Linear relations between polynomial orbits

Abstract

We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f (α), f (f (α)), . . . } with α ∈ 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 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.Date: November 16, 2018. 1 m 1 1 , . . . , f m d d )(L) = L.When Theorem 1.2 applies, we can describe the intersection of L with the product of orbits. Our description involves the following more general notion of orbits: Definition 1.3. If Ω is a set and T is a set of maps Ω → Ω, then for ω ∈ Ω the orbit of ω under T is O T (ω) := {t(ω) : t ∈ T }.

Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture

“…(3) α 2 → α 1 as part of a 3-cycle of type (3) Not necessarily Lattès 2 (2, 2) α 1 , α 2 in disjoint 2-cycles of type (2) Same as (3,1) case 2 (2, 1) α 1 in a 2-cycle of type (2) α 2 of type (1) Not necessarily Lattès 2 (12) α 2 → α 1 as part of structure (12) Lattès of signature (2,2,2,2) 2 (11a) α 2 → α 1 → α 2 as part of structure (11) Lattès of signature (2,2,2,2) 2 (11b) α 2 → α 1 → β → α 1 as part of structure (11) Lattès of signature (2,2,2,2) 2 (11c) α 2 → β → α 1 → β as part of structure (11) Lattès of signature (2,2,2,2) 2 (10a) α 2 → α 1 as part of structure (10) Lattès of signature (2,2,2,2) 2 (10b) α 2 → β → α 1 as part of structure (10) Lattès of signature (2,2,2,2) 2 (8) α 2 → α 1 as part of structure of type (8) Not necessarily Lattès 2 (7, 7) α 1 , α 2 in disjoint structures of type (7) Same as (3,1) case 2 (7, 6) α 1 of type (7), α 2 disjoint of type (6) Not necessarily Lattès Table 2. Enumeration of µ-types for non-trivial maps If a = 2 then α 1 is the root of a tree of type (2), and b ≤ 2.…”

“…In the case where φ has a 3-cycle of type (3), we apply a conjugation if necessary to obtain 0 → 1 → ∞ → 0, giving a map of the form in Table 3. When φ has a fixed point of type (6) and another point of type (7), we take ∞ to be the point of type (6) and 0 to be a fixed point with 1 its only preimage of odd multiplicity. This gives the corresponding form in Table 3.…”

“…Suppose now that O φ (a) ∩ {0, ∞} = ∅. Because A = {0, ∞} is a set of m-branch abundant points for φ, it follows from Table 2 that φ has µ-type (8) or (7,6), since otherwise both 0 and ∞ have finite forward orbits. For both of these µ-types, precisely one of 0, ∞ has infinite forward orbit.…”

“…Depending on the choice of two-element subset A ⊂ V Θ , φ has µ-type (6,6), (7,6), or (7) if a ≡ 1 + i or 0 mod 2Λ, and φ has µ-type (1,1), (2,1), or (2) if a ≡ 1 or i mod 2Λ. Thus µ is either trivial with respect to {0, ∞} or has µ-type (7,6) or (2,1). We give further examples of these kinds in Section 10.7, and postpone their discussion for now.…”

On algebraic integers of bounded house and preperiodicity in polynomial semigroup dynamics

“…Polynomial decompositions. In this section we describe the polynomials for which the second iterate is, up to composition on both sides with linear polynomials, a monomial or Chebyshev polynomial, which is given by [6,Lemma 3.9]. This will motivate the first condition of Definition 1.1.…”