=z: Therefore, for some rational function B and the Galois covering Y D z C 1=z. On the other hand, X is not a rational function of Y , since X is not invariant with respect to z 7 ! 1=z: 3. Bounds for genera of components of A.x/ B.y/ D 0

Fiber products
Let f W C 1 ! C and g W C 2 ! C be holomorphic maps between compact Riemann surfaces. The collection where n.f; g/ is a positive integer and R j are compact Riemann surfaces provided with holomorphic maps is called the fiber product of f and g if and for any holomorphic maps p W R ! C 1 ; q W R ! C 2 between compact Riemann surfaces satisfying f ı p D g ı q there exist a unique index j and a holomorphic map w W R ! R j such that The fiber product exists and is defined in a unique way up to natural isomorphisms. Notice that the universality property implies that the holomorphic maps p j and q j , 1 Ä j Ä n.f; g/; have no non-trivial compositional common right factor in the following sense: the equalities are holomorphic maps between compact Riemann surfaces, imply that deg w D 1: In particular, this implies that Another corollary is that p j , 1 Ä j Ä n.f; g/; is a rational function of q j if and only if deg q j D 1: In practical terms, the fiber product is described by the following algebro-geometric construction. Let us consider the algebraic curve Let us denote by L j ; 1 Ä j Ä n.f; g/, the irreducible components of L and by R j , 1 Ä j Ä n.f; g/, their desingularizations; let be the desingularization maps. Then the compositions extend to holomorphic maps and the collection S n.f;g/ j D1 ¹R j ; p j ; q j º is the fiber product of f and g. Abusing notation we call the Riemann surfaces R j , 1 Ä j Ä n.f; g/; the irreducible components of the fiber product of f and g.
Below we will use the following results, describing the fiber product of f and g ı u through the fiber product of f and g (see [20, Theorem 2.8 and Corollary 2.9]). For better understanding, see diagram (10).
and u W C 3 ! C 2 be holomorphic maps between compact Riemann surfaces. Assume that Corollary 3.2. In the above notation, the fiber products .C 1 ; f / C .C 2 ; g/ and .C 1 ; f / C .C 3 ; g ı u/ have the same number of irreducible components if and only if for every j; 1 Ä j Ä n.f; g/; the fiber product .R j ; q j / C 2 .C 3 ; u/ has a unique irreducible component.

Proof of Theorem 1.3
The proof of Theorem 1.3 uses two results. The first result is the following statement (see [20,Theorem 3.1]), generalizing an earlier result from [17]. Theorem 3.3. Let T; R be compact Riemann surfaces and W W T ! CP 1 a holomorphic map of degree n. Then for any holomorphic map P W R ! CP 1 of degree m such that the fiber product of P and W consists of a unique component E, we have Since .E/ D 2 2g.E/ and .R/ D 2 2g.R/ Ä 2; inequality (11) implies g.E/ m 84n C 168 84 : 1 In [20], instead of g. z S W / Ä 1 the equivalent condition .O W 2 / 0 is used.
In particular, Theorem 3.3 implies the following result proved in [17]: if A and B are rational functions of degrees n and m such that g. z S A / > 1 and the curve C A;B is irreducible, then g.C A;B / satisfies inequality (12). Theorem 1.3 can be considered as an analogue of the last result for reducible curves C A;B , with g. z S A / > 1 replaced by the stronger condition that A is tame.
The second result we need is the following result of Fried (see [6,Proposition 2], or [14, Theorem 3.5]).
