On rational functions sharing the measure of maximal entropy
aa r X i v : . [ m a t h . D S ] M a r ON RATIONAL FUNCTIONS SHARING THE MEASURE OFMAXIMAL ENTROPY
F. PAKOVICH
Abstract.
We show that describing rational functions f , f , . . . , f n sharingthe measure of maximal entropy reduces to describing solutions of the func-tional equation A ◦ X = A ◦ X = · · · = A ◦ X n in rational functions. We alsoprovide some results about solutions of this equation. Introduction
Let f be a rational function of degree d ≥ CP . It was proved by Friere,Lopes, and Ma˜n´e ([11]), and independently by Lyubich ([15]) that there exists aunique probability measure µ f on CP , which is invariant under f , has support equalto the Julia set J ( f ) of f , and achieves maximal entropy log d among all f -invariantprobability measures. In this note, we study rational functions sharing the measureof maximal entropy, that is rational functions f and g such that µ f = µ g , and moregenerally rational functions f , f , . . . , f n such that µ f = µ f = · · · = µ f n . Weassume that considered functions are non-special in the following sense: they areneither Latt`es maps nor conjugate to z ± n or ± T n . In case if f and g are polynomials, the condition µ f = µ g is equivalent to thecondition J ( f ) = J ( g ) . In turn, for non-special polynomials f and g the equality J ( f ) = J ( g ) = J holds if and only if there exists a polynomial h such that J ( h ) = J and(1) f = η ◦ h ◦ s , g = η ◦ h ◦ t for some integers s, t ≥ η , η of J (see [2], [29], andalso [3], [6]-[7] for other related results). Note that a similar conclusion remainstrue if instead of the condition J ( f ) = J ( g ) one were to assume only that f and g share a completely invariant compact set in C (see [18]). Note also that in thepolynomial case any of the conditions J ( f ) = J ( g ) and (1) is equivalent to thecondition that(2) f ◦ k = η ◦ g ◦ l , for some integers k, l ≥ η such that η ( J ( g )) = J ( g ) . Since µ f = µ f ◦ k , the equality µ f = µ g holds whenever f and g share an iterate,that is satisfy(3) f ◦ k = g ◦ l for some integers k, l ≥ . Moreover, µ f = µ g whenever f and g commute. However,the latter condition in fact is a particular case of the former one, since non-specialcommuting f and g always satisfy (3) by the result of Ritt ([27]). Note that in This research was supported by ISF Grant No. 1432/18. distinction with the polynomial case rational solutions of (3) not necessarily havethe form (1) (see [27], [25]).The problem of describing rational functions f and g with µ f = µ g can beexpressed in algebraic terms. Specifically, the results of Levin ([13]) and Levin andPrzytycki ([14]) imply that for non-special f and g the equality µ f = µ g holds if andonly if some of their iterates F = f ◦ k and G = g ◦ l satisfy the system of functionalequations(4) F ◦ F = F ◦ G, G ◦ G = G ◦ F (see [30] for more detail).Examples of rational functions f , g with µ f = µ g , which do not have the form(2), were constructed by Ye in the paper [30]. These examples are based on thefollowing remarkable observation: if X , Y , and A are rational functions such that(5) A ◦ X = A ◦ Y, then the functions F = X ◦ A, G = Y ◦ A satisfy (4). The simplest examples of solutions of (5) can be obtained from rationalfunctions satisfying A ◦ η = A for some M¨obius transformation η , by setting(6) X = η ◦ Y. In this case, the corresponding solutions of (4) have the form (2). However, othersolutions of (5) also exist, allowing to construct solutions of (4) which do not havethe form (2).Roughly speaking, the main result of this note states that in fact all solutions of(4) can be obtained from solutions of (5). More generally, the following statementholds.
Theorem 1.1.
