Portraits of preperiodic points for rational maps
aa r X i v : . [ m a t h . N T ] J u l PORTRAITS OF PREPERIODIC POINTS FOR RATIONALMAPS
D. GHIOCA, K. NGUYEN, AND T. J. TUCKER
Abstract.
Let K be a function field over an algebraically closed field k ofcharacteristic 0, let ϕ ∈ K ( z ) be a rational function of degree at least equal to2 for which there is no point at which ϕ is totally ramified, and let α ∈ K . Weshow that for all but finitely many pairs ( m, n ) ∈ Z ≥ × N there exists a place p of K such that the point α has preperiod m and minimum period n underthe action of ϕ . This answers a conjecture made by Ingram-Silverman [12] andFaber-Granville [7]. We prove a similar result, under suitable modification, alsowhen ϕ has points where it is totally ramified. We give several applications ofour result, such as showing that for any tuple ( c , . . . , c d − ) ∈ k n − and foralmost all pairs ( m i , n i ) ∈ Z ≥ × N for i = 1 , . . . , d −
1, there exists a polynomial f ∈ k [ z ] of degree d in normal form such that for each i = 1 , . . . , d −
1, thepoint c i has preperiod m i and minimum period n i under the action of f . Introduction
Throughout this paper, let K be a finitely generated field of transcendence degree1 over an algebraically closed field k of characteristic 0. By a place p of K , wemean an equivalence class of valuations on K that are trivial on k ; the set of allsuch places is denoted by Ω K . Each such place p gives rise to a valuation ring o p and a maximal ideal denoted (by an abuse of notation) by p . The residuefield is (canonically isomorphic to) k . We let v p and | · | p respectively denote thecorresponding additive and multiplicative valuations normalized by v p ( K ) = Z and | α | p = e − v p ( α ) for every α ∈ K . For a rational map ϕ ∈ K ( x ), for all but finitelymany places p of K we can define the reduction of ϕ modulo p , see Section 3.1 or[19, Chapter 2]. We start with a definition which is central for our paper: Definition 1.1.
Let F be any field. For a rational map ϕ ∈ F ( z ) and for α ∈ F ,we say that ( m, n ) ∈ Z ≥ × N is the preperiodicity portrait, or simply portrait, of α (with respect to ϕ ) if ϕ m ( α ) is periodic of minimum period n for ϕ , while m is thesmallest nonnegative integer such that ϕ m ( α ) is periodic (as always in dynamics,we denote by ϕ k = ϕ ◦ · · · ◦ ϕ composed with itself k times). We call m the preperiodof α and call n the minimum period of α . Let ϕ ∈ K ( z ), α ∈ P ( K ), and p a place of K such that the reduction of ϕ modulo p is well-defined. Assume that α has portrait ( m, n ) under the induced reductionself-map on P ( k ). We call ( m, n ) the preperiodicity portrait of α (under the actionof ϕ ) modulo p . The existence of places p of K for which α has portrait ( m, n ) under Mathematics Subject Classification.
Primary 11G50; Secondary 14G99.
Key words and phrases. heights; abc -conjecture; arithmetic dynamics.The first author was partially supported by an NSERC Discovery Grant. The second authorthanks the Pacific Institute of Mathematical Sciences for its generous support. The third authorwas partially supported by an NSF grant. the action of ϕ modulo p has been studied for more than 100 years (see the papers ofBang [2] and Zsigmondy [24]) and also more recently (see [17, 12, 7, 10, 15]). Someof the results are easier to obtain when the dynamical system induces a group action(see [2, 24], and also more recent results for Drinfeld modules [9, 11]). However,when there is no group action, the problem is much harder. Certain conjectures onthis problem have been made by Ingram-Silverman [12, pp. 300–301] and modifiedby Faber-Granville [7, pp. 190], yet very few general results are known. Faber andGranville [7] have proved that if ϕ ∈ Q ( z ) and α ∈ Q , then for all but finitely many m ∈ Z ≥ there exists a co-finite set S m ⊆ N (i.e., N \ S m is finite) such that foreach n ∈ S m , there exists a prime p such that the preperiod of α modulo p is m ,while its minimum period divides n . On the other hand, Ingram-Silverman [12] andFaber-Granville [7] conjecture that one can obtain a similar conclusion this time forminimum period equal to n . In this paper, we are able to resolve their conjectureover function fields. First we define the set of exceptions from the conjecture ofIngram-Silverman [12] and Faber-Granville [7]. Definition 1.2.
For any rational function ϕ ∈ K ( z ) , let X ( ϕ ) be the set of n such that ϕ is totally ramified at every point of minimum period n . For a rationalfunction ϕ and a point α , we let Y ( ϕ, α ) be the set of positive integers m such that ϕ is totally ramified over ϕ m ( α ) (i.e. ϕ is totally ramified at ϕ m − ( α ) ). Now we can state our main result; for more details on good reduction of a rationalmap ϕ and for the canonical height b h ϕ associated to ϕ , see Section 3. Theorem 1.3.
Let K be a function field. Let ϕ ∈ K ( z ) have degree d > . Let τ, s > and let S be a set of places of K containing all the places of bad reductionfor ϕ such that S ≤ s . Then there is a finite set Z ( τ, s ) ⊂ Z ≥ × N depending onlyon ϕ , K , τ , and s with the following property: for any α ∈ P ( K ) with b h ϕ ( α ) > τ and any ( m, n ) ∈ Z ≥ × N such that ( m, n )
6∈ Z ( τ, s ) , m / ∈ Y ( ϕ, α ) and n / ∈ X ( ϕ ) ,there is a nonarchimedean prime p of good reduction for ϕ such that α has portrait ( m, n ) under the action of ϕ modulo p . Moreover, the set Z ( τ, s ) is effectivelycomputable. The next two remarks explain why the conditions in Theorem 1.3 are necessary.
Remark . Theorem 1.3 without the conditions that m / ∈ Y ( ϕ, α ) and n / ∈ X ( ϕ )was essentially conjectured by Ingram-Silverman [12]. It was Faber and Granville [7]who pointed out that the condition n / ∈ X ( ϕ ) would be necessary. By Lemma 4.24, n ∈ X ( ϕ ) if and only if either ϕ has no point of minimum period n (see Kisaka’sclassification [14] or [7, Appendix B] for a complete list of all rational maps whichhave no point of minimum period n ), or n = 2 and ϕ ( z ) is linearly conjugate to z − . Note that Ingram-Silverman made their conjectures over number fields whileFaber-Granville even restricted further to the field of rational numbers. Theorem1.3 answers completely the same question for function fields. Remark . Theorem 1.3 without the condition that m / ∈ Y ( ϕ, α ) was essentiallyconjectured by Faber-Granville [7, pp. 190]. We briefly explain why this conditionis also necessary. Suppose M > ϕ is totally ramified over ϕ M ( α ). Write β = ϕ M ( α ). We may assume ϕ M − ( α ) = 0 and β = ∞ by making a linear changeof variables. Therefore ϕ has the form: ϕ ( z ) = z d ψ ( z ) + β ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 3 where ψ is a polynomial of degree at most d satisfying ψ (0) = 0. For almost allprimes p of good reduction and for every a ∈ P ( K ), we have that ϕ ( a ) ≡ β mod p if and only if a ≡ p . Hence if ϕ M + n ( α ) ≡ β mod p then ϕ M + n − ( α ) ≡ p . So, for almost all n , there does not exist p such that α has portrait ( M, n )under the action of ϕ mod p .We also note that the hypothesis that the function field has characteristic 0 isused crucially in our proof because we employ in our arguments Mason’s [16] andStothers’ [20] abc -theorem for function fields of characteristic 0 (see also [18]). Itwould be interesting to treat the question in positive characteristic as well, but itappears that new ideas or techniques may be required.We have explained why the conditions m / ∈ Y ( ϕ, α ) and n / ∈ X ( ϕ ) are necessary.Hence the Ingram-Silverman-Faber-Granville conjecture (which was originally madeover number fields) should be modified accordingly. One may also adapt our proofin this paper to resolve their conjecture assuming the abc -conjecture in the contextof number fields. We will treat this in a future paper.The original question that motivates this paper is the “simultaneous portraitproblem” (see Theorem 2.1) over function fields in several parameters; this problemhas no obvious analog over number fields.Since we are excluding places outside S , our conclusion involving the finitenessof Z ( τ, s ) is the best one can hope for . We also note the remarkable uniformity obtained here: the set Z ( τ, s ) only depends on ϕ , K , s , and the lower bound τ onthe canonical height of α under ϕ rather than depending on α .The following result is an immediate consequence of Theorem 1.3 in the casewhen ϕ ∈ K [ z ] is a polynomial which is totally ramified at no point in K ; in thiscase, X ( ϕ ) = ∅ and also Y ( ϕ, α ) = ∅ for any α ∈ K (see Remark 1.4). Corollary 1.6.