commutes. Moreover, the maps X 1 and F have no common non-trivial compositional right factor, and deg Finally, since n.A; B/ n.A; B 1 / n.A 1 ; B 1 /; it follows from n.A; B/ D n.A 1 ; B 1 / that n.A; B/ D n.A; B 1 /: Therefore, n.F; V / D 1 by Corollary 3.2. Now we consider the cases g. z S F / > 1 and g. z S F / Ä 1 separately. In the first case, applying Theorem 3.3 to the fiber product of F and V , we see that Since the order of the monodromy group of a rational function A does not exceed the order of the full symmetric group on n D deg A symbols, it follows from (8) and z Taking into account the second equality in (14), we conclude that if g. z S F / > 1, then m=nŠ 84 n C 168 84 : Assume now that g. z S F / Ä 1: Since X 1 and F have no common non-trivial compositional right factor, X 1 is not a rational function in F , unless deg F D 1. Therefore, if deg F > 1, we can apply Corollary 2.3 to the bottom square in diagram (13), concluding that A is wild, in contradiction with the assumption. Thus, deg F D 1; implying that R D CP 1 and for S D X 1 ı F 1 ı V: Since X and Y have no non-trivial compositional common right factor, the second equality in (15)  Corollary 3.5. Let A be a tame rational function, and X and Y rational functions such that for some s 1: Then there exists a rational function X 0 such that deg X 0 Ä 84.deg A 2/.deg A/Š and X D A ıl ı X 0 ; A ı.s l/ D X 0 ı Y for some l 1: Proof. Equality (16) implies that the curve C A;X has a factor C of genus 0, parametrized by the map t 7 ! .A ı.s 1/ .t /; Y .t //: implying by Theorem 1.3 that X D A ı X 0 and C is the graph x X 0 .y/ D 0 for some rational function X 0 . Since C is parametrized by the map (17), this implies that Applying this reasoning recursively, we obtain the required statement.
The second corollary is the following.
Then A ıs D B ı Q for some rational function Q, and C is the graph Q.x/ y D 0: Proof. Inequality (18) for some rational function R and some d 1, there exists N Ä '.deg A; deg X / and a rational function R 0 such that Proof of Theorem 1.1. By Corollary 3.5, for any decomposition we can find X 0 and l 0 such that deg X 0 Ä 84.n 2/nŠ; and we have X D A ıl ı X 0 and On the other hand, it follows from Theorem 4.1 that there exists N , which depends on n only, such that for any decomposition (22) with d l > N satisfying (21), (22) there exists a rational function Y 0 such that The above implies that any decomposition of A ıd with d N is induced by a decomposition of A ıN . Indeed, if d l Ä N; then decomposition (20) is induced by the decomposition Let F be a rational function of degree at least 2. We define G.F / as the group of Möbius transformations such that  Then the group G.F / is one of the five finite rotation groups of the sphere, A 4 ; S 4 ; A 5 ; C n , D 2n , unless F D Â 1 ı z d ı Â 2 for some Möbius transformations Â 1 and Â 2 : Proof of Theorem 1.4. We recall that functional decompositions F D U ı V of a rational function F into compositions of rational functions U and V , considered up to the equivalence are in a one-to-one correspondence with imprimitivity systems of the monodromy group of F . In particular, the number of such classes is finite. Therefore, if for every i 1 there exist s i 1 and R i 2 C.z/ such that then Theorem 1.1 implies that there exist a rational function U and increasing sequences of non-negative integers f k , k 0; and v k , k 0; such that for some Á k 2 Aut.CP 1 /. In turn, this implies that there exists an increasing sequence of non-negative integers r k , k 1; such that for some k 2 Aut.CP 1 /. Furthermore, since (25) implies that for every k 1 the function B ıf 0 ı k is a compositional right factor of an iterate of B, there exist a rational function V and an increasing sequence of non-negative integers k l ; l 0; such that for some Â l 2 Aut.CP 1 /; implying that for some ı l 2 Aut.CP 1 /: Clearly, the Möbius transformations k l ı 1 k 0 ; l 1; belong to the group G.B ıf 0 /. On the other hand, since the function B is tame, the function B ıf 0 is also tame and hence, by Corollary 2.4, it is not of the form B ıf 0 D Â 1 ı z d ı Â 2 , where Â 1 ; Â 2 2 Aut.CP 1 /: Therefore, by Theorem 4.2, for some l 2 > l 1 , implying that k l 2 D k l 1 : It now follows from (25) that Since l 2 > l 1 and the sequences k l , l 1; and f k ; k 1; are increasing, we know that f k l 2 > f k l 1 , and therefore A and B have a common iterate. This proves (i))(iii). Similarly, if for every i 1 there exist s i 1 and R i 2 C.z/ such that we conclude that there exist increasing sequences f k , k 0; and r k , k 1; such that for some k 2 Aut.CP 1 /. Moreover, there exists an increasing sequence k l ; l 0; such that for some ı l 2Aut.CP 1 /: Finally, for some l 2 >l 1 we have ı l 2 Dı l 1 , implying k l 2 D k l 1 : Since B is tame, the last equality in turn implies (26). This proves (ii))(iii). Finally, it is clear that (iii) implies (i) and (ii).

