The Dynamical Mordell–Lang Conjecture

Abstract

Abstract. We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let φ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of (P 1 ) g has only finite intersection with any curve contained in (P 1 ) g . Our proof uses results from p-adic dynamics together with an integrality argument.

Dynamical Mordell-Lang conjecture in positive characteristic

“…First we note that if α is preperiodic under the action of Φ, i.e., its orbit O Φ (α) is finite, then the conclusion holds easily (see also [BGT16, Proposition 3.1.2.9]). So, from now on, we assume α is not preperiodic under the action of Φ.…”

“…Considering X a semiabelian variety and Φ the translation by a point x ∈ X(K), one recovers the cyclic case in the classical Mordell-Lang conjecture from the above stated dynamical Mordell-Lang Conjecture; we refer the readers to [BGT16] for a survey of recent work on the dynamical Mordell-Lang conjecture.…”

“…On one hand, if we work with an arbitrary quasiprojective variety X, then Conjecture 1.1 is expected to be very difficult since even its counterpart in characteristic 0 is only known to hold in special cases, and generally, when it holds, its proof involves methods which fail in positive characteristic, such as the so-called p-adic arc lemma (see [BGT16,Chapter 4] and its main application from [BGT10] to the dynamical Mordell-Lang conjecture forétale endomorphisms in characteristic 0). On the other hand, even if X is assumed to be G N m and Φ is a group endomorphism, unless one assumes another hypothesis (either on Φ as in [BGT16, Chapter 13], or on V as in Theorem 1.3), Conjecture 1.1 is expected to be very difficult.…”

A Gap Principle for Subvarieties with Finitely Many Periodic Points

“…Then we might not be able to prove the dynamical Mordell-Lang Conjecture using the p-adic interpolation in those cases. However, by a discovery of Bell, Ghioca and Tucker (see [BGT16], Theorem 11.11.3.1 or Theorem 4.1 below), we might expect an approximating function G such that G(n) approximates f n (a) very closely. Suppose Q is a polynomial vanishing on V , then the roots Q • G give much restriction to those n such that f n (a) ∈ V .…”

“…We need Theorem 1.4 to avoid one case in which we cannnot say much about S V . A key technical result is Proposition 4.2, which is a modification of Theorem 11.11.3.1 of [BGT16].…”