Let f , f , . . . , f n be non-special rational functions of degree at leasttwo on CP . Then they share the measure of maximal entropy if and only if someof their iterates F , F , . . . , F n can be represented in the form (7) F = X ◦ A, F = X ◦ A, . . . , F n = X n ◦ A, where A and X , X , . . . , X n are rational functions such that (8) A ◦ X = A ◦ X = · · · = A ◦ X n and C ( X , X , . . . , X n ) = C ( z ) . Theorem 1.1 shows that “up to iterates” describing pairs of rational functions f and g with µ f = µ g reduces to describing solutions of (5). In particular, sincepolynomial solutions of (5) satisfy (6), we immediately recover the result that poly-nomials f, g with µ f = µ g satisfy (2). Nevertheless, the problem of describingsolutions of (5) for arbitrary rational A , X , Y is still widely open. In fact, acomplete description of solutions of (5) is obtained only in the case where A is apolynomial (while X and Y can be arbitrary rational functions) in the paper [5] byAvanzi and Zannier. The approach of [5] is based on describing polynomials A forwhich the genus of an irreducible algebraic curve(9) C A : A ( x ) − A ( y ) x − y = 0 UNCTIONS SHARING THE MEASURE OF MAXIMAL ENTROPY 3 is zero, and analyzing situations where C A is reducible but has a component ofgenus zero. Although the same strategy can be applied to an arbitrary rationalfunction A , both its stages become much more complicated and no general resultsare known to date.Note that the problem of describing solutions of equation (5) for rational A and meromorphic on the complex plane X , Y was posed in the paper of Lyubich andMinsky (see [16], p. 83) in the context of studying the action of rational functionson the “universal space” of non-constant functions meromorphic on C . In algebraicterms, the last problem is equivalent to describing rational functions A such that(9) has a component of genus zero or one .Theorem 1.1 implies an interesting corollary, concerning dynamical characteris-tics of rational functions sharing the measure of maximal entropy. Recall that the multiplier spectrum of a rational function f of degree d is a function which assignsto each s ≥ d s + 1 fixed points of f ◦ s takenwith appropriate multiplicity. Two rational functions are called isospectral if theyhave the same multiplier spectrum. Corollary 1.1.
If non-special rational functions f , f , . . . , f n of degree at least twoshare the measure of maximal entropy, then some of their iterates F , F , . . . , F n are isospectral. The rest of this note is organized as follows. In the second section, we proveTheorem 1.1 and Corollary 1.1. Then, in the third section, we prove two resultsconcerning equation (5) and system (8). The first result states that if the curve C A is irreducible and rational functions X , Y provide a generically one-to-oneparametrization of C A , then X = Y ◦ η for some involution η ∈ Aut ( CP ) . Thesecond result states that if A and X , X , . . . , X n are rational functions such that(8) holds and X , X , . . . , X n are distinct, then n ≤ deg A , and n = deg A only ifthe Galois closure of the field extension C ( z ) / C ( A ) has genus zero or one. In fact,we prove these results in the more general setting, allowing the functions X , Y and X , X , . . . , X n to be meromorphic on C . Functions sharing the measure of maximal entropy
In this section, we deduce Theorem 1.1 and Corollary 1.1 from the criterion (4)and the following four lemmas.
Lemma 2.1.
Let A , A , . . . , A n and Y , Y , . . . , Y n be rational functions such that (10) A i ◦ Y = A i ◦ Y = · · · = A i ◦ Y n , i = 1 , . . . n, and (11) C ( A , A , . . . , A n ) = C ( z ) . Then (12) Y = Y = · · · = Y n . Proof.
By (11), there exists a rational function P ∈ C ( z , z , . . . , z n ) such that z = P ( A , A , . . . , A n ) , implying that(13) Y j = P ( A ◦ Y j , A ◦ Y j , . . . , A n ◦ Y j ) , ≤ j ≤ n. F. PAKOVICH
Now (12) follows from (13) and (10). (cid:3)
Lemma 2.2.
Let F , F , . . . , F n be rational functions such that (14) F i ◦ F = F i ◦ F = · · · = F i ◦ F n , i = 1 , . . . n. Then there exist rational functions A and X , X , . . . , X n such that (15) F i = X i ◦ A, i = 1 , . . . n, (16) C ( X , X , . . . , X n ) = C ( z ) , and (17) A ◦ X = A ◦ X = · · · = A ◦ X n . Proof.
By the L¨uroth theorem, C ( F , F , . . . , F n ) = C ( A )for some rational function A , implying that equalities (15) hold for some rationalfunctions X , X , . . . , X n satisfying (16). Substituting now (15) in (14) we see that X i ◦ ( A ◦ X ) = X i ◦ ( A ◦ X ) = · · · = X i ◦ ( A ◦ X n ) , i = 1 , . . . n. Applying now Lemma 2.1 to the last system we obtain (17). (cid:3)
Lemma 2.3.
Let A and B be rational functions such that the equality A ◦ A = A ◦ B holds. Then A ◦ l ◦ A ◦ l = A ◦ l ◦ B ◦ l for any l ≥ .Proof. The proof is by induction on l . Assuming that the lemma is true for l = k ,we have: A ◦ ( k +1) ◦ B ◦ ( k +1) = A ◦ k ◦ ( A ◦ B ) ◦ B ◦ k = A ◦ k ◦ A ◦ ◦ B ◦ k == A ◦ ◦ A ◦ k ◦ B ◦ k = A ◦ ◦ A ◦ k = A k +2 . (cid:3) Lemma 2.4.