Let K , ϕ , τ , s , and S be as in Theorem 1.3. Assume that ϕ ∈ K [ z ] is a polynomial that is totally ramified at no point in K (equivalently, ϕ ( z ) is notlinearly conjugate over K to a polynomial of the form z d + c ). Then there exists afinite set Z ( τ, s ) ⊂ Z ≥ × N depending only on K , ϕ , τ , and s such that for every α ∈ K satisfying b h ϕ ( α ) > τ and for every ( m, n ) ∈ ( Z ≥ × N ) \ Z ( τ, s ) there existsa place p / ∈ S such that α has portrait ( m, n ) mod p . Theorem 1.3 allows us to prove a result (see Theorem 2.1) for simultaneousportraits of complex numbers realized by polynomials in normal form. Also, Theo-rem 1.3 allows us to prove a strong uniform result for realizing all possible portraitsby almost any constant starting point (see Theorem 2.4). We will state in Section 2these two results, together with other applications of our Theorem 1.3.We sketch briefly the plan of our paper. We state in Section 2 applications ofTheorem 1.3 (see Theorems 2.1, 2.2 and 2.4). In Section 3 we introduce the notationand the basic notions used in the paper. We prove Theorem 1.3 in Section 4 andthen we prove its various applications in Section 5. Finally, we conclude our paperby asking several related questions in Section 6.2.
Applications
We say that a polynomial ϕ ( z ) of degree d is in normal form if it is monic andits coefficient of z d − equals 0. Note that each polynomial ϕ is linearly conjugate toa polynomial in normal form. Therefore, when discussing preperiodicity portraits D. GHIOCA, K. NGUYEN, AND T. J. TUCKER in the family of polynomials of degree d , it makes sense to restrict the analysis tothe case of polynomials in normal form.Let d ≥ k be an algebraically closed field of characteristic 0,let c , . . . , c d − ∈ k , and let ( m i , n i ) ∈ Z ≥ × N for i = 1 , . . . , d −
1. It is naturalto ask whether there exists a polynomial f ∈ k [ z ] in normal form and of degree d such that for each i = 1 , . . . , d −
1, the point c i has preperiodicity portrait ( m i , n i )for the action of f ( z ).Already Theorem 1.3 solves the above question if d = 1. Indeed, one considersthe polynomial f ( z ) = z + t ∈ K [ z ], where K := k ( t ) and then Theorem 1.3yields that at the expense of excluding finitely many portraits (note also that X ( f ) = Y ( f, c ) = ∅ by Remark 1.4), there exists a place p of K such that c has preperiodicity portrait ( m , n ) for the action of f ( z ) modulo p . Reducing f ( z )modulo a place of K is equivalent with specializing t to a value in k , hence providingan answer to the above question if d = 1.As a matter of notation, by a co-finite set of portraits we mean a subset of Z ≥ × N whose complement is finite. Next result answers the above question of“simultaneous multiportraits”, and it follows from Theorem 1.3 coupled with aneasy fact regarding canonical heights of constant points under the action of a non-isotrivial polynomial (see Lemma 5.1). Theorem 2.1.
Let k be an algebraically closed field of characteristic 0, let d ≥ be an integer, and let c , . . . , c d − ∈ k be d − distinct elements. Then thereexists a co-finite set of portraits Z (0) depending on k and d such that for each ( m , n ) ∈ Z (0) , there exists a co-finite set of portraits Z (1) := Z (1) ( c , m , n ) depending on k , d , c , m , and n such that for each ( m , n ) ∈ Z (1) , there existsa co-finite set of portraits Z (2) := Z (2) ( c , m , n , c , m , n ) depending on k , d , c ,..., n such that for each ( m , n ) ∈ Z (2) , and so on ..., there exists a co-finite set of portraits Z ( d − := Z ( d − ( c , ..., n d − ) depending on k , d , c ,..., n d − such that for each ( m d − , n d − ) ∈ Z ( d − there exist a , . . . , a d − ∈ k such thatthe following holds. For ≤ i ≤ d − , the point c i has portrait ( m i , n i ) under z d + a d − z d − + . . . + a z + a . We are interested next in the reverse situation from Theorem 2.1, i.e. given aset of ( d − distinct portraits ( m i , n i ), for which starting points c , . . . , c d − ∈ k is there possible to find a polynomial f ∈ k [ z ] of degree d and in normal formsuch that the preperiodicity portrait of c i with respect to the action of f ( z ) is( m i , n i ) for each i = 0 , . . . , d −
2? In other words, Theorem 2.1 tells us how manyportraits may be missed for a given set of starting points, while the next resultgives information on how many tuples of starting points have to be excluded if acertain set of portraits is to be realized by those starting points.
Theorem 2.2.
Let k be an algebraically closed field of characteristic 0, let d ≥ be an integer, and let ( m , n ) , . . . , ( m d − , n d − ) be distinct elements in Z ≥ × N .Then there exists a co-finite subset T (0) of P ( k ) depending on k and d such thatfor each c ∈ T (0) , there exits a co-finite subset T (1) := T (1) ( m , n , c ) of P ( k ) depending on k , d , m , n , and c such that for each c ∈ T (1) , there exists aco-finite subset T (2) := T (2) ( m , n , c , m , n , c ) depending on k , d , m , . . . , c such that for each c ∈ T (2) , and so on ..., there exists a co-finite subset T ( d − := T ( d − ( m , . . . , c d − ) depending on k , d , m , . . . , c d − such that for each c d − ∈ ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 5 T ( d − there exists a , . . . , a d − ∈ k such that the following holds. For ≤ i ≤ d − ,the point c i has portrait ( m i , n i ) under z d + a d − z d − + . . . + a z + a . Theorem 2.2 follows through an argument similar to the proof of Theorem 2.1once we prove a strong uniform result for the set of possible exceptions of startingpoints which cannot realize a given portrait, i.e. the “dual” statement from Theo-rem 1.3. For this “dual” result we require first the definition of isotrivial rationalmaps . Definition 2.3.
Let k be an algebraically closed field of characteristic , and let K be a finitely generated function field over k of transcendence degree equal to . Let ϕ ∈ K ( z ) of degree d ≥ . We say that ϕ ( z ) is isotrivial if there is σ ∈ K ( z ) ofdegree such that σ − ◦ ϕ ◦ σ ∈ k ( z ) . Let k , K , and ϕ be as in Definition 2.3. We let W ( ϕ ) be the set of ( m, n ) ∈ Z ≥ × N such that the set of points having portrait ( m, n ) with respect to ϕ iseither empty or it is a (proper) subset of P ( k ). Since ϕ is not isotrivial, thereare at most finitely many x ∈ P ( k ) which are preperiodic for ϕ (see [1]); henceTheorem 1.3 yields that there are at most finitely many portraits ( m, n ) ∈ W ( ϕ )such that n / ∈ X ( ϕ ). Theorem 2.4.
Let S be a finite set of places of K containing all places of badreduction. Let ϕ ∈ K ( z ) be non-isotrivial. Then there is a finite set T ( S ) ⊂ P ( k ) depending only on K , ϕ , and S satisfying the following property: for every ( m, n ) ∈ ( Z ≥ × N ) \ W ( ϕ ) and for every α ∈ P ( k ) \ T ( S ) , there is a place p ∈ Ω K \ S suchthat α has portrait ( m, n ) under the action of ϕ modulo p .Remark . From Lemma 4.24 and Corollary 4.25, we have that ( m, n ) / ∈ W ( ϕ ) ifand only if n / ∈ X ( ϕ ) and some point of portrait ( m, n ) is not constant. The firstcondition n / ∈ X ( ϕ ) is necessary for Theorem 1.3 which will be used in the proofof Theorem 2.4. Now, if all points of portrait ( m, n ) are contained in P ( k ), thenfor some α ∈ P ( k ) which is not a point with portrait ( m, n ) for ϕ , we cannot finda place p of K such that the portrait of α for ϕ modulo p is ( m, n ) because thatwould mean that α is in the same residue class modulo p as another point in P ( k )(which has portrait ( m, n ) for ϕ globally).We expect Theorem 2.2 remains valid without the condition that the given por-traits are distinct. In Theorem 2.2, we note that the co-finite set T ( i ) dependson the previously chosen points c , . . . , c i − together with the portraits ( m , n ),...,( m i − , n i − ). It is an interesting problem to relax such dependence on portraits,for which we present the following result for cubic polynomials in normal form. Bya co-countable subset of a set, we mean a subset whose complement is countable. Corollary 2.6.
