aa r X i v : . [ m a t h . N T ] J un STOCHASTIC CANONICAL HEIGHTS
VIVIAN OLSIEWSKI HEALEY AND WADE HINDES
Abstract.
We construct height functions defined stochastically on projective varietiesequipped with endomorphisms, and we prove that these functions satisfy analogs of theusual properties of canonical heights. Moreover, we give a dynamical interpretation of thekernel of these stochastic height functions, and in the case of the projective line, we relatethe size of this kernel to the Julia sets of the original maps. Finally, as an application, weestablish the finiteness of some generalized Zsigmondy sets over global fields. Introduction
The canonical height [3] associated to a smooth projective variety V equipped with anendomorphism φ : V → V is an indispensable tool for studying the arithmetic properties ofthe corresponding discrete dynamical system ( V, φ ). However, many varieties of interest innumber theory (e.g. projective space) possess many such maps, and given that compositionis not commutative in general, the dynamical systems generated by a set of maps can differgreatly from the dynamics of a single map. In this paper, we address this problem andconstruct a new height function that measures the collective action of a set of maps on afixed variety.To accurately characterize the dynamics of a set of maps, one must encode “how often”to expect a particular map to appear at any stage of composition - a slight alteration inthe likelihood of applying a particular map can drastically change the overall dynamics.Moreover, there is no intrinsic reason why all maps should be given equal weight. Wetherefore use the language and tools of probability in our constructions. In so doing, we arein essence analyzing a sort of random walk on the variety (formally a Markov chain), wherethe stochastic motion is generated by random evaluation of maps in some fixed set.
Example . To make this clear, the reader is encouraged to keep in mind the two maps anda fair coin example: suppose that we have two maps S = { φ , φ } on a variety φ i : V → V .Then for any given point P ∈ V , we can flip a coin to determine whether to evaluate φ or φ at P , assigning each outcome an equal probability of 1 /
2. Now repeat this process forthe new point φ ( P ) or φ ( P ) and continue inductively in this way, associating to an infinitesequence of coin flips (a type) of orbit of P .If V is equipped with a height function, then we can ask how the height of P grows aswe move along a particular path. Even more broadly, we can ask about the height growthdistribution as we vary over all possible paths (each path weighted by its probability). Tomake this idea precise, we fix some notation. Let K be a global field, let V /K be a smoothprojective variety over K , and let S be a (finite or infinite) set of endomorphisms on V Mathematics Subject Classification : Primary: 11G50, 37P15. Secondary: 14G05.The first author was partially supported by NSF grant DMS-1246999. defined over K . To define the orbits we will consider, letΦ S,n = n Y i =1 S and Φ S = ∞ Y i =1 S be the set of n -term (and infinite) sequences of elements of S respectively. Given an infinitesequence γ = ( θ n ) n ≥ ∈ Φ S and a positive integer m ≥
1, we let γ m = ( θ i ) mi =1 ∈ Φ S,m anddefine an action of γ m on V by γ m · P = ( θ m ◦ θ m − ◦ · · · ◦ θ )( P ) for P ∈ V .In this way, we define the orbit of a point P ∈ V with respect to a sequence γ ∈ Φ S to be:Orb γ ( P ) = { γ m · P : m ≥ } . Finally, if ν is a probability measure on S , then we define a probability measure ν m on Φ S,m by the product ν m ( γ m ) = m Y i =1 ν ( θ i ), for γ m = ( θ i ) mi =1 .That is, each γ m ∈ Φ S,m is a sequence of m elements of S , and each component of γ m ischosen independently according to ν . Likewise, ν induces a probability measure ν on theset of infinite sequences Φ S ; see [16, Theorem 10.4]. We call (Φ S , F , ν ) the probability spaceof i.i.d sequences of elements of S distributed according to ν ; here F is the σ -algebra of ν -measurable subsets of Φ S .Now for a brief discussion of the relevant material on canonical heights. Let η ∈ Pic( V ) ⊗ R be a divisor class and let h V,η : V ( K ) → R be a corresponding Weil height function; see, forinstance, [17, § § φ : V → V , one requiresthat η is an eigenclass for φ ; the key point in this case is that(1) h V,η ◦ φ = α φ h V,η + O V,η,φ (1)for some α φ ∈ R . With this in mind, we let C ( V, η, φ ) := sup P ∈ V (cid:12)(cid:12)(cid:12) h V,η ( φ ( P )) − α φ h V,η ( P ) (cid:12)(cid:12)(cid:12) be the smallest constant needed for the bound in (1). Then, in order to generalize theconstruction of canonical heights for a single map to a collection of maps (equivalently, fromconstant sequences to arbitrary sequences), we define the following fundamental notion. Definition.
A set of endomorphisms S on a projective variety V is height controlled withrespect to a divisor class η ∈ Pic( V ) ⊗ R if:(1) For all φ ∈ S , there exists α φ such that: φ ∗ ( η ) = α φ η and inf φ ∈ S α φ > φ ∈ S C ( V, η, φ ) is finite.These properties are easily satisfied for any projective space and any finite set S ; however,see Example 2.7 and Remark 2.8 for instances of infinite S . We now state our main con-struction. In what follows, for a finite sequence γ m = ( θ i ) mi =1 ∈ Φ S,m , we define the degreedeg η ( γ m ) = Q mi =1 α θ i , in what we hope is a pardonable (and instructive) abuse of notation. TOCHASTIC CANONICAL HEIGHTS 3
Theorem 1.2.
Let K be a global field, let V /K be a smooth projective variety over K , andlet S be a collection of endomorphisms on V equipped with the following: (1) A common eigendivisor class η ∈ Pic( V ) ⊗ R such that S is height controlled withrespect to η . (2) A probability measure ν on S .Let (Φ S , F , ν ) be the probability space of i.i.d sequences of elements of S distributed accordingto ν , and let h V,η be a Weil height function corresponding to η . Then for all infinite sequences γ ∈ Φ S and all points P ∈ V ( K ) , the canonical height , ˆ h V,η,P ( γ ) := lim n →∞ h V,η ( γ n · P )deg η ( γ n ) , converges. Likewise, the expected canonical height at P for a random γ , E ν (cid:2) ˆ h V,η (cid:3) ( P ) := Z Φ S ˆ h V,η,P ( γ ) dν, converges. Let d ν,η be the (deterministic) constant given by d ν,η := (cid:18) X φ ∈ S ν ( φ )deg η ( φ ) (cid:19) − , and let ν ∗ k be the new probability measure on Φ S,k induced by the pair ( ν , η ) and given by ν ∗ k ( γ k ) := ν k ( γ k )deg η ( γ k ) ( d ν,η ) k . Then the function E ν (cid:2) ˆ h V,η (cid:3) : V ( K ) → R satisfies the following properties: (a) E ν (cid:2) ˆ h V,η (cid:3) = h V,η + O (1) . (b) E ν ∗ k h E ν (cid:2) ˆ h V,η (cid:3) ( γ k · P ) i = ( d ν,η ) k E ν (cid:2) ˆ h V,η (cid:3) ( P ) for all k ≥ and all P ∈ V ( K ) .Remark . The canonical heights ˆ h V,η,P ( γ ), also denoted ˆ h V,η,γ ( P ) depending on whetherwe vary the path γ ∈ Φ S or the basepoint P ∈ V , were also studied in [12], and Theorem 1.2can be viewed as a generalization of [12, Proposition C] in two ways: we allow the generatingset of functions S to be infinite (under suitable conditions), and we allow arbitrary probabilitymeasures on S .There are several reasons why we believe that the expected canonical height E ν (cid:2) ˆ h V,η (cid:3) isthe right height function to study the collective dynamics of the maps in S (by analogy withthe standard canonical height of Call-Silverman). The first reason is that E ν (cid:2) ˆ h V,η (cid:3) = ˆ h V,η,φ whenever S = { φ } is a singleton with trivial probability measure. The second reason is that E ν (cid:2) ˆ h V,η (cid:3) satisfies a transformation law of similar shape to that of the standard canonicalheight. The third reason is that E ν (cid:2) ˆ h V,η (cid:3) detects finite S -invariant subsets of V , an analog ofpreperiodic points of a fixed map. In what follows, we say that a subset F ⊂ V is S -stable if φ ( F ) ⊆ F for all φ ∈ S . Moreover, we say that the probability measure ν is strictly positive if ν ( φ ) > φ ∈ S . VIVIAN O. HEALEY AND WADE HINDES
Corollary 1.4.
