The size of semigroup orbits modulo primes

Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{p}$ of $K$, let $m_{\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\bmod\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\operatorname{\mathsf{N}}_{K/\mathbb{Q}}\mathfrak{p}$ is a set of density zero. For $V=\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.


Introduction
A general expectation in arithmetic dynamics over number fields is that the dynamical systems generated by "unrelated" self-maps f 1 , f 2 : V → V should not be too similar.For example, they should not have identical canonical heights [16], they should not have infinitely many common preperiodic points [2; 8; 11], their orbits should not have infinite intersection [10], and arithmetically their orbits should not have unexpectedly large common divisors [15].It is not always clear what "unrelated" should mean, but in any case it includes the assumption that f 1 and f 2 do not share a common iterate.
Similarly, we expect that the points in semigroup orbits generated by all finite compositions of "unrelated" maps f 1 and f 2 should be asymptotically large [4; 13] when ordered by height, where now unrelated means that the semigroup is not unexpectedly small.For example, the semigroup is small if it contains no free subsemigroups requiring at least 2 generators; cf.[3].
In this note, we study the size of semigroup orbits over finite fields.In particular, we show that, under suitable hypotheses, a free semigroup of maps defined over a number field generates many large orbits when reduced modulo primes.See [1; 6; 7; 21] for additional results in this vein.Definition 1.We set notation that will remain in effect throughout this note.
K ‫ޑ/‬ a number field V /K a smooth projective variety defined over K r ≥ 1 an integer S = { f 1 , . . ., f r } a set of morphisms f i : V → V defined over K d 1 , . . ., d r real numbers satisfying the semigroup generated by S under composition Orb S (P) the orbit { f (P) : f ∈ M S } of a point P ∈ V The following property will play a crucial role in some of our results.
Definition 2. A point P ∈ V is called strongly S-wandering if the evaluation map (1) M S → V, f → f (P), is injective.
Remark 3. If V = ‫ސ‬ 1 and S is any sufficiently generic set of maps as described in Section 3, then the set of points that fail to be strongly S-wandering is a set of bounded height.In particular, it follows in this case that all infinite orbits contain strongly wandering points, and this weaker condition is sufficient for our orbit bounds.
Our goal is to study the number of points in the reduction of Orb S (P) modulo primes.We set some additional notation, briefly recall a standard definition, and then define our principal object of study.
Definition 4. Let p ∈ Spec(R K ), and let R p denote the localization of R K at p, and let k p = R p /pR p denote the residue field.A finite K -morphism f : V → V has good reduction at p if there is a scheme V p /R p that is proper and smooth over R p , and there is an R p -morphism F p : V p → V p whose generic fiber is f : V → V. 1   1 Intuitively, this means that we can find equations for V and for f that have coefficients in R K , and so that when we reduce the equations modulo p, the reduced variety Ṽ mod p is non-singular and the reduced map f : Ṽ → Ṽ is a morphism having the same degree as f .Of course, when we say "find equations", this needs to be done locally on an appropriately fine cover by affine neighborhoods.
We write Ṽp = V p × R p k p for the special fiber of V p .Properness implies that each point Q ∈ V (K ) extends to a section Q p ∈ V p (R p ), and the reduction Qp ∈ Ṽp (k p ) of Q modulo p is the intersection of the image of Q p with the fiber Ṽp , i.e., Similarly, the reduction fp of f modulo p is the restriction of F p to the special fiber Ṽp .
Remark 5. Continuing with notation from Definition 4, we note that if f has good reduction at p, then reduction modulo p commutes with evaluation, Further, composition commutes with reduction for good reduction maps.In other words, if f and g have good reduction at p, then f • g p = fp • gp .Definition 6.Let p ∈ Spec(R K ).Continuing with notation from Definition 4, let f 1 , . . ., f r : V → V be maps that have good reduction modulo p, and let P ∈ V (K ).Then the reduction of the S-orbit of P modulo p is the set Orb S ( P mod p) := fp ( Pp ) : f ∈ M S .

We define
m p := m p (S, P) = # Orb S ( P mod p) to be the size of the mod p reduction of Orb S (P).(If any of the maps f 1 , . . ., f r has bad reduction at p, then we formally set m p = ∞.) Our main result is an analytic formula that implies that m p is not too small on average.Theorem 7. Assume that M S is a free semigroup, that P ∈ V (K ) is a strongly S-wandering point, and that r = #S ≥ 2. Then there exists a constant C 1 = C 1 (K , V, S, P) such that, for all ϵ > 0, (2) Remark 8.The principal result of the paper [21] is an estimate exponentially weaker than (2) in the case that r = #S = 1, while a principal result of the paper [17] is an estimate that exactly mirrors (2) with m p equal to the number of points on the mod p reduction of the multiples of a point on an abelian variety.Thus the present paper, as well as the papers [4; 13], suggest that the analogy arithmetic of points of an abelian variety ⇐⇒ arithmetic of points in orbits of a dynamical system described in [5] and [22, §6.5] may be more accurate when the dynamical system on the right-hand side is generated by at least two non-commuting maps, rather than using orbits coming from iteration of a single map.
