Dynamical Mordell-Lang Research Articles

Backward orbits of critical points

We examine the Galois groups of the extensions K((f′∘fn)−1(0))/K where K is a number field for polynomials f(x)∈K[x]. We use our understanding of this group to study the proportion of primes for which f has a p-adic attracting periodic point for a “typical” f and apply the result to the split case of the Dynamical Mordell-Lang Conjecture.

𝑝-adic interpolation of orbits under rational maps

Let L L be a field of characteristic zero, let h : P 1 → P 1 h:\mathbb {P}^1\to \mathbb {P}^1 be a rational map defined over L L , and let c ∈ P 1 ( L ) c\in \mathbb {P}^1(L) . We show that there exists a finitely generated subfield K K of L L over which both c c and h h are defined along with an infinite set of inequivalent non-archimedean completions K p K_{\mathfrak {p}} for which there exists a positive integer a = a ( p ) a=a(\mathfrak {p}) with the property that for i ∈ { 0 , … , a − 1 } i\in \{0,\ldots ,a-1\} there exists a power series g i ( t ) ∈ K p [ [ t ] ] g_i(t)\in K_{\mathfrak {p}}[[t]] that converges on the closed unit disc of K p K_{\mathfrak {p}} such that h a n + i ( c ) = g i ( n ) h^{an+i}(c)=g_i(n) for all sufficiently large n n . As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps ( h , g ) (h,g) of P 1 × X \mathbb {P}^1 \times X with g g an étale self-map of a quasiprojective variety X X .

Height gap conjectures, D-finiteness, and a weak dynamical Mordell–Lang conjecture

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(\Phi^n(x))$, where $\Phi\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $\Phi$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(\Phi^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.

A Gap Principle for Subvarieties with Finitely Many Periodic Points

AbstractLet $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.

The dynamical Mordell-Lang problem for intersection of two orbits

Let X be an algebraic variety defined over C, let α∈X(C) and Φ:X⟶X be a morphism. The Dynamical Mordell-Lang conjecture describes the intersection of an orbit OrbΦ(α) with a closed subvariety of X. In this paper, we study the intersections of orbits in polynomial dynamics. In particular, various upper bounds are derived for the orbit intersection problem when X is an affine n-space and self maps are polynomial morphisms of special type.

Dynamical Mordell�Lang and automorphisms of blow-ups

We show that if $\phi : X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $\phi^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, \phi)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. These results follow from a non-reduced analogue of the dynamical Mordell-Lang conjecture. Namely, let $\phi : X \to X$ be an etale endomorphism of a smooth projective variety $X$ over a field $k$ of characteristic zero. We show that if $Y$ and $Z$ are two closed subschemes of $X$, then the set $A_\phi(Y,Z) = \{n : \phi^n(Y) \subseteq Z\}$ is the union of a finite set and finitely many residue classes, whose modulus is bounded in terms of the geometry of $Y$.

The Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane

In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane.

Algebraic dynamics of the lifts of Frobenius

We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius. In particular, we prove a version of the dynamical Manin-Mumford conjecture and the dynamical Mordell-Lang conjecture for the coherent backward orbits for such endomorphisms. We also give a new proof of a dynamical version of the Tate-Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedding into a dynamical system for some endomorphism of a projective space.

The dynamical Mordell-Lang conjecture in positive characteristic

LetKKbe an algebraically closed field of prime characteristicpp, letN∈NN\in \mathbb {N}, letΦ:GmN⟶GmN\Phi :\mathbb {G}_m^N\longrightarrow \mathbb {G}_m^Nbe a self-map defined overKK, letV⊂GmNV\subset \mathbb {G}_m^Nbe a curve defined overKK, and letα∈GmN(K)\alpha \in \mathbb {G}_m^N(K). We show that the setS={n∈N:Φn(α)∈V}S=\{n\in \mathbb {N}\colon \Phi ^n(\alpha )\in V\}is a union of finitely many arithmetic progressions, along with a finite set and finitely manypp-arithmetic sequences, which are sets of the form{a+bpkn:n∈N}\{a+bp^{kn}\colon n\in \mathbb {N}\}for somea,b∈Qa,b\in \mathbb {Q}and somek∈Nk\in \mathbb {N}. We also prove that our result is sharp in the sense thatSSmay be infinite without containing an arithmetic progression. Our result addresses a positive characteristic version of the dynamical Mordell-Lang conjecture, and it is the first known instance when a structure theorem is proven for the setSSwhich includespp-arithmetic sequences.

On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture

Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear subvarieties L in P^n such that the intersection of O_F(P)and L is larger than expected. When F is the d'th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are super-spanned by O_F(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P or the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in O_F(P)-S are in linear general position in P^n.

A Euclidean Skolem–Mahler–Lech–Chabauty Method

Using the theory of o-minimality we show that the p-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if f1, . . . , fn is a finite sequence of real analytic functions fi : (−1, 1)→ (−1, 1) for which fi(0) = 0 and | f ′ i (0)| ≤ 1 (possibly zero), a = (a1, . . . , an) is an n-tuple of real numbers close enough to the origin and H(x1, . . . , xn) is a real analytic function of n variables, then the set {m ∈N : H( f ◦m 1 (a1), . . . , f ◦m n (an)) = 0} is either all of N, all of the odd numbers, all of the even numbers, or is finite.