Existence of non-preperiodic algebraic points for a rational self-map of infinite order

Preperiodic points for families of rational maps

Let X be a smooth curve defined over Q̄, let a, b ∈ P(Q̄) and let fλ(x) ∈ Q̄(x) be an algebraic family of rational maps indexed by all λ ∈ X(C). We study whether there exist infinitely many λ ∈ X(C) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q ∈ Q̄[x] such that deg(P ) 2 + deg(Q), and if a, b ∈ Q̄ such that a is periodic for P (x)/Q(x), but b is not preperiodic f...

Portraits of Preperiodic Points for Rational Maps

Let K be a function field over an algebraically closed field k of characteristic 0, let φ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which φ is totally ramified, and let α ∈ K. We show that for all but finitely many pairs (m,n) ∈ Z≥0×N there exists a place p of K such that the point α has preperiod m and minimum period n under the action of φ. Thi...

Uniform Boundedness of Rational Points and Preperiodic Points

We ask questions generalizing uniform versions of conjectures of Mordell and Lang and combining them with the Morton–Silverman conjecture on preperiodic points. We prove a few results relating different versions of such questions.

Distribution of Rational Points: A Survey

The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: a Refined Conjecture

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over Q, assuming the conjecture that it is impossible to have rational points of period 4 or higher. In particular, we show under this assumption that the number of preperiodic points is at most 9. Elliptic curves of small conductor and the genus 2 modular curves X1(13), X1(16), and X1(18...