Estimate (2) can be used to show that there are few primes p for which m p (S, P) is subexponential compared to Np.We quantify this assertion in the following corollary.
Corollary 9. Let S, M S and P be as in Theorem 7, and let δ and δ denote the (upper) logarithmic analytic densities on sets of primes as described in Definition 12.
(a) There is a constant C 2 = C 2 (K , V, S, P) such that holds for all 0 < γ < 1.
(b) Let L(t) be a subexponential function, i.e., a function with the property that In the special case that V = ‫ސ‬ 1 , we show that the conclusions of Theorem 7 and Corollary 9 are true for generic sets of maps.In the statement of the next result, we write Rat d for the space of rational maps of ‫ސ‬ 1 of degree d ≥ 2, so in particular Rat d is an affine variety of dimension 2d + 1; see [20, §4.3] for details.
Theorem 10.Let r ≥ 2, and let d 1 , . . ., d r be integers satisfying Then there is a Zariski dense subset such that the inequality (2) in Theorem 7 and the density estimates in Corollary 9 are true for all number fields K ‫,ޑ/‬ all S ∈ U(K ), and all P ∈ ‫ސ‬ 1 (K ) for which Orb S (P) is infinite.
The contents of this paper are as follows.In Section 2 we build upon prior work [17; 21] of the second author to prove Theorem 7 and Corollary 9. Then in Section 3 we use results from [10; 13; 23] to construct many sets of maps on ‫ސ‬ 1  for which the bounds in Section 2 apply.The key step is to construct a point in every infinite orbit that is strongly wandering.The construction is explicit, and in particular, Theorem 15 describes an explicit set U for which Theorem 10 is true.

The size of orbits modulo p
We start with a key estimate.
Proposition 11.Let S, M S and P be as in Theorem 7.For each m ≥ 2, we define an integral ideal There are constants C i = C i (K , V, S, P) for i = 3, 4 such that the following hold: (b) Assume that S generates a free semigroup, that P ∈ V (K ) is strongly Swandering, and that r = #S ≥ 2. Then Proof.(a) This is [21, Proposition 10].
(b) Next, since V and S are polarized with respect to some line bundle L, we may choose N ≥ 1 and an embedding V ⊆ ‫ސ‬ N such that the f i extend to self-morphisms of ‫ސ‬ N .Next, for notational convenience, we write m p for m p (S, P) and use the standard combinatorics notation [r ] = {1, 2, . . ., r }.Also to ease notation, we write For each good reduction prime p, we consider the map that sends a function f i to the image of P under reduction modulo p, If m p ≤ m, then r k > m p by our choice of k, so the map (4) cannot be injective (pigeonhole principle) and there exist Sine we have assumed that P is strongly wandering, i.e., that the map is injective, it follows that the global points are distinct, so the ideals generated by their "differences" are non-zero.More formally, [21, Lemma 9] says that there is an integral ideal C = C K ⊆ R K with the property that every point Q ∈ ‫ސ‬ N (K ) can be written with homogeneous coordinates Applying [21,Lemma 9] to our situation, for the given P ∈ V (K ), we can write with A 0 (i), . . ., A N (i) ∈ R K and such that the ideal ( 5) Then for p ∤ C we have We define a difference ideal and the product of the difference ideals and hence Since C depends only on K , it remains to estimate the norm of D ′ (m).
Let h( • ) denote the logarithmic Weil height on ‫ސ‬ N , and let A(i) for i ∈ [r ] k be the ideals defined by (5).Then, using [21, Proposition 7], we find for all i and j that 1 where C 5 is an absolute constant.Since NA(i ) and NA( j ) are smaller than NC, this implies that 1 Next we apply the height estimate which is a weak form of [12,Lemma 2.1].This yields 1 This gives Since m ≥ 2, we can absorb C 11 into C 10 , although we remark that if we leave in C 11 (K , V, S, P), then we can take C 10 to depend on only the degrees of the maps in S, This completes the proof of Proposition 11. □ Proof of Theorem 7. To ease notation, we let We start with two elementary estimates.First, the mean value theorem gives Second, an easy integral calculation gives We use these and our other calculations to estimate (by definition of g and G) (by definition of g and G) ≤ C 15 ϵ −1 (from ( 7)).□ Definition 12. Let P ⊂ Spec(R K ) be a set of primes.The upper logarithmic analytic density of P is Similarly, the logarithmic analytic density of P, denoted δ(P), is given by the same formula with a limit, instead of a lim sup.