Let ( V, η, S, ν ) satisfy the conditions of Theorem 1.2. If η is an ampledivisor and ν is strictly positive, then for all P ∈ V ( K ) the following are equivalent: (1) There is a finite, S -stable subset F P ⊂ V containing P . (2) ν (cid:0) { γ ∈ Φ S : Orb γ ( P ) is finite } (cid:1) = 1 . (3) E ν [ˆ h V,η ]( P ) = 0 . With this characterization of the kernel of E ν [ˆ h V,η ] in place, one expects that few points(perhaps at most finitely many) have expected height zero unless the maps in S are dy-namically dependent in some way. As an example of this heuristic, we note the followingapplication of [1, Theorem 1.2] and Corollary 1.4 for the projective line. In what follows,given a rational function φ ∈ Q ( z ), we let PrePer( φ ) denote the set of preperiodic points of φ in P ( Q ). Corollary 1.5.
Let S be a collection of height controlled rational maps on P defined over anumber field K , let ν be a strictly positive probability measure on S , and let h be the absoluteWeil height function on P ( Q ) . Then the following are equivalent: (1) (cid:8) P ∈ P ( Q ) : E ν [ˆ h ]( P ) = 0 (cid:9) is infinite. (2) \ φ ∈ S PrePer( φ ) is infinite. (3) PrePer( φ ) = PrePer( ψ ) for all φ, ψ ∈ S .In particular, if the set of points of expected height zero is infinite, then the Julia sets of allof the maps in S coincide.Remark . For example, if c , c , . . . , c n ∈ Q are distinct algebraic numbers and S is thefinite set of quadratic polynomials, S = { x + c i : 1 ≤ i ≤ n } , equipped with any any strictly positive probability measure, then [2, § .4] and Corollary 1.5imply that (cid:8) P ∈ P ( Q ) : E ν [ˆ h ]( P ) = 0 (cid:9) is finite. On the other hand, it is tempting to guess that there are in fact no points ofexpected height zero for the sets S above. However, this is not true in general, as theexample: S = { x , x − } and the point P = −
1, shows. Therefore, given a generating set S whose elements do not all share the same Julia set, it is perhaps an interesting problem todetermine the finitely many points (over Q ) of expected height zero. Keep in mind that thisis not the same problem as studying the intersection of the preperiodic points of the mapsin S , as the example: S = { x − , x + x } and P = 0, shows.Moreover, when V = P and S is finite, we show that the canonical and expected canonicalheights defined in Theorem 1.2 admit a decomposition into a sum of local heights:(2) ˆ h γ = 1deg( E ) X v ∈ M K n v ˆ λ v, ˜ γ,E and E ν (cid:2) ˆ h (cid:3) = 1deg( E ) X v ∈ M K n v E ν (cid:2) ˆ λ v, ˜ γ,E (cid:3) ; TOCHASTIC CANONICAL HEIGHTS 5 see Theorem 4.3 and Corollary 4.5 in Section 4. We note that similar local Green functionsto those we use in Section 4 seem to have first appeared in [12, § λ v, ˜ γ,E and the decompositions in (2) do not appear in [12]. Furthermore, since we use ˆ λ v, ˜ γ,E and(2) to pose some new questions in number theory from a probabilistic point of view, we carryout our construction of Green functions, instead of just citing [12, § K be a global field, let V = P , andlet S be a finite set of endomorphisms of V . Given a sequence γ ∈ Φ S and a basepoint P ∈ P ( K ), we say that a valuation v of K is a primitive prime divisor of γ n · P if:(a) v ( γ n · P ) > v ( γ m · P ) ≤ ≤ m ≤ n − S are special in some way. To test this heuristic,we define the Zsigmondy set of a pair ( γ, P ) ∈ Φ S × P ( K ) to be: Z ( γ, P ) := { n : γ n · P has no primitive prime divisor } . In particular, when K is a number field we use the abc -conjecture and ideas from [6] and [7]to bound the size of the elements of Z ( γ, P ) under certain mild conditions on the maps in S and the basepoints P ∈ P ( K ). Specifically, we restrict our attention to (good) pairs ( γ, P )in the following sense: G S := n ( γ, P ) ∈ Φ S × P ( K ) : ˆ h γ ( P ) > , ∞ 6∈ Orb γ ( P ) ∪ Orb γ (0) o . Remark . Clearly, Z ( γ, P ) is infinite whenever Orb γ ( P ) is finite (i.e., ˆ h γ ( P ) = 0); seeRemark 2.3. Moreover, certain technicalities arise in our argument when the orbits weconsider contain 0 or ∞ . However, it is possible that these latter conditions can be relaxed.In what follows, CritVal( S ) = S φ ∈ S CritVal( φ ) denotes the union of all critical values ofthe maps φ ∈ S ; see [20, p.353]. Moreover,PrePer(Φ S ) := (cid:8) Q ∈ P ( K ) : ˆ h ψ ( Q ) = 0 for some ψ ∈ Φ S (cid:9) denotes the set of points with finite orbit for some sequence in Φ S (a set of bounded height). Theorem 1.8.
Let K be a number field and suppose that S = { φ , φ , . . . , φ s } is a set ofrational maps on P of degree at least , defined over K , and satisfying: (1) φ − i (0) ≥ , (2) 0 CritVal( S ) .Then the abc -conjecture (3.1) implies that Z ( γ, P ) is finite for all ( γ, P ) ∈ G S . We also examine the case of global function fields of prime characteristic. However, sinceour methods are different in this context, the conditions we impose are also different. Inparticular, we are confined to polynomial dynamics. On the other hand, the result we obtainis unconditional. To formally state this result, let t be an indeterminate, let p be an oddprime, and let K/ F p ( t ) be a finite separable extension. Extending the usual derivative ddt on VIVIAN O. HEALEY AND WADE HINDES F p ( t ) to K via implicit differentiation, we let β ′ denote the derivative of β ∈ K . Given apolynomial φ ( x ) ∈ K [ x ] of degree d ≥
3, write φ ( x ) = A x d + A x d − + . . . A d − x + A d , A i ∈ K. Then we have the following important quantities (c.f. [9, Theorem 1.1]) associated to φ :(1) δ φ := 2 dA A − ( d − A ,(2) b φ := ( dA A ′ − ( d − A A A ′ − dA A A ′ + ( d − A A ′ ) /δ φ (3) f φ = ( d A A ′ − d ( d − A A A ′ − d ( d − A A A ′ + ( d ( d −
2) + 1) A A ′ ) /δ φ Generalizing [9, Theorem 1.1] to dynamical systems generated by a finite set of maps, weprove the following:
Theorem 1.9.
Let K/ F p ( t ) be a function field and suppose that S = { φ , φ , . . . , φ s } is aset of polynomials of degree at least , defined over K , and satisfying: (1) deg( φ ) deg( ψ ) δ φ δ ψ (cid:0) b φ − f ψ (cid:1) = 0 for all φ, ψ ∈ S , (2) 0 PrePer(Φ S ) .Then Z ( γ, P ) is finite for all ( γ, P ) ∈ G S .Remark . Theorem 1.9 is quite broadly applicable from the point of view of algebraicgeometry: let d = max { deg( φ ) : φ ∈ S } and view S as a point of ( A d +1 ) s by associating toeach polynomial in S its ( d + 1)-tuple of coefficients. Then the sets S satisfying condition(1) of Theorem 1.9 are Zariski dense in ( A d +1 ) s ; compare to [9, Remark 1.1]. Moreover,PrePer(Φ S ) ∩ P ( K ) is finite; see Lemma 2.2 or [12, Corollary B]. Acknowledgments:
We thank Patrick Ingram and Joseph Silverman for helpful conver-sations and for making us aware of previous work on random canonical heights in [12].2.
Expected Canonical Heights
To prove Theorem 1.2, we need a generalization of Tate’s telescoping argument for theusual canonical height. To do this, we recall that the functoriality of heights implies that h V,η ◦ φ = α φ h V,η + O V,φ (1) for φ ∈ S .In particular, we have the following bound for height controlled families: Lemma 2.1.
Let S be height controlled with respect to η , let C := sup (cid:8) O V,φ (1) (cid:9) , and let α := inf { α φ } . If ρ r ∈ Φ r for r ≥ , then (cid:12)(cid:12)(cid:12)(cid:12) h V,η ( ρ r ( Q ))deg η ( ρ r ) − h V,γ ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cα − for all Q ∈ V ( K ) . In particular, this bound is independent of ρ r , r , and Q .Proof. Suppose that ρ r = θ r ◦ θ r − · · · ◦ θ for θ i ∈ S , and let θ to be the identity map on V . Then define ρ i := θ i ◦ θ i − · · · ◦ θ ◦ θ for 0 ≤ i ≤ r. Note, that ρ = θ is the identity map. In particular, inspired by Tate’s telescoping argument,we rewrite TOCHASTIC CANONICAL HEIGHTS 7 (cid:12)(cid:12)(cid:12)(cid:12) h V,η ( ρ r ( Q ))deg η ( ρ r ) − h V,γ ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) r − X i =0 h V,η ( ρ r − i ( Q ))deg η ( ρ r − i ) − h V,η ( ρ r − i − ( Q ))deg η ( ρ r − i − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r − X i =0 (cid:12)(cid:12)(cid:12)(cid:12) h V,η ( ρ r − i ( Q ))deg η ( ρ r − i ) − h V,η ( ρ r − i − ( Q ))deg η ( ρ r − i − ) (cid:12)(cid:12)(cid:12)(cid:12) = r − X i =0 (cid:12)(cid:12)(cid:12) h V,η ( ρ r − i ( Q )) − deg η ( θ r − i ) h V,η ( ρ r − i − ( Q )) (cid:12)(cid:12)(cid:12) deg η ( ρ r − i ) ≤ r − X i =0 C deg η ( ρ r − i ) ≤ r X i =1 Cα i ≤ ∞ X i =1 Cα i = Cα − . (3)This completes the proof of Lemma 2.1. (cid:3) It now follows easily that the canonical height ˆ h V,η,P ( γ ) is well-defined and is uniformlybounded (as we vary possible paths γ ∈ Φ S ) by the the Weil height: Lemma 2.2.