Remark 4.3.
It is not the case that Theorem 1.4 is true for all rational functions. For example, it is easy see that for the functions z 6 and z 12 conditions (i) and (ii) are satisfied, while (iii) is not. Nevertheless, one can expect that (i) and (iii) are equivalent for non-special functions. On the other hand, there exist non-special rational functions for which (ii) and (iii) are not equivalent. Specifically, using wild rational functions one can construct A and B such that but A and B have no common iterate (see [23,31]). Since (28) implies that for such A and B any iterate of B is a compositional right factor of an iterate of A.
Our starting point in the proofs of Theorems 1.2, 1.5, and 1.6 is the following lemma. Proof. Recall that by the theorem of Faltings [5], if an irreducible algebraic curve C defined over a finitely generated field K of characteristic 0 has infinitely many K-points, then g. We start by recalling the results of [20], describing pairs of rational functions A and U of degree at least 2 such that for every d 1 the algebraic curve (6) has an irreducible factor of genus 0 or 1. In case A is non-special, the main result of [20] in a slightly simplified form can be formulated as follows (see [20,Theorem 1.2]).
Theorem 5.1. Let A be a non-special rational function of degree at least 2. Then there exist a rational Galois covering X A and a rational function F such that the diagram (29) commutes, and for a rational function U of degree at least 2 the algebraic curve C A ıd ;U has a factor of genus 0 or 1 for every d 1 if and only if U is a compositional left factor of A ıl ı X A for some l 0: The Galois covering X A in Theorem 5.1 can be described explicitly (see [20,Theorem 3.4]). However, we do not need this more explicit description to prove Theorem 1.5 in the case where both functions A and B are non-special. Indeed, since by Lemma 4.4 for every pair .d; i / 2 N N the algebraic curve (5) has a factor of genus 0 or 1, it follows from Theorem 5.1 that for every i 1 there exist d i 1 and S i 2 C.z/ such that Therefore, if for some prime number p, then for every i 1 there exists d i 1 such that implying that ord p .deg A/ > 0: By symmetry, inequality (32) implies (31). Therefore, This proves Theorem 1.5 when A and B are non-special. On the other hand, if A or B is special, then Theorem 1.5 obviously follows from Theorem 1.6 proved below.

5.2.
Proof of Theorem 1.6 for A conjugate to z˙n or˙T n For s 1, we set To prove Theorem 1.6 when A is conjugate to z˙n or˙T n , we use the following result (see [20,Theorem 3.6]).
Theorem 5.2. Let A and U be rational functions of degree at least 2.
(i) If A D z n ; then the algebraic curve C A ıd ;U has a factor of genus 0 or 1 for every d 1 if and only if U D z s ı ; s 2; where is a Möbius transformation.
(ii) If A D T n ; then the algebraic curve C A ıd ;U has a factor of genus 0 or 1 for every d 1 if and only if either U D˙T s ı ; s 2; or U D D s ı ; s 1; where is a Möbius transformation.
