Portraits of preperiodic points for rational maps

Abstract

Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\varphi\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\varphi$ is totally ramified, and let $\alpha\in K$. We show that for all but finitely many pairs $(m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ there exists a place $\mathfrak{p}$ of $K$ such that the point $\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when $\varphi$ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple $(c_1,\dots , c_{d-1})\in k^{n-1}$ and for almost all pairs $(m_i,n_i)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ for $i=1,\dots, d-1$, there exists a polynomial $f\in k[z]$ of degree $d$ in normal form such that for each $i=1,\dots, d-1$, the point $c_i$ has preperiod $m_i$ and minimum period $n_i$ under the action of $f$

Preperiodic portraits for unicritical polynomials

“…The M = 0 case is precisely Theorem 1.1, and the M = 1 case follows from the fact (see [7,Lem. 4.24]) that if a polynomial admits a point of period N , then it also admits a point of portrait (1, N ).…”

Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

“…But this means that (−b, 1 + b) is a solution to the S-unit equation u + v = 1, so there are only finitely many values for b; and then the fact that b −1 (e − 1), b −1 (1 + b − e) is also a solution to the S-unit equation proves that there are only finitely many values for e. This completes the proof that GR 1 2 [P 4,10 ] • (K, S)/ PGL 2 (R S ) is finite. [[P 4,11 ]] We move the points so that 0 and ∞ are fixed by f and f (1) = 0. This puts f in the form f (x) = ax(x − 1)/(bx − c), with Res(f ) = a 2 c(c −b).…”

“…We dehomogenize c = 1, so f (x) = ax(x−1)/(bx−1). Then f −1 (∞) = {∞, b −1 }, and our assumption that we have a good reduction model for P 4,11 requires that b −1 be distinct from {0, 1, ∞} for all primes not in S. Thus b −1 ∈ R * S and b −1 − 1 ∈ R * S . The S-unit equation u − v = 1 has only finitely many solutions, so there are finitely many values for b.…”

Preperiodic portraits for unicritical polynomials over a rational function field

“…Finally, we note that these exceptions have arbitrarily large height: for each k ≥ 0, the points ϕ k 1 (0) and ϕ k 1 (∞) have height k, as do the points ϕ k 2 (0), ϕ k 2 (1/2), and ϕ k 2 (∞) -see Propositions 5.17 and 5.18. As mentioned above, Ghioca This is precisely the case of Theorem 1.2 in which α lies in the constant subfield K. The proof of Theorem 1.4 almost exclusively used the geometry of certain dynamical modular curves associated to the maps f d , whereas the proof of Theorem 1.2 requires Diophantine methods much like those used in [9]. In particular, the argument for the d ≥ 3 case of Theorem 1.2 provides a completely different proof of the d ≥ 3 case of Theorem 1.4 -except for the case M = 0, for which we simply refer to Theorem 1.4 for constant points.…”