Bounded height in families of dynamical systems

TL;DR

This paper proves that in a family of quadratic polynomials, the set of parameters where two points with specific algebraic properties intersect under iteration has bounded height, supporting a broader conjecture in arithmetic dynamics.

Contribution

It establishes a bounded height result for a special case of a new conjecture in arithmetic dynamics involving unlikely intersections.

Findings

The set of parameters t with certain intersection properties has bounded Weil height.

Supports a new bounded height conjecture in the context of arithmetic dynamics.

Fits into the framework of unlikely intersections in dynamical systems.

Abstract

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.