Let P ∈ V ( K ) and let γ ∈ Φ . Then the canonical height, ˆ h V,η,P ( γ ) := lim n →∞ h V,η ( γ n · P )deg η ( γ n ) , is well defined. Moreover, | ˆ h V,η,P ( γ ) − h V,η ( P ) | ≤ Cα − for all P ∈ V ( K ) and γ ∈ Φ .Proof. This is a simple application of Lemma 2.1. Let γ = ( θ n ) ∞ n ≥ , and write γ r := θ r ◦ θ r − · · · ◦ θ for r > . Likewise, for n > m >
0, let ρ = ( γ n K γ m ) := θ n ◦ θ n − . . . θ m +1 . In particular, we see that (cid:12)(cid:12)(cid:12)(cid:12) h V,η ( γ n · P )deg η ( γ n ) − h V,η ( γ m · P )deg η ( γ m ) (cid:12)(cid:12)(cid:12)(cid:12) = 1deg η ( γ m ) (cid:12)(cid:12)(cid:12)(cid:12) h V,η ( ρ · ( γ m · P ))deg η ( ρ ) − h V,η ( γ m · P )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cα m ( α − . (4)Here we apply Lemma 2.1 to the map ρ and the basepoint Q := γ m · P , and we use thatdeg( γ m ) ≥ α m . Letting m grow arbitrarily large, we see that the distance between the n thand m th term of the sequence defining ˆ h V,η,P ( γ ) goes to zero. In particular, this sequenceis Cauchy and therefore converges. As for the bound | ˆ h V,η,P ( γ ) − h V,η ( P ) | ≤ Cα − , let m = 0and n → ∞ in (4). (cid:3) VIVIAN O. HEALEY AND WADE HINDES
Remark . As with the usual canonical height associated to ample divisors η , we have thatˆ h V,η,P ( γ ) = 0 if and only if Orb γ ( P ) is finite. This follows readily from Lemma 2.2 and thesimple identityˆ h V,η,γ m · P ( γ K γ m ) = deg( γ m ) ˆ h V,η,P ( γ ) for all γ ∈ Φ S and m ≥ γ K γ m := ( θ n ) n ≥ m +1 is the m th shift of γ = ( θ n ) n ≥ ∈ Φ S . In particular, if ˆ h V,η,P ( γ ) = 0then ˆ h V,η ( γ m · P ) is absolutely bounded by Lemma 2.2. In particular, the dynamical orbitOrb γ ( P ) = { γ m · P : m ≥ } is finite by Northcott’s Theorem when η is ample. On the otherhand, if Orb γ ( P ) is finite, then ˆ h V,η ( γ m · P ) is bounded and ˆ h V,η,P ( γ ) = 0 as claimed.We now study the expected value of these canonical height functions on the probabilityspace (Φ S , F , ν ) of i.i.d sequences of elements of S distributed according to ν ; see [16,Theorem 10.4]. In what follows, we suppress S and F in the notation and simply write Φand ν where appropriate. (Proof of Theorem 1.2). Fix P ∈ V ( K ) and consider the sequence of random variables h V,η,n,P : Φ → R defined by h V,η,P,n ( γ ) := h V,η ( γ n · P )deg η ( γ n ) . It follows from the definition of (Φ , ν ) that each h V,η,P,n is ν -measurable; see, for instance,[16, Theorem 10.4]. On the other hand, the random variable ˆ h V,η,P : Φ → R is the pointwiselimit of the { h V,η,P,n } n ≥ :ˆ h V,η,P ( γ ) = lim n →∞ h V,η,P,n ( γ ) for all γ ∈ Φ . Moreover, the functions { h V,η,P,n } n ≥ are absolutely bounded; see Lemma 2.1. Hence, theLebegue dominated convergence theorem [16, Theorem 9.1] implies that ˆ h V,η,P is integrableand that(5) E ν (cid:2) ˆ h V,η (cid:3) ( P ) := Z Φ ˆ h V,η,P ( γ ) dν = lim n →∞ Z Φ h V,η,P,n ( γ ) dν. Hence, E ν (cid:2) ˆ h V,η (cid:3) ( P ) is well defined for every P ∈ V ( ¯ K ). As for properties ( a ) and ( b ), notethat Lemma 2.2 and the triangle inequality (for integrals) imply that (cid:12)(cid:12)(cid:12) E ν (cid:2) ˆ h V,η (cid:3) ( P ) − h V,η ( P ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Φ ˆ h V,η,P ( γ ) dν − h V,η ( P ) · Z Φ dν (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Φ ˆ h V,η,P ( γ ) − h V,η ( P ) dν (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Φ (cid:12)(cid:12)(cid:12) ˆ h V,η,P ( γ ) − h V,η ( P ) (cid:12)(cid:12)(cid:12) dν ≤ Cα − . Hence, E ν (cid:2) ˆ h V,η (cid:3) = h V,η + O (1) as claimed. As for the transformation property in (b), wehave first that E ν n (cid:20) E ν [ˆ h V,η ]( γ n · P )deg η ( γ n ) (cid:21) := Z Φ n E [ˆ h V,η ]( γ n · P )deg η ( γ n ) dν n := X γ n ∈ Φ n E [ˆ h V,η ]( γ n · P )deg η ( γ n ) ν n ( γ n ) TOCHASTIC CANONICAL HEIGHTS 9 by definition. In particular, the Lebegue dominated convergence theorem [16, Theorem 9.1],Fubini’s Theorem [16, Theorem 10.3], and [16, Corollary 10.2] together imply that E ν n (cid:20) E [ˆ h V,η ]( γ n · P )deg η ( γ n ) (cid:21) = Z Φ n lim m →∞ Z Φ m h V,η ( γ m · ( γ n · P ))deg η ( γ m ) dν m deg η ( γ n ) dν n (5) and [16, Corollary 10.2]= lim m →∞ Z Φ n Z Φ m h V,η ( γ m · ( γ n · P ))deg η ( γ m ) deg η ( γ n ) dν m dν n [16, Theorem 9.1]= lim m →∞ Z Φ n Z Φ m h V,η (( γ m ◦ γ n ) · P ))deg η ( γ m ◦ γ n ) dν m dν n = lim m →∞ Z Φ n + m h V,η ( γ n + m · P )deg η ( γ n + m ) dν n + m [16, Theorem 10.3]= lim m →∞ Z Φ h V,η,P,n + m ( γ ) dν [16, Corollary 10.2]= Z Φ ˆ h V,η,P ( γ ) dν (5)= E (cid:2) ˆ h V,η (cid:3) ( P ) . On the other hand, ν n ( γ n ) / deg η ( γ n ) = ( d ν ,η ) − n ν ∗ n ( γ n ) for all γ n ∈ Φ n by definition of ν ∗ n .Therefore, we deduce that E (cid:2) ˆ h V,η (cid:3) ( P ) = E ν n (cid:20) E [ˆ h V,η ]( γ n · P )deg η ( γ n ) (cid:21) = X γ n ∈ Φ n E [ˆ h V,η ]( γ n · P )deg η ( γ n ) ν n ( γ n )= ( d ν ,η ) − n X γ n ∈ Φ n E [ˆ h V,η ]( γ n · P ) ν ∗ n ( γ n ) = ( d ν ,η ) − n E ν ∗ n h E [ˆ h V,η ]( γ n · P ) i (6)as claimed. This is the analog of the usual transformation property for canonical heightsdefined by a fixed endomorphism. (cid:3) Remark . We note that if deg( φ ) = d for all φ ∈ S , then d ν,η = d , ν ∗ k = ν k , and we obtainthe transformation formula E ν k h E ν (cid:2) ˆ h V,η (cid:3) ( γ k · P ) i = d k E ν (cid:2) ˆ h V,η (cid:3) ( P ) . Remark . We note that for each P ∈ V the variance of the distribution of ˆ h V,η,P ( γ ) as wevary over sequences γ ∈ Φ S is absolutely bounded by Lemma 2.2 and Popoviciu’s inequality: σ V,η,ν,P := Z Φ S (cid:16) ˆ h V,η,P − E ν [ˆ h V,η ]( P ) (cid:17) dν ≤ (cid:16) Cα − (cid:17) . Example . Let V := P N , let S = { φ , φ , . . . , φ s } be any finite col-lection of endomorphisms of degree at least two, and let ν be any probability measure on S . Then any divisor class η ∈ Pic( P N ) ∼ = Z satisfies φ ∗ i ( η ) = deg( φ i ) η for all 1 ≤ i ≤ s .Moreover, there are constants O η,i (1) for each 1 ≤ i ≤ s such that h P N ,η ◦ φ i = deg( φ i ) h V,η + O η,i (1) . Therefore, S is height controlled with respect to any divisor: C η := max { O η,i (1) } is finite.In particular, one could take: V := P , S := { φ , φ } to be any two rational maps in one-variable (of degree at least 2), and ν ( φ ) = 1 / ν ( φ ): that is, we flip a fair coin todetermine which map to apply at every stage of composition. Example . Let V := P , let η = ∞ , let B >