(b) We let The assumption that L is subexponential means that for all µ > 0 there exists a constant C 16 (L , µ) depending only on L and µ such that We also note that We now fix a µ > 0 and estimate (since µ is fixed, so we can discard finitely many terms) This estimate holds for all µ > 0, so we find that which completes the proof that δ(P L ) = 0. □

Orbits of generic families of maps of ‫ސ‬ 1
In this section, we show that there are many sets of endomorphisms of ‫ސ‬ 1 for which Theorem 7 holds.To make this statement precise, we need some definitions.
Definition 13.Let f be a non-constant rational map of ‫ސ‬ 1 defined over The map f is critically simple if all of its critical values are simple.
Definition 14.Let f and g be non-constant rational maps of ‫ސ‬ 1 with respective critical value sets CritVal f and CritVal g .We say that f and g are critically separated if CritVal f ∩ CritVal g = ∅.
Our first result says that the conclusions of Theorem 7 and Corollary 9 hold for certain sets S that contain a pair of critically simple and critically separated maps and initial points P with infinite orbit.Theorem 15.Let K ‫ޑ/‬ be a number field, let S be a set of endomorphisms of ‫ސ‬ 1 defined over K containing a pair of critically simple and critically separated maps of degree at least 4, and let P ∈ ‫ސ‬ 1 (K ) be a point with infinite S-orbit.Then there is a constant C 17 = C 17 (K , S, P) such that for all ϵ > 0, Remark 16.In particular, there is a constant C 18 (S) such that Theorem 15 holds for all P ∈ ‫ސ‬ 1 (K ) satisfying h(P) > C 18 (S); see Lemma 18.
We start with a definition and some basic height estimates.In what follows, we fix an embedding V ⊆ ‫ސ‬ N and extend the maps f i to self-morphisms of ‫ސ‬ N ; here we use our assumption that S is polarizable with respect to some line bundle L.Moreover, h( • ) denotes the logarithmic Weil height on ‫ސ‬ N .
Lemma 18.Let V ‫ޑ/‬ be a variety, and let S = { f 1 , . . ., f r } be a set of polarized endomorphisms as described in Definition 1. Then there exists a constant C 19 = C 19 (S, V, L) such that the following statements hold for all Q ∈ V ‫:)ޑ(‬ (a) If Q is moderately S-preperiodic as described in Definition 17, then h(Q)≤C 19 .
In particular, this is true if Orb S (Q) is finite.
Proof.These estimates are proven in [4,Lemma 2.11].□ We combine Lemma 18 with the techniques in [13; 19] to obtain the following result for pairs of maps that are critically simple and critically separated.Proposition 19.Let f 1 and f 2 be endomorphisms of ‫ސ‬ 1 of degree at least 4, let S = { f 1 , f 2 }, and suppose that f 1 and f 2 are critically simple and critically separated.Then the following statements hold: (a) The semigroup M S is free.
(b) Let P ∈ ‫ސ‬ 1 ‫)ޑ(‬ be a point whose S-orbit Orb S (P) is infinite.Then there exists a point Q ∈ Orb S (P) such that Q is strongly S-wandering as described in Definition 2.
Proof.(a) See [13, Proposition (b) We fix a number field K over which P, f 1 , and f 2 are defined.Letting ⊂ ‫ސ‬ 1 × ‫ސ‬ 1 be the diagonal, we define three curves Then the main results in [19] (see also [13,Proposition 4.6]) imply that the curves 1 and 2 are each the union of and an irreducible curve of geometric genus ≥ 2, while 1,2 is itself an irreducible curve of geometric genus ≥ 2.More specifically, the assumption that f 1 and f 2 are critically simple implies from [19,Corollary 3.6] that C 1 ∖ and C 2 ∖ are irreducible, while the assumption that f 1 and f 2 are critically separated implies from [19, Proposition 3.1] that C 1,2 is irreducible.It then follows from [19, pages 208 and 210] that the geometric genera of these curves are given by the formulas In particular, the assumption that f 1 and f 2 have degree at least 4 ensures that these genera are at least 2. We now invoke Faltings's theorem [9], [14, Theorem E.0.1] to deduce that the set is finite.We note that the definition of says that for all P, Q ∈ ‫ސ‬ 1 (K ), we have (10) Let π 1 , π 2 : ‫ސ‬ 1 × ‫ސ‬ 1 → ‫ސ‬ 1 be the two projection maps, and let The fact that Orb S (P) ⊆ ‫ސ‬ 1 (K ) is infinite, combined with Northcott's theorem [18] saying that ‫ސ‬ 1 (K ) has only finitely many points of bounded height, implies that there exists a point Q ∈ Orb S (P) satisfying (11) h(Q) > C 21 .