Let us prove Theorem 1.6 when A is conjugate to˙T n . Clearly, without loss of generality we may assume that A D T n if n is even, or A D˙T n if n is odd. Since by Lemma 4.4 for every pair .d; i / 2 N N the algebraic curve (5) has a factor of genus 0 or 1, it follows from Theorem 5.2 (ii) that if A D T n ; then for any i 1 either B ıi D˙T s ı ; s 2; 2 Aut.CP 1 /; or B ıi D D s ı ; s 1; 2 Aut.CP 1 /: The same is true if A D T n ; since we can apply Theorem 5.
for some 2 Aut.CP 1 /. Furthermore, since finite critical values of Chebyshev polynomials are˙1, and the local multiplicity of˙T s at each of the points in T 1 s ¹ 1; 1º distinct from 1 and 1 is 2, equality (37) implies by the chain rule that whenever m > 2 the equalities .1/ D .1/ and ¹ 1; 1º D ¹ 1; 1º hold. Thus, D˙z and hence B D˙T m : Let now O A .z 1 / and O B .z 2 / be orbits having an infinite intersection. Evidently, without loss of generality we may assume that z 1 D z 2 D z 0 , and it is clear that z 0 ¤ 1. The equalities A D˙T n and B D˙T m imply that there exist a linear function˛A of the form nz or nz C 1=2 and a linear function˛B of the form mz or mz C 1=2 such that the diagrams 0 is a point of C such that cos.2 z 0 0 / D z 0 and k; l 1 are integers such that then .˛ı k A˙˛ı l B /.z 0 0 / is an integer. Taking into account the form of˛A and˛B , this implies that either z 0 0 is a rational number, or˛ı k A D˙˛ı l B . In the first case, however, z 0 0 is a preperiodic point both for˛A modulo 1 and for˛B modulo 1, implying that the orbits O A .z 1 / and O B .z 2 / are finite, and therefore cannot have an infinite intersection. Thus, ık A D˙˛ı l B , implying that A ık D B ıl . This finishes the proof of Theorem 1.6 when A is conjugate to˙T n .
In case A is conjugate to z˙n, the proof can be done in a similar way using Theorem 5.2 (i) and the family of semiconjugacies

Proof of Theorem 1.6 when A is a Lattès map
In this section, we need some further definitions and results concerning Riemann surface orbifolds; in particular, the definition of the orbifold O A 0 associated with a rational function A, and the description of Lattès maps as self-covering maps of orbifolds of zero Euler characteristic. All the necessary information can be found in [20,Sections 2.1 and 2.4].
The first result we need is the following (see [20,Theorem 3.5]). In addition, we need the following two facts (see [20,Theorem 2.4] and [21, Lemma 3.5]).  6. The polynomial case

Polynomial decompositions
First of all, we recall that if A is a polynomial, and A D U ı V is a decomposition into a composition of rational functions, then there exists a Möbius transformation such that U ı and 1 ı V are polynomials. Thus, when studying decompositions of A ıd we can restrict ourselves to considering decompositions into compositions of polynomials. We also mention that since a polynomial cannot be a Lattès map, a polynomial is special if and only if it is conjugate to z n or˙T n : The following result follows easily from the fact that the monodromy group of a polynomial of degree n contains a cycle of length n: such that Notice that where d > N; we conclude that For a polynomial T we denote by Aut.T / the set of polynomial Möbius transformations commuting with T . The following result classifies polynomials commuting with a given non-special polynomial (see [28] and [24, Section 6.2]). Theorem 6.2. Let A be a polynomial of degree at least 2, not conjugate to z n or˙T n : Then there exists a polynomial T such that A D ı T ık ; where 2 Aut.A/ and k 1; and any polynomial B commuting with A has the form B D ı T ıl ; where 2 Aut.A/ and l 1: Corollary 6.3. Let A be a polynomial of degree at least 2, not conjugate to z n or˙T n : Assume that B is a polynomial commuting with A such that deg B deg A: Then B D A ı S for some polynomial S .
Proof. Applying Theorem 6.2 and taking into account that ; 2 Aut.A/; we see that B D A ı S for the polynomial S D ı 1 ı T ı.l k/ :

Equivalence relation
Let A be a rational function. Following [19], we say that a rational function y A is an elementary transformation of A if there exist rational functions U and V such that A D V ı U and y A D U ı V . We say that A and B are equivalent and write A B if there exists a chain of elementary transformations between A and B. Notice that any pair A, y A as above gives rise to the semiconjugacies The first part of Theorem 6.4 was proved in [19], using the McMullen theorem about isospectral rational functions [11]. This approach however provides no bound for d.A/. The fact that d.A/ can be bounded in terms of n was proved in [21, Theorem 1.1 and Remark 5.2]). Lemma 6.5. Let A be a special function, and A 0 A: Then A 0 is special.