0, andconsider the set of functions S B := (cid:8) φ d,c ( z ) = z d + c : c ∈ Z , | c | ≤ B, d > (cid:9) . Then, [10, Lemma 12] implies that(7) | h η ( φ d,c ( P )) − dh η ( P ) | ≤ log | c | ≤ log | B | for all P ∈ P ( Q ) K { η } = Q .In particular, (for simplicity) we consider only rational basepoints. Hence, (7) implies that S B is height controlled with respect to η . From here, we can take any probability measure ν we like on S B (note that S B is set theoretically just N B +1 ). For instance, one could considerany of the following well-known probability measures: Geometric:
Let r ∈ (0 ,
1) and let ν B,r, be the measure on ( S B , S B ) generated by ν B,r, ( φ d,c ) = (1 − r ) r d − B + 1 . Then one checks via the summation formula for geometric series that P ν B,r, ( φ d,c ) = 1. Inparticular, the data (cid:0) P ( Q ) , η, S B , ν B,r, (cid:1) satisfies the hypothesis of Theorem 1.2. Poisson:
Let λ > ν B,λ, be the measure on ( S B , S B ) generated by ν B,λ, ( φ d,c ) = e − λ λ d − (2 B + 1)( d − . Then one checks via the exponential summation formula that P ν B,λ, ( φ d,c ) = 1. In partic-ular, the data (cid:0) P ( Q ) , η, S B , ν B,λ, (cid:1) satisfies the hypothesis of Theorem 1.2. Remark . Likewise, given any probability measure on the set of integers | c | ≤ B , one cantwist the usual Geometric and Poisson distributions to form new probability measures on S B and study the corresponding expected canonical heights. Even more generally, for any finiteset of rational maps S , the set ¯ S = { φ ◦ x d : φ ∈ S, d ≥ } is an infinite, height controlledfamily (generalizing the unicritical maps of bounded height).We now prove a dynamical application of the expected canonical height function (analo-gous to the characterization of preperiodic points as the kernel of the usual canonical height). (Proof of Corollary 1.4). If there exists a finite, S -stable subset F P containing P , thenOrb γ ( P ) is contained in F P for all γ ∈ Φ S ; hence, ν (cid:0) { γ ∈ Φ S : Orb γ ( P ) is finite } (cid:1) = 1 . Therefore, (1) = ⇒ (2). On the other hand, if (2) holds, then we see immediately that ν (cid:0) { γ ∈ Φ S : ˆ h V,η,P ( γ ) = 0 } (cid:1) = 1 . In particular, the expected canonical height must vanish: E ν [ˆ h V,η ]( P ) = R Φ S ˆ h V,η,P ( γ ) dν = 0,and (2) = ⇒ (3). Finally, suppose that E ν [ˆ h V,η ]( P ) = 0. Then the transformation law in TOCHASTIC CANONICAL HEIGHTS 11 property (b) of Theorem 1.2 implies that E ν ∗ k h E ν (cid:2) ˆ h V,η (cid:3) ( γ k · P ) i = ( d ν,η ) k E ν (cid:2) ˆ h V,η (cid:3) ( P ) = 0for all k ≥
1. However, Φ
S,k is countable, so that(8) 0 = E ν ∗ k h E ν (cid:2) ˆ h V,η (cid:3) ( γ k · P ) i = X γ k ∈ Φ k E [ˆ h V,η ]( γ k · P ) ν ∗ k ( γ k ) . Moreover, since η is ample, [12, Theorem 2.3 (3)] implies that ˆ h V,η,Q is non-negative for all Q ∈ P ( K ). Therefore, E ν [ˆ h V,η ]( γ k · P ) is non-negative for all γ k ∈ Φ S,k and all k ≥
1. On theother hand, ν ∗ k ( γ k ) is positive, since deg η ( γ k ) is positive and ν is a strictly positive measure.Hence, (8) implies that E ν [ˆ h V,η ]( γ k · P ) = 0 for all γ k ∈ Φ S,k and all k ≥
1. However, E ν [ˆ h V,η ] = h V,η + O (1), so that(9) F P := [ γ ∈ Φ S [ k ≥ ( γ k · P ) ⊆ V ( K ( P ))must be a set of bounded height (with respect to h V η ); here K ( P ) /K is the field of definitionof P . In particular, Northcott’s theorem and the fact that η is ample imply that F P is finite.Moreover, F P is S -stable. Therefore (3) = ⇒ (1), completing the argument. (cid:3) Remark . To summarize, if η is an ample divisor, then we have defined a height function E ν [ˆ h V,η ] on V that characterizes the existence of a finite subset that is simultaneously stablefor all maps in S . In fact, we could formally define a map E ν [ˆ h V,η ] : Fin( V ) → R on theset Fin( V ) of finite subsets of V given by E ν [ˆ h V,η ]( F ) = P P ∈ F E ν [ˆ h V,η ]( P ), and note that E ν [ˆ h V,η ]( F ) = 0 if and only if F is S -stable. Remark . Note that Corollary 1.4 does not depend on the probability measure ν on S . (Proof of Corollary 1.5). If P ∈ P ( Q ) is such that E ν [ˆ h ]( P ) = 0, then Corollary 1.4 im-plies that P ∈ PrePer( φ ) for all φ ∈ S ; however, note that the converse is false in general: P = 0 and S = { x − , x + x } ; In any case, if (cid:8) P ∈ P ( Q ) : E ν [ˆ h ]( P ) = 0 (cid:9) is infinite,then T φ ∈ S PrePer( φ ) is infinite, and (1) implies (2). On the other hand, if T φ ∈ S PrePer( φ )is infinite, then PrePer( φ ) ∩ PrePer( ψ ) is infinite for all φ, ψ ∈ S . Hence, we deduce thatPrePer( φ ) = PrePer( ψ ) for all φ, ψ ∈ S by [1, Theorem 1.2]. Therefore, (2) implies (3). Fi-nally, suppose that PrePer( φ ) = PrePer( ψ ) for all ψ ∈ S , and take any point P ∈ PrePer( φ ).Then F P := [ γ ∈ Φ S [ k ≥ ( γ k · P ) ⊆ PrePer( φ ) ∩ P ( K ( P ));here we use that PrePer( ψ ) is ψ -stable and that PrePer( φ ) = PrePer( ψ ) for all ψ ∈ S .However, PrePer( φ ) is a set of bounded height by Northcott’s theorem. Therefore, F P is afinite set since [ K ( P ) : K ] is a finite extension. On the other hand, F P is clearly S -stable,so that Corollary 1.4 implies that E ν [ˆ h ]( P ) = 0. Hence, we have shown thatPrePer( φ ) ⊆ (cid:8) P ∈ P ( Q ) : E ν [ˆ h ]( P ) = 0 (cid:9) . In particular, since PrePer( φ ) is an infinite set by [20, Exercise 1.18], we se that (3) implies(1) as claimed. (cid:3) Primitive prime divisors in generalized orbits
To begin this section, let K be a number field and let o K be the ring of integers of K .Given a prime ideal p ⊂ o K , we let k p := o K / po K and N p := (log k p ) / [ K : Q ] be the residuefield and local (logarithmic) degree of p respectively, and define the Weil height [20, § α ∈ K ∗ by h ( α ) = − X p min( v p ( α ) , N p + 1[ K : Q ] X σ : K → C max(log | σ ( α ) | , . In particular, it follows from the product formula [20, Proposition 3.3] that(10) X v p ( α ) > v p ( α ) N p ≤ h ( α ) and − X v p ( α ) < v p ( α ) N p ≤ h ( α )for all α ∈ K ∗ . As in [6] and [8], the main tool we use to study the prime factors of elementsof orbits is the (Roth) abc -conjecture. Conjecture 3.1 ( abc -conjecture) . Let K be a number field. Then for any ǫ > , there existsa constant C K,ǫ such that for all a, b, c ∈ K ∗ satisfying a + b = c , we have that h ( a, b, c ) < (1 + ǫ )( rad ( a, b, c )) + C K,ǫ . Here rad(a,b,c) is a suitably defined radical of the tuple ( a, b, c ) ; see [6, § . In fact, the key idea we use here is that the abc -conjecture implies that polynomial valuesare reasonably square-free in the following sense; see [6, Proposition 3.4]
Proposition 3.2.