Suppose that k is algebraically closed of characteristic 0. Thenthere exist a co-finite subset U (1) of k such that for every c ∈ U (1) , there exists aco-countable subset U (2) ( c ) of k depending on c such that for every c ∈ U (2) ( c ) ,the following holds. For every pair of portraits ( m , n ) and ( m , n ) , there exist a, b ∈ k such that for each i = 1 , , c i has portrait ( m i , n i ) under z + az + b .As a consequence, when k is uncountable there exist uncountably many ( c , c ) ∈ k such that for every pair of portraits ( m , n ) and ( m , n ) , there exist a, b ∈ k such that for each i = 1 , , c i has portrait ( m i , n i ) under z + az + b . D. GHIOCA, K. NGUYEN, AND T. J. TUCKER Preliminaries
Good reduction of rational maps. If ϕ : P → P is a morphism definedover K , then (fixing a choice of homogeneous coordinates) there are relatively primehomogeneous polynomials F, G ∈ K [ X, Y ] of the same degree d = deg ϕ such that ϕ ([ X, Y ]) = [ F ( X, Y ) : G ( X, Y )]. In affine coordinates, ϕ ( z ) = F ( z, /G ( z, ∈ K ( z ) is a rational function in one variable. Note that by our choice of coordinates, F and G are uniquely defined up to a nonzero constant multiple.Let p be a place of K with valuation ring o p and residue field k . We define asfollows the reduction modulo p of a point P ∈ P ( K ). We let x, y ∈ o v not both inthe maximal ideal of o p such that P = [ x : y ] and then the reduction of P modulo p is defined to be r p ( P ) := [ x : y ], where z ∈ k is the reduction modulo p of theelement z ∈ o p .Let ϕ : P −→ P be a morphism over K , given by ϕ ([ X, Y ]) = [ F ( X, Y ) : G ( X, Y )], where
F, G ∈ o p [ X, Y ] are relatively prime homogeneous polynomials ofthe same degree such that at least one coefficient of F or G is a p -adic unit. Let ϕ p := [ F p : G p ], where F p , G p ∈ k [ X, Y ] are the reductions of F and G modulo p .We say that ϕ has good reduction at p if ϕ p : P ( k ) −→ P ( k ) is a morphism ofthe same degree as ϕ . Equivalently, ϕ has good reduction at p if ϕ extends as amorphism to the fibre of P o p ) above p . For all but finitely many places p of K ,the map ϕ has good reduction at p (for more details, see the comprehensive bookof Silverman [19, Chapter 2]).If ϕ ∈ K [ z ] is a polynomial, we can give the following elementary criterionfor good reduction: ϕ has good reduction at v if and only if all coefficients of ϕ are v -adic integers, and its leading coefficient is a v -adic unit. For simplicity, wewill always use this criterion when we choose a place v of good reduction for apolynomial ϕ .3.2. Absolute values and heights in function fields.
For any finite extension
L/K we let Ω L be the set of places of L . For q ∈ Ω L and p ∈ Ω K , if q | K = p thenwe write q | p and let v q and | · | q respectively denote the extension of v p and of | · | q on L . For every q ∈ Ω q , we let e ( q ) be the ramification index for the extension ofplaces q | p where p = q | K .For each x ∈ K we define its Weil height as h K ( x ) = 1[ K ( x ) : K ] X p ∈ Ω K X q ∈ Ω K ( x ) q | p e ( q ) · log + | x | q , where always log + ( z ) := log max { , z } for any real number z . We prefer to usethe notation h K for the Weil height (normalized with respect to K ) in order toemphasize the dependence on the ground field K for our definition of the height. Forexample, if L/K is a finite field extension, and x ∈ K , then h L ( x ) = [ L : K ] · h K ( x ).We extend h K on P ( K ) by h K ( ∞ ) = 0.Let x and y be distinct elements of P ( K ), we have the following inequality:(3.1) { p ∈ Ω K : r p ( x ) = r p ( y ) } ≤ h K ( x ) + h K ( y ))To prove this, we assume that x, y ∈ K since the case x = ∞ or y = ∞ is easy.The set in the left-hand side of (3.1) is contained in: { p ∈ Ω K : | x − y | p < } ∪ { p ∈ Ω p : | x | p > | y | p > } ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 7 whose cardinality is bounded above by: h K ( x − y ) + h K ( x ) + h K ( y ) ≤ h K ( x ) + h K ( y )) . Canonical heights for rational maps. If ϕ ∈ K ( z ) is a rational map ofdegree d ≥
2, then for each point x ∈ P ( K ), following [5] we define the canonicalheight of x under the action of ϕ by: b h ϕ ( x ) = lim n →∞ h K ( ϕ n ( x )) d n . According to [5], there is a constant C ϕ depending only on K and ϕ such that | h K ( z ) − b h ϕ ( z ) | < C ϕ for all z ∈ P ( K ).4. Proof of Theorem 1.3
Throughout this section, k is an algebraically closed field of characteristic 0, K is a finitely generated function field over k of transcendence degree equal to 1, and ϕ ∈ K ( z ) is a rational function of degree d >
1. Throughout this section, unlessstated otherwise all constants depend on K and ϕ . If a constant depends on otherarguments, our notation will clearly indicate them. For example, C , B , D , . . . de-note constants depending on K and ϕ only, while C ( α, β, γ, . . . ) denotes a constantdepending on K, ϕ, α, β, γ, . . . .4.1.
A preliminary estimate.
At each place p of K , we use the chordal metric d p ( · , · ) defined as d p ([ x : y ] , [ a : b ]) = | xb − ya | p (max( | x | p , | y | p )) (max( | a | p , | b | p )) . We see then that for any place p , we have r p ([ x : y ]) = r p ([ a : b ]) if and only if d p ([ x : y ] , [ a : b ]) < Proposition 4.1.
Let τ, δ > be real numbers, let i ≥ be an integer, let α, β ∈ K such that b h ϕ ( α ) ≥ τ > , and let F ( z ) be a monic, separable polynomial withcoefficients in K whose roots γ j satisfy the following conditions: (1) ϕ i ( γ j ) = β for each j ; (2) each γ j is not periodic; and (3) for each j and for each ℓ = 0 , . . . , i − , ϕ ℓ ( γ j ) = β .For each positive integer n ≥ i , we let Z n be the set of places p of K such thateither ϕ has bad reduction at p or (4.2) max( d p ( ϕ m ( α ) , β ) , | F ( ϕ n − i ( α )) | p ) < for some positive integer m < n . Then there are constants C ( δ, i, τ ) , C ( δ, i, τ ) ,and B ( δ, i, τ ) depending only on i , δ , τ , and ϕ such that for all positive integers n > B ( δ, i, τ ) , we have (4.3) Z n ≤ δh K ( ϕ n ( α )) + ( C ( δ, i, τ ) + 2 n ) h K ( β ) + C ( δ, i, τ )Informally, the set Z n consists of all places p such that ϕ n − i ( α ) is in the sameresidue class modulo p as one of the roots γ j of F (see Remark 4.4) and β is inthe same residue class modulo p as an iterate ϕ m ( α ) with m < n . In other words,we are looking at places p such that ϕ m ( α ), ϕ n ( α ), and β have the same reduction D. GHIOCA, K. NGUYEN, AND T. J. TUCKER modulo p . The conclusion of Proposition 4.1 is that Z n is bounded above by anexplicit quantity whose major term (as n grows) is δh K ( ϕ n ( α )). Remark . Let the notation be as in Proposition 4.1. Let L be the splitting fieldof F ( z ) over K . Let Γ be the set of places p of K such that there is a place q | p of L and a root γ j of F such that | γ j | q >
1. We have: ≤ X j h K ( γ j ) . Using ϕ i ( γ j ) = β so that deg( F ) ≤ d i and h K ( γ j ) = d i h K ( β ) + O (1), there existconstants C ( i ) and C ( i ) such that: ≤ X j h K ( γ j ) ≤ C ( i ) h K ( β ) + C ( i ) . For every place p of K outside Γ, for every x ∈ K , the inequality | F ( x ) | p < γ j of F and a place q | p of L suchthat r q ( x ) = r q ( γ j ). Proof of Proposition 4.1.