Let d i ≥ , ≤ i ≤ n, and n i,j ≥ , ≤ i, j ≤ n, i = j, be integerssuch that d n i,j i = d n j,i j , ≤ i, j ≤ n, i = j. Then there exist integers l i ≥ , ≤ i ≤ n, such that d l = d l = · · · = d l n n . Proof.
The proof is by induction on n . For n = 2, we obviously can set l = n , , l = n , . Assuming that the lemma is true for n = k , we can find integers a i , ≤ i ≤ k, and b i , ≤ i ≤ k + 1 , such that d a = d a = · · · = d a k k and d b = d b = · · · = d b k +1 k +1 , implying that d a b = d a b = · · · = d a k b k UNCTIONS SHARING THE MEASURE OF MAXIMAL ENTROPY 5 and d b a = d b a = · · · = d b k +1 a k +1 . Therefore, d a b = d b a = d b a = · · · = d b k +1 a k +1 , and hence the lemma is true for n = k + 1 . (cid:3) Proof of Theorem 1.1.
For any rational functions A and X , X , . . . , X n satisfying(8) the corresponding functions (7) satisfy system (14). In particular, for any pair i, j ≤ i, j ≤ n, i = j, the equalities F i ◦ F i = F i ◦ F j , F j ◦ F j = F j ◦ F i , ≤ i, j ≤ n, hold, implying that the functions f i , f j share the measure of maximal entropy.Therefore, all f , f , . . . , f n share the measure of maximal entropy.In the other direction, if µ f = µ f = · · · = µ f n , then using the criterion (4) wecan find integers n i,j , ≤ i, j ≤ n, i = j, such that(18) f ◦ n i,j i ◦ f ◦ n i,j i = f ◦ n i,j i ◦ f ◦ n j,i j , f ◦ n j,i j ◦ f ◦ n j,i j = f ◦ n j,i j ◦ f ◦ n i,j i . Suppose first that(19) deg f = deg f = · · · = deg f n . Then (19) and (18) imply that n i,j = n j,i , ≤ i, j ≤ n. Applying now Lemma 2.3to (18), we see that for any integer number M divisible by all the numbers n i,j , ≤ i, j ≤ n, the equalities f ◦ Mi ◦ f ◦ Mi = f ◦ Mi ◦ f ◦ Mj , ≤ i, j ≤ n, hold. Thus, the functions F i = f ◦ Mi , ≤ i ≤ n, satisfy system (14), implying byLemma 2.2 that equalities (15), (16), and (17) hold.For arbitrary rational functions f , f , . . . , f n sharing the measure of maximalentropy, we still can write system (18), implying that(deg f i ) n i,j = (deg f j ) n j,i , ≤ i, j ≤ n, i = j. Applying Lemma 2.4, we can find l i , ≤ i ≤ n, such that the rational functions f ◦ l i i , ≤ i ≤ n, have the same degree. Since these functions along with thefunctions f , f , . . . , f n share the measure of maximal entropy, we can write system(18) for these functions. Using now the already proved part of the theorem, weconclude that there exist m i , ≤ i ≤ n, such that the rational functions F i = f ◦ m i i , ≤ i ≤ n, satisfy (14), implying (15), (16), and (17). (cid:3) Proof of Corollary 1.1.