Let F ( x ) ∈ o K [ x ] be a polynomial of degree at least without repeatedfactors. Then for all ǫ > there exists a constant C f,ǫ such that: X v p ( F ( z )) > N p ≥ (deg( F ) − − ǫ ) h ( z ) + C f,ǫ for all z ∈ K . As a reminder, in our study of prime factors in dynamical orbits, we restrict our attentionto pairs ( γ, P ) ∈ Φ S × P ( K ) in the following set: G S := n ( γ, P ) ∈ Φ S × P ( K ) : ˆ h γ ( P ) > , ∞ 6∈ Orb γ ( P ) ∪ Orb γ (0) o . (Proof of Theorem 1.8). For each φ ∈ S , fix a representation φ ( x ) = F φ ( x ) G φ ( x ) for some coprimepolynomials F φ , G φ ∈ o K [ x ], and let s := (cid:8) p : v p (Res ( F φ , G φ )) = 0 , φ ∈ S (cid:9) ;here Res ( F φ , G φ ) denotes the resultant of F φ and G φ . In particular, s is finite and γ m hasgood reduction at all p s for all m ≥ γ ∈ Φ S ; see [20, Theorem 2.15]. To easenotation, for a given sequences γ = ( θ n ) n ≥ , write F n = F θ n and G n = G θ n for any n ≥ ǫ > C S,ǫ = min { C F n ,ǫ } such that:(11) (deg( F n ) − − ǫ ) h ( γ n − · P ) + C S,ǫ ≤ X v p ( F n ( γ n − · P )) > N p . Here we use assumptions (1) and (2) of Theorem 1.8 to deduce that F n is square-free andhas degree at least 3; see also Remark 3.3. Moreover, (11) holds for all P , γ , and n . TOCHASTIC CANONICAL HEIGHTS 13
Now assume that n ∈ Z ( γ, P ). To study the prime ideals defining the right hand side of(11), assume that v p ( F n ( γ n − · P )) >
0. If in addition p s and v p ( γ n − · P ) ≥
0, then(12) v p ( γ n · P ) = v p ( F n ( γ n − · P )) − v p ( G n ( γ n − · P )) = v p ( F n ( γ n − · P )) > n ∈ Z ( γ, P ), so that (12) implies that v p ( γ m · P ) > ≤ m ≤ n − S have good reduction at p , so that that(13) ( γ n K γ m ) · ≡ ( γ n K γ m ) · ( γ m · P ) ≡ γ n · P ≡ p );see [20, Theorem 2.18]. Therefore, (10), (11) and (13) together imply that(14) (deg( F n ) − − ǫ ) h ( γ n − · P ) + C S,ǫ ≤ ⌊ n ⌋ X m =1 v p ( γ m · P ) > N p + n − X m = ⌊ n ⌋ v p (( γ n K γ m ) · > N p + X p ∈ s N p + X v p ( γ n − · P ) < N p ≤ ⌊ n ⌋ X m =1 h ( γ m · P ) + n − X m = ⌊ n ⌋ h (( γ n K γ m ) ·
0) + h ( γ n − · P ) + C S, . This is similar to the bound in [7, (15)]. Moreover, it is here that we use that ∞ and 0are not in the orbit of P and 0, in order to apply (10). Now we use properties of canonicalheights. Specifically, Lemma 2.2 and Remark 2.3 together imply that(15) (cid:12)(cid:12)(cid:12) h ( ρ r · Q ) − deg( ρ r )ˆ h ρ ( Q ) (cid:12)(cid:12)(cid:12) ≤ C S for all r ≥
1, all ρ ∈ Φ S , and all Q ∈ P ( K ); here C S is the height control constant fromTheorem 1.2. In particular, by letting ǫ = 1 / F n ) ≥
4, wededuce from (14) and (15) thatdeg( γ n − )ˆ h γ ( P ) ≤ (cid:18) ˆ h γ ( P ) ⌊ n ⌋ X m =1 deg( γ m ) + n − X m = ⌊ n ⌋ deg( γ n K γ m )ˆ h γ K γ m (0) (cid:19) + C S, n + C S, . (16)Therefore, after dividing both sides of (16) by deg( γ n − )ˆ h γ ( P ) and noting that ˆ h γ K γ m (0) ≤ C S for all m , we achieve a bound of the form(17) ≤ C S, ,γ,P (cid:16) P ⌊ n ⌋ m =1 deg( γ m )deg( γ ⌊ n/ ⌋ ) (cid:17) deg (cid:0) γ n − K γ ⌊ n/ ⌋ (cid:1) + deg( θ n ) C S, ,γ,P (cid:16) P n − m = ⌊ n ⌋ deg( γ n K γ m )deg( γ n K γ ⌊ n/ ⌋ ) (cid:17) deg( γ ⌊ n/ ⌋ ) + C S, ,γ,P n deg( γ n − ) + C S, ,γ,P deg( γ n − ) ≤ C S, ,γ,P deg (cid:0) γ n − K γ ⌊ n/ ⌋ (cid:1) + C S, ,γ,P deg (cid:0) γ ⌊ n/ ⌋ (cid:1) + C S, ,γ,P n deg( γ n − ) + C S, ,γ,P deg( γ n − ) . However, the right-hand side of (17) goes to zero as n grows. Hence, n is bounded and Z ( γ, P ) is finite as claimed. (cid:3) Remark . Conditions (1) and (2) of Theorem 1.8 are equivalent to writing φ ( x ) = F φ ( x ) G φ ( x ) with disc( F φ ) = 0 and deg( F φ ) ≥ φ ∈ S . We now study dynamical Zsigmondy sets when K/ F p ( t ) is a finite separable extensionand p is odd. To do this, we translate the proof of [9, Theorem 1.1] to the non-autonomoussetting and use properties of canonical heights from Section 2. For completeness, we remindthe reader of the definition of the Weil height in this setting. Given a valuation v on K withresidue field k v and local degree N v [21, Definition 1.1.14], we define the height of α ∈ K ∗ to be(18) h ( α ) = − X v min { v ( α ) , } N v = X v max { v ( α ) , } N v , where the equality above follows from the fact that on a curve, the number of zeros of anon-constant function equals the number of poles when counted with multiplicity; see, forinstance, [21, Theorem 1.4.11].Before we begin our proof of Theorem 1.9, we record a few auxiliary results, includingLemma 2.1 from [9] stated below. However, given the technical nature of the material thatfollows, the reader is encouraged to keep in mind that our overall strategy (as in the proofof [9, Theorem 1.1]) is to show that for any sufficiently long string ψ m ∈ Φ S,m there existsa root of ψ m that fails to satisfy a Riccati equation. Here a Riccati equation is a first orderdifferential equation of the form y ′ = ay + by + c. In what follows, K sep denotes the separable closure of K . Lemma 3.4.
Let K/ F p ( t ) , let φ ( x ) ∈ K [ x ] have degree d ≥ , and write φ ( x ) = A x d + A x d − + . . . A d − x + A d . If d ∈ K ∗ and the quantity δ φ := 2 dA A − ( d − A is non-zero, then for all β ∈ K sep such that β and φ ( β ) both satisfy a Riccati equation, i.e. (19) β ′ = aβ + bβ + c and φ ( β ) ′ = eφ ( β ) + f φ ( β ) + g for some a, b, c, e, f, g ∈ K , either [ K ( β ) : K ] ≤ d, or the coefficients in (19) are uniquely determined by φ : (20) a = 0 , b = ( dA A ′ − ( d − A A A ′ − dA A A ′ + ( d − A A ′ ) /δ φ ,e = 0 , f = ( d A A ′ − d ( d − A A A ′ − d ( d − A A A ′ + ( d ( d −
2) + 1) A A ′ ) /δ φ ,c = ( A A A ′ − A A A ′ + A A A ′ ) /δ φ , g = A d − c − A d f + A ′ d . We gather another result that uses our technical assumptions from Theorem 1.9. Inparticular, we show that not all roots of sufficiently long strings of elements of S can satisfya Riccati equation over K . It is in this step especially where we encounter the subtlety ofiterating multiple maps. Lemma 3.5.