Recall that we have b h ϕ ( ϕ ( z )) = d b h ϕ ( z ) for all z ∈ K andthat there is a constant C ϕ such that | h K ( z ) − b h ϕ ( z ) | < C ϕ for all z ∈ K . Thestrategy of our proof is to divide Z n into sets denoted Y , Y , and Y below in whichthe inequality (4.2) holds when n − m is respectively large, small, and moderate.Choose B ( δ, i, τ ) such that the inequalities:(4.5) 1 /d B ( δ,i,τ )+ i < min { , δ } / n + 1) C ϕ ≤ δ d n τ − C ϕ ) ≤ δ h K ( ϕ n ( α ))hold for every n > B ( δ, i, τ ). We note that the first inequality in (4.6) is possiblesince d n dominates other terms when n grows, and that the second inequality in(4.6) is always true since h K ( ϕ n ( α )) + C ϕ ≥ b h ϕ ( ϕ n ( α )) ≥ d n τ. ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 9
For any α ∈ P ( K ) and all n > B ( δ, i, τ ), we have n − B ( δ,i,τ ) − i X ℓ =0 { p : r p ( ϕ ℓ ( α )) = r p ( β ) }≤ n − B ( δ,i,τ ) − i X ℓ =0 (cid:0) h K ( ϕ ℓ ( α )) + h K ( β ) (cid:1) (by (3.1)) ≤ n ( C ϕ + h K ( β )) + 2 n − B ( δ,i,τ ) − i X ℓ =0 b h ϕ ( ϕ ℓ ( α ))= 2 n ( C ϕ + h K ( β )) + 2 d B ( δ,i,τ )+ i n − B ( δ,i,τ ) − i X r =0 b h ϕ ( ϕ n ( α )) d r ≤ d B ( δ,i,τ )+ i ∞ X r =0 d r ! b h ϕ ( ϕ n ( α )) + 2 n ( C ϕ + h K ( β )) ≤ min { , δ } b h ϕ ( ϕ n ( α )) + 2 n ( C ϕ + h K ( β )) (by (4.5)) ≤ δ h K ( ϕ n ( α )) + 2( n + 1)( C ϕ + h K ( β )) ≤ δh K ( ϕ n ( α )) + 2( n + 1) h K ( β ) (by (4.6))(4.7)Thus, if Y is the set of primes such that d p ( ϕ ℓ ( α ) , β ) < ℓ ≤ n − B ( δ, i, τ ) − i , then(4.8) Y ≤ δh K ( ϕ n ( α )) + 2( n + 1) h K ( β ) . Let Y be the set of primes of K for which ϕ does not have good reduction. Thenclearly,(4.9) Y ≤ C where C depends only on K and ϕ .Now, let L be the splitting field for F ( z ). Since ϕ i ( γ j ) = β , we have:(4.10) b h ϕ ( γ j ) = 1 d i b h ϕ ( β ) . Let Y be the set of primes p outside Y and the set Γ in Remark 4.4 such thatmax (cid:0) d p ( ϕ m ( α ) , β ) , | F ( ϕ n − i ( α )) | p (cid:1) < n − i ≤ m < n . For each such prime we have r q ( ϕ m − ( n − i ) ( γ j )) = r q ( β ) for someroot γ j of F and some prime q of L with q | p . From m − ( n − i ) < i , condition (3)and (4.10), we have ϕ m − ( n − i ) ( γ j ) = β and b h ϕ ( ϕ m − ( n − i ) ( γ j )) ≤ b h ϕ ( β ). This latterinequality implies:(4.11) h K ( ϕ m − ( n − i ) ( γ j )) ≤ h K ( β ) + 2 C ϕ . Using (3.1) and (4.11) we have(4.12) Y ≤ i (4 h K ( β ) + 4 C ϕ ) . Let Y be the set of primes p outside Y and the set Γ in Remark 4.4 such thatmax (cid:0) d p ( ϕ m ( α ) , β ) , | F ( ϕ n − i ( α )) | p (cid:1) < for some positive integer m with n − i > m > n − i − B ( δ, i, τ ). If p ∈ Y , then ϕ m ( α ) ≡ ϕ n ( α ) ≡ β (mod p ) , so β modulo p is in a cycle of period dividing n − m .There is a prime q | p of L and a root γ j of F ( z ) such that γ j ≡ ϕ n − i ( α ) ≡ ϕ ( n − i ) − m ( β ) (mod p ), we see that γ j is in the same cycle modulo q . This impliesthat γ j modulo q has period dividing n − m . From (4.11) and n − m < B ( δ, i, τ ) + i ,we have: h K ( ϕ n − m ( γ j )) + h K ( γ j ) ≤ b h ϕ ( ϕ n − m ( γ j )) + b h ϕ ( γ j ) + 2 C ϕ ≤ (cid:18) d B ( δ,i,τ ) + 1 d i (cid:19) b h ϕ ( β ) + 2 C ϕ ≤ (cid:18) d B ( δ,i,τ ) + 1 d i (cid:19) ( h K ( β ) + C ϕ ) + 2 C ϕ (4.13)Note that each γ j has degree at most d i over K since ϕ i ( γ j ) = β for each j .Using (3.1) and (4.13), we have:(4.14) Y ≤ B ( δ, i, τ ) d i (cid:18) (cid:18) d B ( δ,i,τ ) + 1 d i (cid:19) ( h K ( β ) + C ϕ ) + 4 C ϕ (cid:19) . Since Z n is contained in Y ∪ Y ∪ Γ ∪ Y ∪ Y , we see that (4.3) is a consequence ofRemark 4.4, (4.8), (4.9), (4.12), and (4.14). (cid:3) A consequence of the abc -theorem for function fields.
The followingresult is crucial for the proof of Theorem 1.3 and it is a consequence of the abc -theorem for function fields by Mason-Stothers (see Silverman’s formulation [18]).We will treat the number field case in a future work in which a similar result holdsassuming the abc -conjecture.
Proposition 4.15.
Let K be a function field. Let e ≥ be a positive integer. Thenfor any monic f ( z ) ∈ K [ z ] of degree e without repeated roots, and for any γ ∈ K we have (4.16) { primes p of K | v p ( f ( γ )) > } ≥ h K ( γ ) − e e X i =1 h K ( η i ) + 2 g K ! where η , . . . , η e are the roots of f in K , and we denote by g L the genus of anyfunction field L .Proof. The given inequality holds trivially if γ is a root of f ( z ). We now assume γ is not a root of f ( z ). Let L = K ( η , . . . , η e ). We have(4.17)[ L : K ] { primes p of K | v p ( f ( γ )) > } ≥ [ i =1 { primes q of L | v q ( γ − η i ) > } Now, when L ramifies over K at a prime p , we must have that either some η i hasa pole at a prime lying over p for 1 ≤ i ≤ e , or η i − η j has a zero at a prime lyingover p for 1 ≤ i < j ≤ e . Thus, the number of primes p of K that are ramified in L/K is bounded by e X i =1 h K ( η i ) + X ≤ i We now apply the Riemann-Hurwitz theorem for L/K . Note that for each p of K where L/K is ramified, the total ramification contribution of primes of L lyingabove p in the Riemann-Hurwitz formula is at most [ L : K ] − 1, hence:(4.18) 2 g L − ≤ e e X i =1 h K ( η i ) + 2 g K − ! [ L : K ] . Now we construct a change of coordinates σ that takes η , η , η to 0, 1, ∞ , i.e. σ ( z ) = η − η η − η · z − η z − η . Then for any z , we have(4.19) h K ( z ) − X i =1 h K ( η i ) ≤ h K ( σ ( z )) ≤ h K ( z ) + 4 X i =1 h K ( η i ) . Let B be the set of primes q of L such that v q ( η i ) = 0 for some 1 ≤ i ≤ v p ( η i − η j ) = 0 for some 1 ≤ i < j ≤ 3. Then we have(4.20) B ≤ X i =1 h L ( η i ) + X ≤ i 3, combining equations (4.17), (4.18), (4.19), (4.20), and (4.21) givesthe desired inequality (4.16). (cid:3) Proof of Theorem 1.3: small m or small n . Assume the notation inTheorem 1.3, we prove the existence of p such that r p ( α ) has portrait ( m, n ) foralmost all ( m, n ) where n / ∈ X ( ϕ ), m / ∈ Y ( ϕ, α ), and either m or n is small. Notethat the constants that appear here may depend on the finite set of places S . Asbefore, we will always indicate such dependence. We will use the following verysimple lemmas repeatedly. Lemma 4.22. Let p be a prime of good reduction for ϕ . Suppose that r p ( γ ) = r p ( γ ) but r p ( ϕ ( γ )) = r p ( ϕ ( γ )) . If γ is periodic for ϕ modulo p , then γ is notperiodic for ϕ modulo p .Proof. We write r p ( ϕ n ( γ )) = r p ( γ ) for some n > 0. Suppose that γ was alsoperiodic modulo p ; then we can write r p ( ϕ n ( γ )) = r p ( γ ) for some n > 0. Since r p ( ϕ ( γ )) = r p ( ϕ ( γ )) we then must have r p ( γ ) = r p ( ϕ n n ( γ )) = r p ( ϕ n n ( γ )) = r p ( γ ) , a contradiction. (cid:3) The next lemma is immediate since for any finite extension L/K , and for anyplace q of L that lies above the place p of K , two points of K have the samereduction modulo p if and only if they have the same reduction modulo q . Lemma 4.23. Let L be an algebraic extension of K . Let p be a prime of goodreduction for ϕ and suppose that α has portrait ( m, n ) modulo q for some q | p .Then α has portrait ( m, n ) modulo p . We begin by considering the case where n is small. First, another lemma. Lemma 4.24. Assume that ϕ has a point of minimum period n and assume oneof the following two conditions: (i) ϕ ( z ) is not linearly conjugate to z − ; or (ii) ϕ ( z ) is linearly conjugate to z − and n = 2 .Then there exists a point β of minimum period n such that ϕ − ( β ) contains a pointthat is not periodic.Proof. Note that if