The corollary follows from the statement of the theoremand the fact that for any rational functions U and V the rational functions U ◦ V and V ◦ U are isospectral (see [24], Lemma 2.1). (cid:3) Functional equation A ( ϕ ) = A ( ψ ) . Equation (5) is a particular case of the functional equation A ◦ X = B ◦ Y, which, under different assumptions on A, B and
X, Y , has been studied in manypapers (see e. g. [1], [4], [10], [12], [17], [19], [20], [22], [26]). Nevertheless, to ourbest knowledge precisely equation (5) was the subject of only two papers. One ofthem is the paper of Avanzi and Zannier cited in the introduction. The other one is
F. PAKOVICH the paper [28] by Ritt, written eighty years earlier, where some partial results wereobtained. In particular, Ritt observed that solutions of (5) with X = Y can beobtained using finite subgroups of Aut ( CP ) as follows. Let Γ be a finite subgroupof Aut ( CP ) and θ Γ its invariant function, that is a rational function such that θ Γ ( x ) = θ Γ ( y ) if and only if y = σ ( x ) for some σ ∈ Γ . Then for any subgroupΓ ′ ⊂ Γ the equality(20) θ Γ = ψ ◦ θ Γ ′ holds for some ψ ∈ C ( z ), implying that ψ ◦ θ Γ ′ = ψ ◦ ( θ Γ ′ ◦ σ )for every σ ∈ Γ. Nevertheless, θ Γ ′ = θ Γ ′ ◦ σ unless σ ∈ Γ ′ . For example, for thedihedral group D n , generated by z → /z and z → εz, where ε = e πin , and itssubgroup D equality (20) takes the form12 (cid:18) z n + 1 z n (cid:19) = T n ◦ (cid:18) z + 1 z (cid:19) giving rise to the solution T n ◦ (cid:18) z + 1 z (cid:19) = T n ◦ (cid:18) εz + 1 εz (cid:19) of (5) not satisfying to (6). Ritt also constructed solutions of (5) using rationalfunctions arising from the formulas for the period transformations of the Weierstrassfunctions ℘ ( z ) for lattices with symmetries of order greater than two.In this note, we do not make an attempt to obtain an explicit classificationof solutions of (5) in spirit of [5]. Instead, we prove two general results whichemphasize the role of symmetries in the problem. Theorem 3.1.
Let A be a rational function and ϕ, ψ distinct functions meromor-phic on C such that A ◦ ϕ = A ◦ ψ. Assume in addition that the algebraic curve C A is irreducible. Then the desingu-larization R of C A has genus zero or one and there exist holomorphic functions ϕ : R → CP , ψ : R → CP and h : C → R such that ϕ = ϕ ◦ h, ψ = ψ ◦ h, and the map from R to C A given by z → ( ϕ ( z ) , ψ ( z )) is generically one-to-one.Moreover, (21) ϕ = ψ ◦ η for some involution η : R → R .Proof. The first conclusion of the theorem holds for any parametrization of analgebraic curve by functions meromorphic on C (see e. g. [9], Theorem 1 andTheorem 2), so we only must show the existence of an involution µ satisfying (21).Since the equation of C A is invariant under the exchange of variable, along withthe meromorphic parametrization z → ( ϕ , ψ ) the curve C A admits the meromor-phic parametrization z → ( ψ , ϕ ). Since the desingularization R is defined up toan automorphism, it follows now from the first part of the theorem that ϕ = ψ ◦ η, ψ = ϕ ◦ η UNCTIONS SHARING THE MEASURE OF MAXIMAL ENTROPY 7 for some η ∈ Aut ( R ), implying that(22) ϕ = ϕ ◦ ( η ◦ η ) , ψ = ψ ◦ ( η ◦ η ) . Finally, η ◦ η = z since otherwise (22) contradicts to the condition that the map z → ( ϕ ( z ) , ψ ( z )) is generically one-to-one. (cid:3) Theorem 3.2.
Let A be a rational function of degree d and ϕ , ϕ , . . . , ϕ n distinctmeromorphic functions on C such that (23) A ◦ ϕ = A ◦ ϕ = · · · = A ◦ ϕ n . Then n ≤ d . Moreover, if n = d , then the Galois closure of the field extension C ( z ) / C ( A ) has genus zero or one.Proof. Since for any z ∈ CP the preimage A − ( z ) contains at most d distinctpoints, if (23) holds for n > d , then for every z ∈ CP at most d of the val-ues ϕ ( z ) , ϕ ( z ) , . . . , ϕ n ( z ) are distinct, implying that at most d of the functions ϕ , ϕ , . . . , ϕ n are distinct.The second part of the theorem is the “if” part of the following criterion (see[23], Theorem 2.3). For a rational function A of degree d , the Galois closure of thefield extension C ( z ) / C ( A ) has genus zero or one if and only if there exist d distinctfunctions ψ , ψ , . . . , ψ d meromorphic on C such that A ◦ ψ = A ◦ ψ = · · · = A ◦ ψ d . (cid:3) Note that rational functions A for which the genus g A of the Galois closure ofthe field extension C ( z ) / C ( A ) is zero are exactly all possible “compositional leftfactors” of Galois coverings of CP by CP and can be listed explicitly. On theother hand, functions with g A = 1 admit a simple geometric description in termsof projections of maps between elliptic curves (see [21]). The simplest examples ofrational functions with g A ≤ z n , T n , (cid:0) z n + z n (cid:1) , and Latt`es maps. References
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Department of Mathematics, Ben Gurion University of the Negev, Israel
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