Let K/ F p ( t ) be a function field and suppose that S = { φ , φ , . . . , φ s } satisfiesthe hypothesis of Theorem 1.9. Then there exists a constant M = M ( S ) such that for anystring ψ M = ( θ n ) Mn =1 ∈ Φ S,M of length M there is an integer ≤ m ψ ≤ and a root β of thesubstring ( ψ M K ψ m ψ ) = ( θ n ) Mn = m ψ +1 that satisfies: TOCHASTIC CANONICAL HEIGHTS 15 (a) β ∈ K sep , (b) [ K ( β ) : K ] > d , (c) β does not satisfy a Riccati equation over K .Proof. To find the pair m ψ and β , let d := max { deg( φ ) : φ ∈ S } and note thatˆ h minΦ S ,K (2 d ) := inf n ˆ h ψ ( α ) : [ K ( α ) : K ] ≤ d, α PrePer(Φ S ) , ψ ∈ Φ S o is a positive number; this follows from Lemma 3.8 below. Now let m ≥ α ∈ P ( K ) is a root of a string ψ m ∈ Φ S,m and that [ K ( α ) : K ] ≤ d . Extend ψ m to someinfinite sequence ψ ∈ Φ S , and note that Remark 2.3 implies that(21) ˆ h ψ K ψ m (0) = ˆ h ψ K ψ m ( ψ m ( α )) = deg( ψ m ) ˆ h ψ ( α ) ≥ deg( ψ m ) ˆ h minΦ S ,K (2 d ) . Here we use our assumption that 0 PrePer(Φ S ), so that α PrePer(Φ S ) also. Now let C S be the height control constant from Theorem 1.2, and note that Lemma 2.2 implies thatˆ h ψ K ψ m (0) ≤ C S , independent of the extension ψ of ψ m . Therefore, in light of (21), we definethe constant(22) M := log +3 (cid:0) C S (cid:14) ˆ h minΦ S ,K (2 d ) (cid:1) + 3 , where log +3 ( z ) = max { log ( z ) , } . To see that M has the desired properties, let ψ M ∈ Φ S,M be any string. Note first that we may choose a separable root β M of ψ M : for if every rootof ψ M is inseparable, then deg( ψ M ) is divisible by char( K ); see [4, § ψ M = ( θ n ) Mn =1 explicitly. Then we claim that ( m ψ , β ) in Lemma 3.5 can be chosen to be one of the pairsbelow: ( m ψ , β ) = (cid:0) , β M (cid:1) , (cid:0) , θ ( β M ) (cid:1) or (cid:0) , θ ( θ ( β M )) (cid:1) . To see this, note that condition (a) of Lemma 3.5 is easily satisfied: since β M is separable,both θ ( β M ) and θ ( θ ( β M )) are also separable. On the other hand, the relative degrees (cid:2) K ( β M ) : K (cid:3) , (cid:2) K ( θ ( β M )) : K (cid:3) and (cid:2) K ( θ ( θ ( β M ))) : K (cid:3) are all at least 2 d by (21) andthe definition of M in (22). Therefore, it remains to show that at least one of the algebraicfunctions β , θ ( β M ) or θ ( θ ( β M )) does not satisfy a Riccati equation over K . Suppose, fora contradiction, that all three satisfy a Riccati equation over K . Then Lemma 3.4 appliedseparately to the point β = β M with the map φ = θ and the point β = θ ( β M ) with themap φ = θ implies that β ′ M = b β M + c and θ ( β M ) ′ = f θ ( β M ) + g ,θ ( β M ) ′ = b θ ( β M ) + c and θ ( θ ( β )) ′ = f θ ( θ ( β )) + g (23)for the unique solutions ( b , c , f , g ) ∈ K and ( b , c , f , g ) ∈ K in (20) correspondingto φ = θ and φ = θ respectively. In particular, 0 = ( b − f ) θ ( β M ) + ( c − g ). But[ K ( θ ( β M )) : K ] > d , so that ( b − f ) = 0. However, this again contradicts condition (1)of Theorem 1.9. (cid:3) The importance of Riccati equations to the arithmetic of function fields is partially ex-plained by their ability to detect the isotriviality of hyperelliptic curves [9, Lemma 2.2].
Lemma 3.6.
Let K/ F p ( t ) and suppose that ρ ( x ) ∈ K [ x ] is an irreducible (and separable)polynomial of degree d ≥ . If β ∈ K sep is such that ρ ( β ) = 0 and β does not satisfy aRiccati equation over K , i.e. β ′ = aβ + bβ + c for all a, b, c ∈ K , then the hyperelliptic curve C : Y = ρ ( X ) is non-isotrivial. As a final preparation for the proof of Theorem 1.9, we replace Proposition 3.2 witheffective forms of the Mordell conjecture over global function fields [13, 18, 22].
Theorem 3.7. (Effective Mordell Conjecture) Let X be a non-isotrivial curve of genus atleast defined over a function field K of characteristic p > . Then there are constants B X, and B X, depending on X such that for all Q ∈ X ( K ) , h κ X ( Q ) ≤ B X, d ( Q ) + B X, ; here h κ X is a height function with respect to the canonical divisor κ X of X and d ( Q ) = 2 genus( K ( Q )) − K ( Q ) : K ] is the relative discriminant of the extension K ( Q ) /K .Proof. Strictly speaking, the bounds in [13, 18, 22] are for curves with non-zero Kodaira-Spencer class. However, the non-isotrivial case follows from this one; see [14, Theorem 5] or[23, Theorem 2]. For if X is a non-isotrivial curve, then there is an r (a power of p ) and aseparable extension L/K such that X is defined over L r and that the Kodaira-Spencer classof X over L r is non-zero [23, Appendix]. Now, if we apply any of the bounds in [13, 18, 22]to X over L r , then we achieve the bound in Theorem 3.7. (cid:3) Having completed the requisite preparations, we can now prove the finiteness of somepolynomial Zsigmondy sets over global function fields. (Proof of Theorem 1.9).
Let γ ∈ Φ S , let P ∈ P ( K ) be such that ˆ h γ ( P ) >
0, and supposethat n ∈ Z ( γ, P ). Without loss of generality, we may assume that n > M ; see (22) for adefinition of M . Write γ n = ( θ m ) nm =1 so that γ n = ψ M ( γ ) ◦ γ n − M , where ψ M ( γ ) := ( θ m ) nm = n − M +1 and γ n − M := ( θ m ) n − Mm =1 . Now since ψ M ( γ ) is a string of length M , Lemma 3.5 implies that ψ M ( γ ) = ρ M ( γ ) ◦ ψ m ψ ( γ ),where ψ m ψ is a string of length 0, 1 or 2 and ρ M ( γ ) has a root β satisfying conditions (1)-(3)of Lemma 3.5. In particular, the polynomial ρ M ( γ ) has a factorization ρ M ( γ ) = f γ,M, ( x ) e f γ,M, ( x ) e . . . f γ,M,r ( x ) e r , satisfying: the f γ,M,i ∈ K [ x ] are distinct and irreducible, deg( f γ,M, ) ≥
6, and f γ,M, hasa separable root β M,γ that does not satisfy a Riccati equation over K . In particular, thehyperelliptic curve C γ,M : Y = f γ,M, ( X )is non-isotrivial by Lemma 3.6. Moreover, there are only finitely many such curves C γ,M toconsider since ρ M ( γ ) ∈ Φ S,M − i for some 0 ≤ i ≤