ϕ − ( β ) contains only periodic points then ϕ − ( β ) is a singlepoint and thus ϕ ramifies completely at ϕ − ( β ). Since ϕ has at most two totallyramified points, we see then that if n ≥ 3, then ϕ ramifies completely over atmost two points in any cycle of size n , and we are done. If n = 1, then afterchange of coordinates, ϕ is a polynomial, which we call it f . Then the fixed pointsof f are solutions to f ( x ) − x = 0, which has at least one solution (note thatdeg( f ) = deg( ϕ ) = d > f is totally ramified at one of these solutions then f is conjugate to f ( z ) = z d , and the fixed point 1 has a non-periodic image ξ d of1, where ξ d is a primitive d -th root of unity, and we are done. Similarly, if n = 2and we have a point γ of period 2 such that ϕ ramifies completely at both γ and ϕ ( γ ) then ϕ ( z ) is conjugate to z − d . The given condition implies that d > 2. Since ϕ ( x ) − x = x d − x , we see that ϕ has exactly d − d > ϕ cannot ramify completely over all of them, so at least one of themhas non-periodic inverse. (cid:3) Lemma 4.24 together with the Kisaka’s classification [14] gives the completedescription of X ( ϕ ): Corollary 4.25. One of the following holds: (i) X ( ϕ ) = { n } for some n ∈ N , and moreover, ϕ does not have a point ofminimum period n . (ii) ϕ ( z ) is linearly conjugate to z − and X ( ϕ ) = { } . (iii) X ( ϕ ) = ∅ . Our next result yields the conclusion of Theorem 1.3 when the period n is small: Proposition 4.26. Let τ , s , and S be as in Theorem 1.3. Fix a positive integer n / ∈ X ( ϕ ) . Then there is a constant M ( n, τ, s ) depending on K , ϕ , n , S , and τ such that for any α ∈ K with b h ϕ ( α ) > τ and any m > M ( n, τ, s ) , there is a prime p / ∈ S such that α has portrait ( m, n ) under the action of ϕ modulo p .Proof. By Lemma 4.24 and Corollary 4.25, there is β ∈ P ( K ) of minimum period n and non-periodic ζ ∈ P ( K ) such that ϕ ( ζ ) = β . Let L = K ( ζ ) and note that β ∈ P ( L ). We will occasionally apply previous results for L in place of K . Since β , ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 13 ζ , and L depend on K , ϕ , and n , constants depending on L and ϕ will ultimatelydepend on K , ϕ , and n . Let S L be the places of L lying above those in S .We see that ϕ − ( ζ ) contains at least four points (see [19, pp. 142]), none of whichare periodic. Thus, there is a monic separable polynomial F of degree greater than2 with coefficients in L such such that for every root γ of F , we have ϕ ( γ ) = ζ and γ is not periodic. Because ζ is not periodic, then ϕ ℓ ( γ ) = ζ for each root γ of F , and for each ℓ = 0 , . . . , α ∈ P ( K ) such that b h ϕ ( α ) > τ . There is a constant C ( τ ) ≥ ϕ m − ( α ) = ∞ for m > C ( τ ) since we can simply require d C ( τ ) τ > b h ϕ ( ∞ ). ByProposition 4.15, there is a constant C ( n ) such that:(4.27) { q ∈ Ω L : v q ( F ( ϕ m − ( α ))) > } ≥ h L ( ϕ m − ( α )) − C ( n )for every m > C ( τ ).Let β j = ϕ j ( β ) for 0 ≤ j ≤ n − β .Let E be the set of primes q ∈ Ω L of good reduction satisfying one of the followingtwo conditions:(4.28) There are 0 ≤ i < j ≤ n − r q ( β i ) = r q ( β j ).(4.29) r q ( ζ ) = r q ( β n − )By (3.1), there is a constant C ( n ) such that E ≤ C ( n ). This last inequalitytogether with (4.27) and Remark 4.4 imply the existence of a constant C ( n, τ, s )such that for every m > C ( n, τ, S ) the following holds. There exists a place q ∈ Ω L \ ( E ∪ S L ) such that ϕ m − ( α ) and some root γ of F have the same reduction moduloa place of L ( γ ) lying above q . Therefore ϕ m ( α ) and ζ have the same reductionmodulo q . Since q / ∈ E , conditions (4.28) and (4.29) together with Lemma 4.22imply that ϕ m +1 ( α ) ≡ β is periodic of minimum period n and that ϕ m ( α ) ≡ ζ isnot periodic modulo q . Therefore α has portrait ( m + 1 , n ) modulo q . Lemma 4.23finishes our proof. (cid:3) We now consider small values of m : Proposition 4.30. Let τ , s , and S be as in Theorem 1.3. Let m ≥ be an integer. (1) If m = 0 , then there is a constant N ( τ, s ) such that for any n > N ( τ ) andany α with b h ϕ ( α ) ≥ τ , there is a prime p / ∈ S such that α has portrait ( m, n ) modulo p . (2) If m > , then there is a constant N ( m, τ, s ) such that for any n >N ( m, τ, s ) and any α satisfying b h ϕ ( α ) ≥ τ and ϕ is not totally ramifiedat ϕ m − ( α ) , there is a prime p / ∈ S such that α has portrait ( m, n ) underthe action of ϕ modulo p .Proof. We may assume that ϕ m ( α ) = ∞ . Otherwise, we can make the change ofvariables z z .If m = 0, then let F be a monic separable polynomial of degree greater than twosuch that every root γ of F satisfies ϕ ( γ ) = α . Since α is not preperiodic (becauseit has positive canonical height), then also each root γ of F is not periodic, andmoreover, F ℓ ( γ ) = α for ℓ = 0 , . . . , 4. If m > 0, and ϕ is not totally ramified at ϕ m − ( α ), then let F be a monic separable polynomial of degree greater than twosuch that each γ of F satisfies: (i) ϕ ( γ ) = ϕ m ( α ) and ϕ ℓ ( γ ) = ϕ m ( α ) for ℓ = 0 , . . . , γ is not periodicbecause ϕ m ( α ) is not periodic); and(ii) ϕ ( γ ) = ϕ m − ( α ).The fact that deg( F ) > F ) ≥ 4) follows from [19, pp. 142].As before, there is a constant C ( τ ) ≥ n > C ( τ ), wehave ϕ m + n − ( α ) = ∞ . Then, by Proposition 4.15 there is a constant C ( m ) suchthat:(4.31) { p : v p ( F ( ϕ m + n − ( α ))) > } ≥ h K ( ϕ m + n − ( α )) − C ( m )for every n > C ( τ ).For n > C ( τ ), we let W n be the set of primes p of K such that either ϕ hasbad reduction at p ormax( d p ( ϕ ℓ ( ϕ m ( α )) , ϕ m ( α )) , | F ( ϕ n − ( ϕ m ( α ))) | p ) < ℓ < n . Proposition 4.1 (for β = ϕ m ( α ), i = 5, and δ = d )shows that there are constants C ( m, τ ) > C ( τ ) and C ( m, τ ) such that for all n > C ( m, τ ), we have(4.32) W n ≤ d h K ( ϕ m + n ( α )) + nC ( m, τ ) . If m > 0, let E be the set places p ∈ Ω K of good reduction such that r q ( ϕ ( γ )) = r q ( ϕ m − ( α )) for some root γ of F and some place q of K ( γ ) above p . If m = 0, welet E be the empty set. There is a constant C ( m, τ ) such that:(4.33) E ≤ C ( m, τ ) . Note that h K ( ϕ r ( α )) = d r b h ϕ ( α ) + O (1) for every integer r ≥ α ∈ P ( K ), where O (1) only depends on K and ϕ . From the right-hand sides of (4.31),(4.32), and (4.33) together with Remark 4.4, there is a constant C ( m, τ, s ) >C ( m, τ ) such that for every n > C ( m, τ, s ) and every α ∈ P ( K ) satisfying b h ϕ ( α ) ≥ τ , there is a place p ∈ Ω K satisfying the following conditions:(I) p / ∈ W n ∪ E ∪ S .(II) | F ( ϕ m + n − ( α )) | p < γ of F and a place q | p of K ( γ )such that r q ( ϕ m + n − ( α )) = r q ( γ ).Now the condition (I) implies r p ( ϕ m + n ( α )) = r q ( ϕ m ( α )). The conditions p / ∈ W n and | F ( ϕ m + n − ( α )) | p < d p ( ϕ m + ℓ ( α ) , ϕ m ( α )) ≥ ℓ < n . Hence r p ( ϕ m ( α )) has minimum period n . Finally when m > 0, by thedefinition of E and Lemma 4.22, the condition p / ∈ E implies that ϕ m − ( α ) is notperiodic modulo p . Therefore α modulo p has portrait ( m, n ). (cid:3) Proof of Theorem 1.3: large m and n . Let τ , s , and S be as in Theorem1.3. We now show that there are constants C ( τ, s ) and C ( τ, s ) such that forevery m ≥ C ( τ, s ), n ≥ C ( τ, s ), and α ∈ P ( K ) satisfying b h ϕ ( α ) ≥ τ , there isa place p ∈ Ω \ S such that α has portrait ( m, n ) modulo p . Combining this withPropositions 4.26 and 4.30, we finish the proof of Theorem 1.3.There is a constant C ( τ ) ≥ m ≥ C ( τ ) and every α ∈ P ( K ) satisfying b h ϕ ( α ) ≥ τ , we have ϕ m ( α ) = ∞ and:(4.34) ϕ − ( ϕ m ( α )) contains neither ∞ nor any ramification points of ϕ . ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 15 Note that the above conditions are satisfied when d m − τ is greater than the canon-ical heights of ∞ and of the ramification points of ϕ .Fix any m ≥ C ( τ ) and α ∈ P ( K ) satisfying b h ϕ ( α ) ≥ τ . Now, there are d − ≥ η i ∈ K such that η i = ϕ m − ( α ) and ϕ ( η i ) = ϕ m ( α ) for i = 1 , . . . , d − 1. Let e = d − F ( z ) = Q ei =1 ( z − η i ). Applying Proposition 4.15and using the fact that | h K − b h ϕ | is uniformly bounded, we have that there existconstants C and C such that the following inequality holds for every positiveinteger n : { p : v p ( F ( ϕ m + n − ( α ))) > } ≥ b h ϕ ( ϕ m + n − ( α )) − C e X i =1 b h ϕ ( η i ) − C = b h ( α ) (cid:0) d m + n − − eC d m − (cid:1) − C . (4.35)Hence there is a constant C ( τ, s ) such that for every n > C ( τ, s ), we have:(4.36) { p ∈ Ω K \ S : v p ( F ( ϕ m + n − ( α ))) > } ≥ b h ϕ ( α ) d m + n − Let L be the splitting field of K . We now argue as in Remark 4.4 as follows. LetΓ be the set of places p ∈ Ω K such that there are some q | p in Ω L and some root η i of F satisfying | η | q > 1. Then we have:(4.37) ≤ e X i =1 h K ( η i ) ≤ e b h ϕ ( α ) d m − + O (1)where O (1) only depends on K and ϕ . As in Remark 4.4, if p / ∈ Γ, ϕ has goodreduction at p , and if | F ( ϕ m + n − ( α )) | p < η i of F , anda place q of L lying above p such that r q ( η i ) = r q ( ϕ m + n − ( α )). This implies r p ( ϕ m ( α )) = r p ( ϕ m + n ( α )). Therefore, from (4.36) and (4.37) there is a constant C ( τ, s ) > C ( τ, s ) such that for every n > C ( τ, s ), we have:(4.38) { p ∈ Ω K \ S : r p ( ϕ m + n ( α )) = r p ( ϕ m ( α )) } ≥ b h ϕ ( α ) d m + n − Let E be the set of primes p ∈ Ω K such that there are a prime q | p of L and aroot η i of F satisfying r q ( ϕ ( η i )) = r q ( ϕ m − ( α )).If r q ( ϕ ( η i )) = r q ( ϕ m − ( α )) then we have either v q ( ϕ m − ( α )) < v q ( ϕ ( η i ) − ϕ m − ( α )) > 0. Therefore: E ≤ h K ( ϕ m − ( α )) + e X i =1 h K ( ϕ ( η i ) − ϕ m − ( α )) ≤ h K ( ϕ m − ( α )) + e X i =1 ( h K ( ϕ ( η i )) + h K ( ϕ m − ( α )))= ( e + 1) b h ϕ ( α ) d m − + e b h ϕ ( α ) d m − + O (1)(4.39)where O (1) depends only on K and ϕ .Hence there is C ( τ, s ) such that for every n > C ( τ, s ),we have:(4.40) E ≤ b h ϕ ( α ) d m + n − . Let E be the set of primes p of good reduction such that r p ( ϕ n ′ ( ϕ m ( α ))) = r p ( ϕ m ( α )) for n ′ a proper divisor of n . As before, we either have v p ( ϕ m ( α )) < v p ( ϕ m + n ′ ( α ) − ϕ m ( α )) > 0. Hence, we have(4.41) E ≤ h K ( ϕ m ( α )) + X p | n h K (cid:16) ϕ n/p ( ϕ m ( α )) − ϕ m ( α ) (cid:17) where p ranges over the distinct prime factors of n . There is an absolute constant C such that for all n ≥ C , the number of distinct prime factors of n is less thanlog n . Then for n ≥ C : E ≤ b h ϕ ( ϕ m ( α )) + X p | n (cid:16)b h ϕ (cid:16) ϕ n/p ( ϕ m ( α )) (cid:17) + b h ϕ ( ϕ m ( α )) (cid:17) + C log( n ) ≤ (log n + 1) b h ϕ ( α ) d m + (log n ) b h ϕ ( α ) d m + n + C log n (4.42)where C depends only on K and ϕ .Since d n dominates both (log( n )) d n/ and (log n + 1) when n grows sufficientlylarge (and independently from m ), there exists a constant C > C dependingonly on K and ϕ such that for n > C , we have:(4.43) E ≤ b h ϕ ( α ) d m + n − For m > C ( τ ) and for n > max { C ( τ, s ) , C ( τ, s ) , C } , from (4.38), (4.40),and (4.43) there exist at least b h ϕ ( α ) d m + n − many primes p ∈ Ω K \ S such thatthe following conditions hold:(I) r p ( ϕ m + n ( α )) = r p ( ϕ m ( α )).(II) p / ∈ E ∪ E .Condition (I) together with p / ∈ E imply that ϕ m ( α ) has minimum period n under the action of ϕ modulo p . Condition p / ∈ E together with Lemma 4.22 implythat ϕ m − ( α ) is not periodic modulo p . Hence α has portrait ( m, n ) modulo p ,which finishes the proof of Theorem 1.3.5. Proof of the applications of Theorem 1.3 Using Theorem 1.3 we can prove now its applications. First we prove Theo-rem 2.1, and then we will prove Theorems 2.4 and 2.2.5.1. Simultaneous multiple portraits. We begin with a few simple lemmas. Lemma 5.1. Let ϕ ( z ) ∈ K ( z ) be a rational function of degree d > and let α ∈ P ( K ) be a preperiodic point with portrait ( m, n ) . Then for all but finitelymany places p ∈ Ω K of good reduction, α modulo p has portrait ( m, n ) .Proof. Let E be the set of places p ∈ Ω K of good reduction such that the followingtwo conditions hold:(i) If m > 0, we have r p ( ϕ m − ( α )) = r p ( ϕ m + n − ( α )).(ii) For some prime divisor ℓ of n , we have r p ( ϕ m + nℓ ( α )) = r p ( ϕ m ( α )).By (3.1), E is finite. By Lemma 4.22, for every place p ∈ Ω K \ E , we have α modulo p has portrait ( m, n ). (cid:3) We have the following lemma for determining when polynomials in normal formare isotrivial. ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 17 Lemma 5.2. Let k be an algebraically closed field of characteristic , and let K be a finitely generated function field over k of transcendence degree equal to . Let ϕ ( z ) = z d + a d − z d − + · · · + a ∈ K [ z ] where d ≥ . Then ϕ is isotrivial if andonly if ϕ ∈ k [ z ] .Proof. Suppose that σ − ◦ ϕ ◦ σ ∈ k ( z ). Let ˜ ϕ denote σ − ◦ ϕ ◦ σ . Then h ϕ ( x ) = h ˜ ϕ ( σ − ( x )) = h ( σ − ( x )) for all x . Let β denote the point at infinity. Then h ( σ − ( β )) = 0, so β ∈ P ( k ). Hence, after composing σ with a degree one ele-ment of k ( z ), we may suppose that σ ( β ) = β , which means that σ is a polynomial b z + b ∈ K [ z ]. Since σ − ◦ ϕ ◦ σ ∈ k ( z ) = b d − z d + db d − b z d − + lower order terms , we see that b ∈ k and that b must therefore be in k as well. Thus, σ ∈ k [ z ], so ϕ ∈ k [ z ]. (cid:3) The following lemma is crucial for the proof of Theorem 2.1. Lemma 5.3. Let k be an algebraically closed field of characteristic , let K be afinitely generated function field over k of transcendence degree equal to , let d ≥ be an integer and let m be an integer such that ≤ m ≤ d − . Let f ( z ) = z d + a d − z d − + · · · + a where (1) a i ∈ K for all i ; (2) a i ∈ k for i > m ; and (3) there is some j ≤ m such that a j ∈ K \ k .Then there are at most m distinct constants x ∈ k such that b h f ( x ) < d .Proof. By (3), there is some place p of K such that | a j | p > j ≤ m ; fixthis p . Take any x ∈ K such that | x | p ≥ max i | a i | p . Then, for all i ≤ d − 2, we have | a i x i | p ≤ | x i +1 | p < | x d | p , so | f ( x ) | p = | x | d p . By induction, we then have | f n ( x ) | p = | x | d n p for all n . Thus, in particular for any α ∈ k such that | f ( α ) | p ≥ max i | a i | p , wehave b h f ( α ) = 1 d b h f ( f ( α )) ≥ d . Thus, it suffices to show that there are at most m constants x ∈ k such that | f ( x ) | p < max i | a i | p . Let N = − min i v p ( a i ); then N > 0. Let π ∈ K be agenerator for the maximal ideal p . Then it suffices to show that at there are atmost m constants x ∈ k such that | π N f ( x ) | p < 1. Now, for each a i , we have that π N a i is in the local ring at p . We let b i denote the image of π N a i in the residuefield k p of p which is canonically isomorphic to k . If | π N f ( x ) | p < x ∈ k , thenwe have(5.4) b m x m + · · · + b = 0since b i = 0 for all i > m by (2) (note that N ≥ b i = 0 for some i ≤ m ,we see that (5.4) has at most m solutions x , and our proof is complete. (cid:3) Proof of Theorem 2.1. When d = 2, we have f ( z ) = z + a defined over thefunction field k ( a ). By Corollary 4.25 and the Kisaka’s classification [14], we havethat X ( f ) = ∅ . For every constant c ∈ k , there does not exist a positive integer m such that f m ( c ) = 0, hence the set Y ( f, c ) is empty. By Lemma 5.3, b h f ( c ) ≥ / From now on, assume d ≥ 3. Every polynomial of degree d in normal form whosecoefficient a d − is nonzero is not totally ramified at any point (other than ∞ ). Wewill repeatedly use this observation and apply Theorem 1.3 (see also Corollary 1.6).We prove Theorem 2.1 by showing inductively the existence of the a i ’s realizing theportraits for the c i ’s. The a i ’s will be first independent variables, and then we makea series of specializations of the a i ’s which we call in turn a i, , a i, , · · · , until wespecialize all the variables to values in k .First, we let k := k ( a , a , . . . , a d − ), and we apply Theorem 1.3 (see Corol-lary 1.6) to the polynomial f ( z ) = z d + a d − z d − + · · · + a z + a defined over the function field K := k ( a ) with the starting point c . We notethat by Lemma 5.3, we know that b h f ( c ) ≥ d . Take the exceptional set of of places S to be the set containing only the place “at infinity” of K = k ( a ) which is theonly pole of a . Hence we obtain the existence of a co-finite set Z (0) ⊂ Z ≥ × N ofportraits such that for all ( m , n ) ∈ Z (0) , there exists a , ∈ k such that c hasportrait ( m , n ) with respect to f ( z ) := z d + a d − z d − + · · · + a z + a z + a , . This is the polynomial f obtained after specializing a to a , ∈ k . This special-ization is equivalent with reducing f modulo a place of K . Fix ( m , n ) ∈ Z (0) and a corresponding a , .Next we let k := k ( a , . . . , a d − ) and we regard f ( z ) above as a polynomialdefined over the function field K := k ( a , a , ) (note that trdeg k K = 1 because a , ∈ k ( a )). Applying Lemma 5.3 to f ( z ) we conclude that there exists atmost one constant point c ∈ k such that b h f ( c ) < d . Since, by construction, c ∈ k ⊂ k is preperiodic of portrait ( m , n ) for f , we must have b h f ( c ) ≥ d .Let the exceptional set of places S consist of places p of K where a or a , has apole, or when c does not have portrait ( m , n ) modulo p . The set S is finite byLemma 5.1. Therefore we can apply Theorem 1.3 (see Corollary 1.6) and obtain aco-finite set Z (1) ⊂ Z ≥ × N of portraits such that for each ( m , n ) ∈ Z (1) , thereexists a , ∈ k (and in turn a , ∈ k ) such that c has portrait ( m , n ) and c has portrait ( m , n ) under the action of f ( z ) := z d + a d − z d − + · · · + a z + a , z + a , . This comes from reducing a and a , modulo a place in K outside S . Fix( m , n ) ∈ Z (1) and also fix corresponding a , and a , .The above process could be done inductively as follows. Let i ∈ { , . . . , d − } ,and assume we previously found Z (0) , Z (1) ,..., Z ( i − and fixed ( m j , n j ) ∈ Z ( j ) for 0 ≤ j ≤ i − a ,i − , a ,i − , . . . , a i − ,i − ∈ k i − where k i − := k ( a i , . . . , a d − ) such that the following hold. Let k i := k ( a i +1 , . . . , a d − ) with theunderstanding that k i = k when i = d − 2. Let f i ( z ) := z d + a d − z d − + . . . + a i z i + a i − ,i − z i − + . . . + a ,i − z + a ,i − , which is a polynomial defined over the function field K i := k i ( a i , a i − ,i − , . . . , a ,i − ).We now have that for 0 ≤ j ≤ i − 1, the point c j has portrait ( m j , n j ) under theaction of f i . ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 19 Lemma 5.3 now asserts that there are at most i constants c ∈ k i such that b h f i ( c ) < d . Since c , . . . , c i − are such constants, we must have b h f i ( c i ) ≥ d . Let S i be the set of places p of K i such that a i has a pole, or for some 0 ≤ j ≤ i − a j,i − has a pole, or for some 0 ≤ j ≤ i − c j modulo p does nothave portrait ( m j , n j ). This set S i is finite by Lemma 5.1. By Theorem 1.3 (seealso Corollary 1.6), there exists a co-finite set Z ( i ) ⊂ Z ≥ × N of portraits such thatfor every ( m i , n i ) ∈ Z ( i ) there exist a ,i , . . . , a i,i ∈ k i satisfying the following. For0 ≤ j ≤ i , the point c j has portrait ( m j , n j ) under: f i +1 ( z ) := z d + a d − z d − + . . . + a i +1 z i +1 + a i,i z i + . . . + a ,i z + a ,i . We continue the above process until i = d − 2, which finishes the proof of Theo-rem 2.1. (cid:3) Almost any portrait is realized by almost any starting point. Let ϕ ( z ) ∈ K ( z ) having degree d ≥ 2. In the proof of Theorem 2.4 we will use thefollowing easy fact. Lemma 5.5. Let ϕ ( z ) ∈ K ( z ) be non-isotrivial. Then the set { α ∈ P ( K ) : Y ( ϕ, α ) = ∅} is finite.Proof. We use the following two properties following from the fact that ϕ is non-isotrivial (see [1]):(i) The set Prep ϕ ( K ) of preperiodic points in P ( K ) is finite.(ii) There is a positive lower bound τ for the canonical height of points in P ( K ) \ Prep ϕ ( K ).Let R be the finite set (possibly empty) of points in P ( K ) where ϕ is totallyramified at. There exists M such that for every m > M and every α ∈ P ( K ) \ Prep ϕ ( K ), ϕ is not totally ramified at ϕ m ( α ). To see this, we simply require that b h ϕ ( ϕ m ( α )) > d M τ is greater than the canonical height of any point in R . Thenthe given set in the lemma is contained in the finite set:Prep ϕ ( K ) ∪ R ∪ ϕ − ( R ) . . . ∪ (cid:0) ϕ M (cid:1) − ( R ) . (cid:3) Proof of Theorem 2.4. Let T be the finite set of points in P ( k ) consisting of eitherpreperiodic points or the points in the set in Lemma 5.5. We now have Y ( ϕ, α ) = ∅ for every α ∈ P ( k ) \ T . Note that if ( m, n ) / ∈ W ( ϕ ) then n / ∈ X ( ϕ ) (see Remark2.5).There is a lower bound τ on the canonical heights of points in P ( k ) \ T (see[1]). By Theorem 1.3, there is a finite set Z ( τ, | S | ) such that for every ( m, n ) ∈ ( Z ≥ × N ) \ ( Z ( τ, | S | ) ∪ W ( ϕ )) and for every α ∈ P ( k ) \ T there exists a place p / ∈ S such that α has portrait ( m, n ) modulo p .Hence it suffices to fix an ( m, n ) ∈ Z ( τ, | S | ) \ W ( ϕ ) and prove that there existsa finite subset T of P ( k ) (possibly depending on K , ϕ , ( m, n ), and S ) such thatthe following holds. For every α ∈ P ( k ) \ T , there exists a place p / ∈ S such that α has portrait ( m, n ) modulo p .Now let C be a nonsingular projective curve over k whose function field is K .We identify places of K with points in C . Choose a Zariski open subset V of C such that V ⊆ C \ S and ϕ extends to a morphism from P k × k V to itself. For every ( µ, η ) ∈ Z ≥ × N , the equation ϕ µ + η ( z ) = ϕ µ ( z ) defines a Zariski closed subset V µ,η of P k × k V which is equidimensional of dimension 1. Define: U = V m,n \ m − [ µ =0 V µ,n ∪ [ p | n V m,n/p where p ranges over all prime factors of n . We have that U is a Zariski open subsetof V m,n . Since ϕ has a point of portrait ( m, n ), the set U is non-empty.Let ρ denote the projection from P k × k V to P k . We prove that the image ρ ( U ) cannot be a finite subset of P k . Assume otherwise, say ρ ( U ) = { u , . . . , u r } .Then this implies that all points of portrait ( m, n ) under ϕ are the constant points u , . . . , u r contradicting the assumption ( m, n ) / ∈ W ( ϕ ). Hence ρ ( U ) is infinite.Since ρ ( U ) is constructible in P k