In full generality Lemma 6.5 is proved in [21,Lemma 2.11]. Below we use this lemma only in the polynomial case, in which it follows from the well known description of decompositions of z n and˙T n .

Polynomial orbits and iterates
We start by re-proving the main result of [32], basing merely on the results of Sections 6.1-6.2. 2 Theorem 6.6. Let A be a polynomial of degree n 2 not conjugate to z n or˙T n : Then there exists an integer N , depending on n only, such that any decomposition of A ıd with d N is induced by a decomposition of A ıN .
Proof. It is enough to show that if a polynomial A is not conjugate to z d or˙T d , then equality (16) for some polynomials X and Y with deg X large enough with respect to deg A implies that X D A ı R (43) for some polynomial R. Indeed, in this case without loss of generality we may assume that A ı.s 1/ D R ı Y by Theorem 6.1, and applying this argument inductively, we obtain an analogue of Corollary 3.5, which holds for any non-special polynomial A. The rest of the proof is similar to the proof of Theorem 1.1.
Since (16) implies that P.X / Â P.A/, we have gcd.deg X; deg A/ > 1. Therefore, by Theorem 6.1, there exists a polynomial V 1 of degree at least 2 such that for some polynomials U 1 and X 1 : In turn, (44) implies where Applying now the same reasoning to (45) we can find polynomials U 2 ; V 2 ; X 2 , deg V 2 2; such that A 1 D V 2 ı U 2 ; X 1 D V 2 ı X 2 ; and Continuing in the same way and taking into account that deg V i 2; we see that there exist an integer p 1 and a sequence of elementary transformations Since a polynomial cannot be a Lattès map, the equivalence class OEA contains at most finitely many conjugacy classes by Theorem 6.4. Setting M D n d.A/K ; where K is a natural number to be defined later, assume that deg X > M: Since deg V i Ä n; this implies that p d.A/K C 1. Therefore, there exist indices 0 Ä s 0 < s 1 < < s K Ä p such that A s 0 , A s 1 ; : : : ; A s K are conjugate to each other. We now consider the commutative diagram and W i D V s i 1 C1 ı V s i 1 C2 ı ı V s i ; 1 Ä i Ä K: Since A s K D 1 ı A s 0 ı for some 2 Aut.CP 1 /; the polynomial W D W 1 ı W 2 ı ı W K ı 1 commutes with A s 0 : Moreover, since A is non-special, so is A s 0 by Lemma 6.5. Assume now that K log 2 n. Since deg V i 2; in this case deg W n, and hence W D A s 0 ı S for some polynomial S , by Corollary 6.3. Therefore, Summarizing, we see that the condition deg X > n d.A/ log 2 n implies (43) for some polynomial R: Now we re-prove the main result of [7,8], relying on Theorems 5.1 and 6.6. Theorem 6.7. Let A and B be polynomials of degree at least 2 such that an orbit of A has an infinite intersection with an orbit of B. Then A and B have a common iterate.
Proof. By Theorem 1.6, we may assume that A and B are not special. Arguing as in Section 5.1, we see that there exist a Galois covering X A and a rational function F such that diagram (29) commutes and for every i 1 there exist d i 1 and S i 2 C.z/ such that equality (30) holds. Moreover, P.B/ Â P.A/; implying that for every i 1 there exist s i d i such that deg.B ıi / j deg.A ıs i /: Equality (30) implies which in turn implies Applying now Theorem 6.1 to (47) and taking into account (46), we conclude that for every i 1 there exist R i 2 COEz such that (24) holds. Finally, arguing as in the proof of Theorem 1.4, but using Theorem 6.6 instead of Theorem 1.1, we conclude that A and B have a common iterate.
Funding. This research was supported by ISF Grant No. 1432/18.