2. In particular, there are uniform heightbound constants B S, := max γ ∈ Φ S { B C γ,M , } and B S, := max γ ∈ Φ S { B C γ,M , } TOCHASTIC CANONICAL HEIGHTS 17 from Theorem 3.7. From here, the proof that n is bounded is (mutatis mutandis) the sameas the proof of [9, Theorem 1.1] and similar to the proof of Theorem 1.8 above. Specifically,we consider the algebraic point Q n,γ,P = (cid:16) ( ψ m ψ ( γ ) ◦ γ n − M )( P ) , q ( f γ,M, ◦ ( ψ m ψ ( γ ) ◦ γ n − M ))( P ) (cid:17) ∈ C γ,M ( K ) , so that Theorem 3.7 implies that h κ C γ,M ( Q n,γ,P ) ≤ B S, d ( Q n,γ,P ) + B S, . On the other hand, let x n,γ ( P ) = X ( Q n,γ,P ) and y n,γ ( P ) = Y ( Q n,γ,P ) respectively. Then(24) h ( x n,γ ( P )) ≤ B S, (cid:18) X v ( y n,γ ( P )) > N v + X v ( y n,γ ( P )) < N v (cid:19) + B S, ;here we use [21, Proposition 3.7.3] to calculate d ( Q n,γ,P ) and use [19, Theorem III.10.2] tocompare the two heights h κ C γ,M ( Q n,γ,P ) and h ( x n,γ ( P )). However, S is a set of polynomials,so that v ( y n,γ ( P )) < P , the coefficients of the polynomials in S , orthe coefficients of f γ,M, have negative valuations. Therefore, we deduce from (24) that(25) h ( x n,γ ( P )) ≤ B S, X v ∈ s P N vv ( y n,γ ( P )) > + B S, h ( P ) + B S, ;here s P is the set valuations v satisfying the following: • v ( P ) ≥ • every polynomial φ ∈ S has good reduction at v, • f γ,M,i has good reduction at v or all i and all γ ∈ Φ S , (26)Now we use our assumption that n ∈ Z ( γ, P ). In particular, if v ∈ s P and v ( y n,γ ( P )) > v ( γ n ( P )) >
0. Therefore, v ( γ m ( P )) > ≤ m ≤ n − v ( γ n ( P )) > v ( γ m ( P )) > v (( γ n K γ m )(0)) >
0. In particular, we see that (18) and (25) imply that(27) h ( x n,γ ( P )) ≤ B S, (cid:18) ⌊ n ⌋ X m =1 v ( γ m · P ) > N v + n − X m = ⌊ n ⌋ v (( γ n K γ m ) · > N v (cid:19) + B S, h ( P ) + B S, ≤ B S, (cid:18) ⌊ n ⌋ X m =1 h ( γ m · P ) + n − X m = ⌊ n ⌋ h (( γ n K γ m ) · (cid:19) + B S, h ( P ) + B S, ; compare to (14) over number fields. However, (cid:12)(cid:12) h ( ρ r · Q ) − deg( ρ r )ˆ h ρ ( Q ) (cid:12)(cid:12) ≤ C S for all r ≥ ρ ∈ Φ S , and all Q ∈ P ( K ), where C S is the height control constant from Theorem 1.2;see Lemma 2.2. Therefore,deg( γ n − M )ˆ h γ ( P ) ≤ B S, (cid:18) ˆ h γ ( P ) ⌊ n ⌋ X m =1 deg( γ m ) + n − X m = ⌊ n ⌋ deg( γ n K γ m )ˆ h γ K γ m (0) (cid:19) + B S, ˆ h γ ( P ) + B S, n + B S, ;(28) compare this to the estimate (16) in the number field setting. Here we use also the trivialbound deg( γ n − M ) ≤ deg(( ψ m ψ ( γ ) ◦ γ n − M )), in an attempt to make the expressions lesscumbersome.Finally, ˆ h γ K γ m (0) ≤ C S for all m by Lemma 2.2, and we deduce that n is bounded as in(17) over number fields. Namely, divide both sides of (28) by deg( γ n )ˆ h γ ( P ) and see that theright hand goes to zero as n grows while the left hand side is bounded below by 1 /d M , where d = max { deg( φ ) : φ ∈ S } . This completes the proof of Theorem 1.9. (cid:3) We conclude this section by noting that there is an absolute (positive) lower bound onthe canonical height of all non-preperiodic points of bounded degree, a fact used to establishLemma 3.5.
Lemma 3.8.
Let K be a global field and let S = { φ , φ , . . . , φ s } be a set of endomorphismson P all defined over K and of degree at least . Then for all D > , the quantity ˆ h minΦ S ,K ( D ) := inf n ˆ h ψ ( α ) : [ K ( α ) : K ] ≤ D, α PrePer(Φ S ) , ψ ∈ Φ S o is strictly positive.Proof. Choose any Q ∈ P such that [ K ( Q ) : K ] ≤ D and Q PrePer(Φ S ); this ispossible by Northcott’s Theorem and Lemma 2.2. Then for any ρ ∈ Φ S , we have that(29) ˆ h minΦ S ,K ( D ) = inf n ˆ h ψ ( α ) : ˆ h ψ ( α ) < ˆ h ρ ( Q ) , [ K ( α ) : K ] ≤ D, α PrePer(Φ S ) , ψ ∈ Φ S o . On the other hand, if ˆ h ψ ( α ) < ˆ h ρ ( Q ), then h ( α ) < ˆ h ρ ( Q ) + C S by Lemma 2.2. Moreover,the set of points(30) T D := n α ∈ P ( K ) : [ K ( α ) : K ] ≤ D, h ( α ) < ˆ h ρ ( Q ) + C S , α PrePer(Φ S ) o is finite by Northcott’s theorem. In particular, sinceˆ h minΦ S ,K ( D ) ≥ min α ∈ T D inf ψ ∈ Φ S { ˆ h ψ ( α ) } by (29) and (30) above and T D is finite, it suffices to show that inf ψ ∈ Φ S { ˆ h ψ ( α ) } is strictlypositive for any non-preperiodic α to prove that ˆ h minΦ S ,K ( D ) >
0: the minimum value of afinite set of positive numbers is positive. To do this, we note first that (cid:12)(cid:12) ˆ h Q ( γ ) − ˆ h Q ( ψ ) (cid:12)(cid:12) ≤ C S ∆( γ, ψ ) for all Q ∈ P ( K ) and all γ, ψ ∈ Φ S , where ∆( γ, ψ ) = 2 − min { n : γ n = ψ n } gives a metric on Φ S ; this follows easily from Lemma 2 . α the canonical height map, ˆ h α : Φ S → R ≥ given by γ → ˆ h γ ( α ), is continuous; in fact, ˆ h α is uniformly continuous. In particular, since Φ S is compact ( S is finite), ˆ h α must attain its minimum value somewhere on Φ S . In particular,this minimum value must be positive when α PrePer(Φ S ) by definition. (cid:3) Remark . In fact, for non-preperiodic basepoints, we have established uniform bounds onZsigmondy sets: assuming the hypothesis of Theorem’s 1.8 and 1.9, there exists an integer N = N ( S, K ) such that if n ∈ Z ( γ, P ) for some ( γ, P ) ∈ G S with P PrePer(Φ S ), then n ≤ N . This follows from (16), (28), and Lemma 3.8. TOCHASTIC CANONICAL HEIGHTS 19 Local Height Decomposition
As with the usual canonical height (associated to a single endomorphism), the canonicaland expected canonical heights defined in Theorem 1.2 can be written as a sum of localheights. To do this, we follow the construction given in [20, § V = P and that our generating set of self maps S = { φ , φ , . . . , φ s } is finite. Moreover,since most of the results in this section are straightforward adaptations of Lemma 2.1, Lemma2.2, and results in [20, § M K be a complete set of absolute values on K . To each place | · | v ∈ M K , we define a norm k·k v on A ( K v ) given by k ( x, y ) k v = max {| x | v , | y | v } , where K v is the completion of K at | · | v ; let n v be its local degree [20, § φ i : A → A over K for each φ i ∈ S , and associate a “lift” to a sequence γ = ( θ n ) n ≥ ∈ Φ S by ˜ γ = (˜ θ n ) n ≥ . Then we define an associated Green function to ˜ γ and | · | v by analogy with the canonical height:(31) G v, ˜ γ ( x, y ) = lim n →∞ log k ˜ γ n ( x, y ) k v deg(˜ γ n ) , ( x, y ) ∈ A ∗ ( K v );here A ∗ ( K v ) = { ( x, y ) ∈ A ( K v ) : x = 0 or y = 0 } . We collect some facts about theseGreen functions below; compare to [20, Proposition 5.58 and Theorem 5.59]. Proposition 4.1.
