Linear relations between polynomial orbits

Abstract

We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and 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^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture.

27 pages

References in corpus (6)

• The dynamical Mordell-Lang problem for etale maps
• Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture
• On Ritt's polynomial decomposition theorems
• Remarks on endomorphisms and rational points
• Naive Noncommutative Blowing Up
• Heights and preperiodic points of polynomials over function fields

Cited by in corpus (13)

• On Ritt's polynomial decomposition theorems
• A gap principle for dynamics
• Dynamics on Berkovich spaces in low dimensions
• The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps
• Growth rate of ample heights and the dynamical Mordell-Lang type conjecture
• Functional equations in polynomials
• Analogues of the Jordan-Holder theorem for transitive G-sets
• Periods of rational maps modulo primes
• The size of semigroup orbits modulo primes
• Unit equations on quaternions
• On intersections of polynomial semigroups orbits with plane lines
• Semigroups of rational functions: some problems and conjectures
• Lower bounds for genera of fiber products