We claim that Q is strongly wandering for S. To see this, suppose that ( 12) where without loss of generality we may assume that n ≥ m.Our goal is to prove that m = n and i k = j k for all 1 ≤ i ≤ n.
To ease notation, we let be the compositions with the initial map omitted.Thus ( 12) and ( 13) say that ( 14) It follows from ( 14) and ( 10) that one of the following is true: On the other hand, we know that by (11).But this contradicts Lemma 18. Hence (ii) and (iii) are false, so (i) is true.We recall that m ≤ n, so repeating this argument, we conclude that If m < n is a strict inequality, then we see that But then Lemma 18 implies that h(Q) ≤ C 19 ≤ C 21 , and we obtain a contradiction of (11).Thus m = n and i k = j k for all 1 ≤ k ≤ n, which completes the proof that Q is a strongly S-wandering point.□ We now have the tools in place to prove Theorem 15.
Proof of Theorem 15.Let S be the given set of endomorphisms of ‫ސ‬ 1 , and let f 1 and f 2 be the given maps in S that have degree at least 4 and that are critically simple and critically separated.We let We are given that the point P ∈ ‫ސ‬ 1 (K ) has infinite S-orbit, and hence by Northcott's theorem [18], there are points of arbitrarily large height in Orb S (P).We choose a point where C 19 (S ′ ) is the constant associated to the set S ′ appearing in Lemma 18.In particular, it follows from Lemma 18(b) and Northcott's theorem that Orb S ′ (Q ′ ) must be infinite.Then Proposition 19 implies that M S ′ is free and that there is a point that is strongly S ′ -wandering.Applying Theorem 7 to the set S ′ and the point Q, we deduce that for some constant C 17 depending on S, Q (and so P) and K .For this last conclusion, we have also used the fact that m p (S ′ , Q) ≤ m p (S, P), which is immediate from the inclusion Orb S ′ (Q) ⊆ Orb S (P).□ Proof of Theorem 10.We recall that Rat d denotes the space of rational maps of degree d.Then it follows from Theorems 1.1-1.4 in [19] that if d 1 , d 2 ≥ 4, then the set and Theorem 15 gives us that the desired inequality (2) for every S generated by a set of maps We conclude with a variant of Theorem 15 in which the maps are polynomials.We start with a definition.
C(x) = a power map or a Chebyshev polynomial.
Theorem 21.Let K ‫ޑ/‬ be a number field, let S be a set of endomorphisms of ‫ސ‬ 1 defined over K , and let P ∈ ‫ސ‬ 1 (K ) be a point such that Orb S (P) is infinite.Suppose further that S contains polynomials f 1 (x), f 2 (x) ∈ K [x] having the following properties: (1) Neither f 1 nor f 2 is power-like; see Definition 20.
(2) For all g ∈ ‫[ޑ‬x] satisfying deg(g) ≥ 2, we have Then there is a constant C 22 = C 22 (K , S, P) such that for all ϵ > 0, The proof of Theorem 21 is similar to the proof of Theorem 15, except that we use [10; 23] instead of [13; 19].As a first step, we need the following result, which is a polynomial analogue of Proposition 19.(b) The proof is very similar to the proof of Proposition 19, so we just give a brief sketch, highlighting the differences.We note that we have picked a coordinate function x on ‫ސ‬ 1 .We let ∞ ∈ ‫ސ‬ 1 be the pole of x and let ‫ށ‬ 1 = ‫ސ‬ 1 ∖ {∞}.Replacing P with another point in Orb S (P) if necessary, we may assume that P ̸ = ∞ is not the point at infinity.We choose a set S of primes of K so that the ring of Sintegers R K ,S satisfies P ∈ ‫ށ‬ 1 (R K ,S ) and f 1 (x), f 2 (x) ∈ R K ,S [x].
We use the map f 1 × f 2 : ‫ށ‬ 2 → ‫ށ‬ 2 to define three affine curves, Then [13,Proposition 4.5], itself a consequence of the main results of [10; 23], tells us that these are geometrically irreducible curves of geometric genus at least 1.
(This is where we use the assumptions (1) and ( 2) of Theorem 21 on f 1 and f 2 .)

Proposition 22 .
Let f 1 and f 2 be polynomials satisfying the hypotheses of Theorem 21, and let S = { f 1 , f 2 }.Then the following statements hold: (a) The semigroup M S is free.(b) Let P ∈ ‫ސ‬ 1 ‫)ޑ(‬ be a point whose S-orbit Orb S (P) is infinite.Then there exists a strongly S-wandering point Q ∈ Orb S (P).Proof.(a) See [13, Proposition 4.5].
Rat d 2 : f 1 and f 2 are critically simple and critically separated is Zariski dense in Rat d 1 × Rat d 2 .Then for any d 3 , . . ., d r ≥ 2, the set