The functions G v, ˜ γ are well defined and satisfy the following properties: (1) Let ( x, y ) ∈ A ∗ ( K v ) . Then G v, ˜ γ ( x, y ) = log k ( x, y ) k v + O v (1) for all v ∈ M K . Moreover, O v (1) = 0 for all but finitely many v ∈ M K . (2) Let ( x, y ) ∈ A ∗ ( K ) be any representative of P = [ x, y ] ∈ P ( K ) . Then ˆ h γ ( P ) = X v ∈ M K n v G v, ˜ γ ( x, y ) for any choice of lifts (over K ) of the elements of S .Proof. The argument that the limit defining G v, ˜ γ exists is, mutatis mutandis, the same asthat given to establish that ˆ h γ is well defined: one uses Tate’s telescoping argument plus thefact that S is finite (and thus height controlled). In particular, the key fact in this setting isthat there exist positive constants c i,v such that(32) (cid:12)(cid:12)(cid:12)(cid:12) log k ˜ φ i ( x, y ) k v − deg( φ i ) log k ( x, y ) k v (cid:12)(cid:12)(cid:12)(cid:12) ≤ c i,v for all ( x, y ) ∈ A ∗ ( K v );see [20, Proposition 5.57(a)]. Hence, (32) and the adapted version of Tate’s telescopingargument given in (3) and (4) imply that(33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log k ˜ γ n ( x, y ) k v deg(˜ γ n ) − log k ˜ γ m ( x, y ) k v deg(˜ γ m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max i { c i,v } α m ( α − , < m < n ; here α = min { deg( φ i ) } ≥
2. Hence, the limit defining G v, ˜ γ ( x, y ) is Cauchy and thereforeconverges. Moreover, letting m = 0 and n → ∞ establishes statement (1). On the otherhand, if v ∈ M K satisfies:(34) v is nonarchimedean, k ˜ φ i k v = 1, and | Res ( ˜ φ i ) | v = 1 , then c i,v = 0; see [20, Proposition 5.57(b)]. Hence, for all but finitely v ∈ M K , we have that G v, ˜ γ ( x, y ) = log k ( x, y ) k v for all ( x, y ) ∈ A ∗ ( K v ); to see this, note that (34) holds for all i forall but finitely many places (recall that we fix the lifts ˜ φ i at the outset). This completes theproof of statement (1). As for statement (2), we note first that if c ∈ K ∗ v , then(35) G v, ˜ γ ( cx, cy ) = G v, ˜ γ ( x, y ) + log | c | v ;the proof is identical to [20, (5.38) pp.289]. In particular, if c ∈ K ∗ , then(36) X v ∈ M K n v G v, ˜ γ ( cx, cy ) = X v ∈ M k n v G v, ˜ γ ( x, y ) for all ( x, y ) ∈ A ∗ ( K ) , by the product formula [20, Proposition 3.3]. Hence, the right hand side of statement (2)of Proposition 4.1 is independent of the choice of representative for P = [ x, y ]. Now fix P ∈ P ( K ) and a representative ( x, y ), and note that the sum in (36) is a finite sum, since G v, ˜ γ ( x, y ) = log k ( x, y ) k v = 0 for all but finitely many v ∈ M K ; of course this time the finitelymany primes depend on ( x, y ). In fact, the stronger statement holds: there exists a finiteset T ˜ γ, ( x,y ) ⊂ M K , depending on ˜ γ and ( x, y ), such that for all | · | v T ˜ γ, ( x,y ) , G n,v, ˜ γ ( x, y ) := log k ˜ γ n ( x, y ) k v deg(˜ γ n ) = log k ( x, y ) k v = 0 for all n ≥ X v ∈ M K n v G v, ˜ γ ( x, y ) = X v ∈ T ˜ γ, ( x,y ) n v G v, ˜ γ ( x, y ) = X v ∈ T ˜ γ, ( x,y ) n v lim n →∞ G n,v, ˜ γ ( x, y )= lim n →∞ X v ∈ T ˜ γ, ( x,y ) n v G n,v, ˜ γ ( x, y ) = lim n →∞ X v ∈ M K n v G n,v, ˜ γ ( x, y )= lim n →∞ P v ∈ M K n v log k ˜ γ n ( x, y ) k v deg(˜ γ n ) = lim n →∞ h ( γ n · P )deg(˜ γ n )= ˆ h γ ( P )as claimed. This completes the proof of Proposition 4.1. (cid:3) Remark . We note that G v, ˜ γ : A ∗ ( K ) → R is continuous, since the sequence of continuousfunctions G n,v, ˜ γ converges uniformly to G v, ˜ γ by (33); compare to [20, Proposition 5.58(e)].Following [20, § v -adic distance from points to divisors. Moreover, these functions aredefined on Zariski open subsets of P (the ambient space), unlike the Green functions.To do this, let E ∈ K [ x, y ] be a homogenous polynomial of degree deg( E ) = e (whichdetermines a divisor of P ). For a lift ˜ γ of γ ∈ Φ S , determined by fixed lifts of the elementsof S , we define the local canonical height at v associated to the pair (˜ γ, E ) to be the function:(37) ˆ λ v, ˜ γ,E ([ x, y ]) := e G v, ˜ γ ( x, y ) − log | E ( x, y ) | v TOCHASTIC CANONICAL HEIGHTS 21 for all [ x, y ] ∈ P ( K v ) with E ( x, y ) = 0. We collect some properties of these local canonicalheight functions below; compare to [20, Theorems 5.60, 5.61]. Theorem 4.3.
Let E ∈ K [ x, y ] be a homogenous polynomial of degree e and let ˜ γ be a liftof γ ∈ Φ S . Then the following statements hold: (1) ˆ λ v, ˜ γ,E : P ( K v ) K { E = 0 } → R is a well defined function. (2) The function P → ˆ λ v, ˜ γ,E ( P ) + log | E ( P ) | v k P k ev extends to a bounded continuous functionon all of P ( K v ) . (3) The canonical height has a decomposition as a sum of local canonical heights: ˆ h γ ( P ) = 1deg( E ) X v ∈ M K n v ˆ λ v, ˜ γ,E ( P ) for all P ∈ P ( K ) K { E = 0 } .Proof. The proof of statement (1) follows directly from (35). As for statement (2), note thatˆ λ v, ˜ γ,E ( P ) + log | E ( P ) | v k P k ev = e (cid:0) G v, ˜ γ ( P ) − log k P k v (cid:1) is bounded by Proposition 4.1 part (1). Moreover, both G v, ˜ γ and log k·k v are continuous func-tions on A ∗ ( K v ), and hence their difference is also continuous on A ∗ ( K v ). Furthermore, sincethis difference is invariant under scaling, the map P → ˆ λ v, ˜ γ,E ( P ) + log | E ( P ) | v k P k ev is continuousas claimed. Finally, if P = [ x, y ] ∈ P ( K ) K { E = 0 } , thenˆ h γ ( P ) = X v ∈ M k n v G v, ˜ γ ( x, y ) = X v ∈ M k n v deg( E ) (cid:16) ˆ λ v, ˜ γ,E ([ x, y ]) + log | E ( x, y ) | v (cid:17) = 1deg( E ) X v ∈ M K n v ˆ λ v, ˜ γ,E ( P ) + 1deg( E ) X v ∈ M K n v log | E ( x, y ) | v = 1deg( E ) X v ∈ M K n v ˆ λ v, ˜ γ,E ( P )(38)as claimed; here, we use (37), Proposition 4.1 part (2), and the product formula, whichimplies that P v ∈ M K n v log | E ( x, y ) | v vanishes. (cid:3) We now fix the point P ∈ P ( K ) and vary its orbit in P by varying γ ∈ Φ S . In particular,we consider the functions γ → ˆ λ v,E,P ( γ ) := ˆ λ v, ˜ γ,E ( P ) for each v ∈ M K . As a first observation,note that we may interpret Theorem 4.3 part (3) as a way of writing the random variable γ → ˆ h P ( γ ) as a sum of (local) random variables:ˆ h P ( γ ) = X v ∈ M K n v ˆ λ v,E,P ( γ ) . Since it is often a useful technique in probability theory to decompose a complicated randomvariable into a sum of independent random variables, we ask the following question:
Question 4.4.
For what points P ∈ P ( K ) , is n ˆ λ v,E,P ( γ ) o v ∈ M K a collection of independentrandom variables on Φ S ? Finally, we note that E ν [ˆ h ] : P → R , a height function that in some sense packagestogether the collective dynamics of the functions in S at the point P , also has a decompositioninto a sum of local pieces. Corollary 4.5.
Let K be a global field, let E ∈ K [ x, y ] be a homogenous polynomial, and let S = { φ , φ , . . . , φ s } be a finite set of endomorphisms (over K ) on P , all of degree at least . Fix lifts ˜ φ i : A → A for each φ i , and extend these to lifts of each element in Φ S . Then (1) For all v ∈ M K , the local expected canonical height, E ν (cid:2) ˆ λ v, ˜ γ,E (cid:3) ( P ) := Z Φ S ˆ λ v,E,P ( γ ) dν, is well defined for all P ∈ P ( K v ) K { E = 0 } . (2) The expected canonical height has a decomposition as a sum local expected canonicalheights: E ν (cid:2) ˆ h (cid:3) ( P ) = 1deg( E ) X v ∈ M K n v E ν (cid:2) ˆ λ v, ˜ γ,E (cid:3) ( P ) for all P ∈ P ( K ) K { E = 0 } .Proof. The first statement follows from the Lebegue dominated convergence theorem: foreach n define the random variables λ v,E,P,n : Φ S → R given by λ v,E,P,n ( γ ) = deg( E ) log k ˜ γ n ( x, y ) k v deg(˜ γ ) − log | E ( x, y ) | v ;here ( x, y ) ∈ A ∗ ( K v ) is any representative of P . Then (33) implies that λ v,E,P,n is a sequenceof uniformly bounded functions (take m = 0). Furthermore, ˆ λ v,E,P is the pointwise limit ofthe λ v,E,P,n . Hence, [16, Theorem 9.1] implies that ˆ λ v,E,P is integrable and E ν (cid:2) ˆ λ v, ˜ γ,E (cid:3) ( P ) := Z Φ S ˆ λ v,E,P ( γ ) dν = lim n →∞ Z Φ S λ v,E,P,n ( γ ) dν. Moreover, the decomposition in statement (2) follows directly from Theorem 4.3 part (3)and the linearity of the integral: for any fixed P , the sum is in fact a finite sum; see (34). (cid:3) References [1] M. Baker and L. DeMarco,
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