Nonlinear complex same-degree polynomials with infinite orbit intersection must have a common iterate

Analysis 1 Minute

You need to know: Complex numbers, set {\mathbb C}[X] of polynomials f(x) in complex variable x with complex coefficients, degree \text{deg}(f) of a polynomial f, notation f^n(z) for f(f(\dots f(z)\dots)), where f is repeated n times.

Background: For f \in {\mathbb C}[X], and initial point x_0 \in {\mathbb C}, the orbit O_f(x_0) is the set \{x_0, f(x_0), f(f(x_0)), \dots, f^n(x_0), \dots\}. We say that same degree polynomials f, g \in {\mathbb C}[X] have a common iterate if latex f^n = g^n for some n.

The Theorem: On 14th May 2007, Dragos Ghioca, Thomas Tucker, and Michael Zieve submitted to arxiv a paper in which they proved the following result. Let x_0, y_0 \in {\mathbb C} and f, g \in {\mathbb C}[X] with \text{deg}(f) = \text{deg}(g) > 1. If O_f(x_0) \cap O_g(y_0) is infinite, then f and g have a common iterate.

Short context: The study of orbits of polynomial maps is one of the main topics in complex dynamics. One natural question to ask is under what conditions two orbits may have infinite intersections. This may obviously be the case if the polynomials have the common iterate. The Theorem states that, for same-degree non-linear polynomials, this obvious sufficient condition is in fact necessary. The Theorem has applications in arithmetic geometry.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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