Dragos Ghioca

Intersections of polynomial orbits, and a dynamical Mordell–Lang conjecture

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell–Lang conjecture.

Linear relations between polynomial orbits

We study the orbits of a polynomial f 2CŒX , namely, the sets 1 ;f . /; f .f . //; : : :o with 2 C. We prove that if two nonlinear complex polynomials f;g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C with a d -tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.

Preperiodic points for families of rational maps

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.

The Dynamical Mordell–Lang Conjecture

* Introduction* Background material* The dynamical Mordell-Lang problem* A geometric Skolem-Mahler-Lech theorem* Linear relations between points in polynomial orbits* Parametrization of orbits* The split case in the dynamical Mordell-Lang conjecture* Heuristics for avoiding ramification* Higher dimensional results* Additional results towards the dynamical Mordell-Lang conjecture* Sparse sets in the dynamical Mordell-Lang conjecture* Denis-Mordell-Lang conjecture* Dynamical Mordell-Lang conjecture in positive characteristic* Related problems in arithmetic dynamics* Future directions* Bibliography* Index

Equidistribution and integral points for Drinfeld modules

We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegels integral points theorem over number fields and to an analog of a theorem of Schinzels regarding the order of a point modulo certain primes.

The Mordell-Lang theorem for Drinfeld modules

We study the quasi-endomorphism ring of infinitely definable subgroups in separably closed fields. Based on the results we obtain, we are able to prove a Mordell-Lang theorem for Drinfeld modules of finite characteristic. Using specialization arguments we are able to prove also a Mordell-Lang theorem for Drinfeld modules of generic characteristic.

A gap principle for dynamics

Let f(1), ... , f(g) is an element of C(z) be rational functions, let Phi = (f(1), ... ,f(g)) denote their coordinate-wise action on (P-1)(g), let V subset of (P-1)(g) be a proper subvariety, and let P be a point in (P-1)(g)(C). We show that if S = {n >= 0 : Phi(n)(P) is an element of V(C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n <= N such that Phi(n)(P) is an element of V(C) is less than log(k)N, where log(k) denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport-koth and Mumford.

A dynamical version of the Mordell-Lang conjecture for the additive group

We prove a dynamical version of the Mordell‐Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to those employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

THE ISOTRIVIAL CASE IN THE MORDELL-LANG THEOREM

We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety G defined over a finite field with a closed subvariety X C G. We also study a related question in the context of a power of the additive group scheme.

Portraits of preperiodic points for rational maps

Let