by Chevalley’s theorem, we must have that ρ ( U )is co-finite in P k .Now for every α ∈ ρ ( U ), pick any P ∈ ρ − ( α ), and let p ∈ V be the image of P under the projection from P k × k V to V . We have that α has portrait ( m, n ) underthe action of ϕ modulo p . This finishes the proof of Theorem 2.4. (cid:3) We now prove of Theorem 2.2, which in turn relies on Theorem 2.4. Proof of Theorem 2.2. The proof uses an inductive process which is “dual” to theproof of Theorem 2.1. First, we let k := k ( a , a , . . . , a d − ), and we apply Theo-rem 2.4 to the polynomial f ( z ) = z d + a d − z d − + · · · + a z + a defined over the function field K := k ( a ). Then Lemma 5.2 shows that f isnon-isotrivial. By Lemma 5.3, there does not exist c ∈ k which is preperiodic under f . Therefore by Remark 2.5 and Kisaka’s list [14], we have W ( f ) = ∅ , hence( m , n ) / ∈ W ( f ). Take the exceptional set of of places S to be the set containingonly the place “at infinity” of K = k ( a ) which is the only pole of a . Hence weobtain the existence of the co-finite set T (0) ⊂ P ( k ) such that for all c ∈ T (0) ,there exists a , ∈ k such that c has portrait ( m , n ) under the action of f ( z ) := z d + a d − z d − + · · · + a z + a z + a , . This is the polynomial f obtained after specializing a to a , ∈ k . This special-ization is equivalent with reducing f modulo a place of K . Fix c ∈ T (0) and acorresponding a , .The above process could be done inductively as follows. Let i ∈ { , . . . , d − } ,and assume we previously found co-finite sets T (0) , T (1) ,. . . , T ( i − ⊂ P ( k ) andfixed c j ∈ T ( j ) for 0 ≤ j ≤ i − a ,i − , a ,i − , . . . , a i − ,i − ∈ k i − where k i − := k ( a i , . . . , a d − ) such that the following hold. Let k i := k ( a i +1 , . . . , a d − )with the understanding that k i = k when i = d − 2, and let f i ( z ) := z d + a d − z d − + . . . + a i z i + a i − ,i − z i − + . . . + a ,i − z + a ,i − , which is as a polynomial defined over the function field K i := k i ( a i , a i − ,i − , . . . , a ,i − ).Also, for each j = 0 , . . . , i − 1, the point c j has portrait ( m j , n j ) under the actionof f i .Lemma 5.3 now asserts that there are at most i constants c ∈ k i such that b h f i ( c ) < d . Since c , . . . , c i − are such constants and since ( m i , n i ) is distinct from( m , n ) , . . . , ( m i − , n i − ), there exists no c ∈ k such that c has portrait ( m i , n i ) ORTRAITS OF PREPERIODIC POINTS FOR RATIONAL MAPS 21 under f i . Therefore we have ( m i , n i ) / ∈ W ( f i ). Let S i be the set of places p of K i such that a i has a pole, or for some 0 ≤ j ≤ i − a j,i − has a pole, orfor some 0 ≤ j ≤ i − c j does not have portrait ( m j , n j ) under the actionof f i modulo p . This set S i is finite by Lemma 5.1. By Theorem 1.3, there exists aco-finite set T ( i ) ⊂ P ( k ) such that for every c i ∈ T ( i ) there exist a ,i , . . . , a i,i ∈ k i satisfying the following. For 0 ≤ j ≤ i , the point c j has portrait ( m j , n j ) under theaction of f i +1 ( z ) := z d + a d − z d − + . . . + a i +1 z i +1 + a i,i z i + . . . + a ,i z + a ,i . Continuing the above process until i = d − 2, we finish the proof of Theorem 2.2. (cid:3) We conclude this section by proving Corollary 2.6. Proof of Corollary 2.6. As in the proof of Theorem 2.2, let k = k ( a ), K = k ( b ).We apply Theorem 2.4 to obtain a co-finite subset U (1) of k such that for every c ∈ U (1) , the following holds. For every ( m , n ) ∈ Z ≥ × N , there exists b ∈ k ( a )such that c has portrait ( m , n ) under: ϕ c ,m ,n ( z ) := z + az + b regarded as a polynomial in K [ z ]. Here K := k ( a, b ) is a function field over k := k .We claim that W ( ϕ c ,m ,n ) is empty. By Lemma 5.3, c is the only constantpreperiodic point of ϕ c ,m ,n . Hence for every portrait ( m , n ) = ( m , n ), wehave ( m , n ) / ∈ W ( ϕ c ,m ,n ). It suffices to show ( m , n ) / ∈ W ( ϕ c ,m ,n ) byproving that ϕ c ,m ,n has a point γ = c of portrait ( m , n ). The case n > m = 0 we pick γ = ϕ c ,m ,n ( c ), while if m > γ = c suchthat ϕ c ,m ,n ( γ ) = ϕ c ,m ,n ( c ). Note that this is possible since ϕ c ,m ,n has nototally ramified point other than infinity. We now consider the case n = 1. Sincethe polynomial: ϕ c ,m ,n ( z ) − z = z + ( a − z + b is not the cube of a linear polynomial in K [ z ], we have that ϕ c ,m ,n has at leasttwo distinct points α , α having portrait (0 , ϕ has no totallyramified point (other than infinity) and looking at appropriate backward orbits of α and α , we get at least 2 points having portrait ( m , W ( ϕ c ,m ,n ) = ∅ .Let S ( c , m , n ) be the set of places p of K such that ϕ c ,m ,n has bad reduc-tion at p or c does not have portrait ( m , n ) modulo p . Then Applying Theo-rem 2.4, we see then that for each ( m , n ) there is a co-finite set U ( c , m , n , m , n )such that for all c ∈ U ( c , m , n , m , n ), there is a polynomial f ( z ) = z +˜ az +˜ b ∈ k [ z ] such that, for i = 1 , c i has portrait ( m i , n i ) under f . Define: U (2) ( c ) := \ (( m ,n ) , ( m ,n )) U ( c , m , n , m , n )which is a co-countable subset of k . From our construction, the sets U (1) and U (2) ( c ) for every c ∈ U (1) satisfy the assertion in Corollary 2.6.For the second assertion in the corollary, we simply pick the elements ( c , c ) ∈ k satisfying c ∈ U (1) and c ∈ U (2) ( c ). (cid:3) Future directions One might ask if something much stronger than Theorem 2.1 and Corollary 2.6is true. It is possible that if d ≥ 3, then for any distinct points c , . . . , c d − ∈ C and any (not necessarily distinct) ( m , n ) , . . . , ( m d − , n d − ) ∈ Z ≥ × N , there is apolynomial f ( z ) = z d + a d − z d − + · · · + a ∈ C [ z ] such that c i has portrait ( m i , n i )under f . We know of no counterexamples. Note, however, that in the case d = 2,there is no polynomial f ( z ) = z + a such that 0 has portrait (1 , m ); this followsimmediately from the fact that the general quadratic z + t ramifies completely at0. Since for d ≥ 3, the general degree d polynomial in normal form has no totallyramified points (other than the point at infinity), this particular example has noanalog in degree greater than 2. On the other hand, it is also true that here is nopolynomial f ( z ) = z + a such that − / , k ( a , . . . , a d − ). Oneissue that arises here is the possibility that some c i is an iterate of another c j underthe general degree d polynomial, something that cannot happen when all of the c i are in k .The multi-portrait problem studied here was inspired by work of Douady, Hub-bard, and Thurston [6], who treated the problem of portraits of critical points ofrational functions. Their work yielded not only existence results, but also informa-tion about finiteness (up to change of variables) and transversality (of intersectionsof hypersurfaces corresponding to portraits of marked critical points). We hope totreat constant point analogs of these results in future work. References [1] M. 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Dragos Ghioca, Department of Mathematics, University of British Columbia, Van-couver, BC V6T 1Z2, Canada E-mail address : [email protected] Khoa Nguyen Department of Mathematics, University of British Columbia, Vancou-ver, BC V6T 1Z2, Canada E-mail address : [email protected] Thomas Tucker, Department of Mathematics, University of Rochester, Rochester,NY 14627, USA E-mail address ::