Which sequences are orbits?

Daniel A. Nicks; David J. Sixsmith

arxiv: 1907.11006 · v1 · pith:AURUZFWEnew · submitted 2019-07-25 · 🧮 math.DS · math.CV

Which sequences are orbits?

Daniel A. Nicks , David J. Sixsmith This is my paper

Pith reviewed 2026-05-24 16:03 UTC · model grok-4.3

classification 🧮 math.DS math.CV

keywords complex dynamicsorbits under iterationholomorphic functionsexistenceuniquenessinverse problems

0 comments

The pith

Given a sequence of complex numbers, existence and uniqueness of a holomorphic function having it as an orbit under iteration are delicate questions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reverses the usual direction in dynamical systems by starting with a sequence rather than a function and asking which functions, if any, make the sequence an orbit under iteration. Restricting to the complex plane, this produces questions of existence and uniqueness for holomorphic functions. The authors resolve some of these questions while showing that the issues are quite delicate. A sympathetic reader would care because this inverts the standard construction of orbits and exposes how analytic constraints shape which sequences can arise.

Core claim

Starting from an arbitrary sequence of complex numbers, the problem of finding holomorphic functions on domains in the complex plane such that the sequence equals x, f(x), f(f(x)), … is one of existence and uniqueness; both properties turn out to depend sensitively on the sequence itself, and the paper resolves specific instances of these questions.

What carries the argument

The orbit condition x_{n+1} = f(x_n) for a function f, treated as the defining relation that must be solved for f given the sequence.

If this is right

Certain sequences admit no holomorphic function for which they are orbits.
For other sequences the function, when it exists, is unique.
Analytic continuation and identity theorems in the plane impose strong restrictions on possible orbits.
The same sequence may be an orbit for different functions when the domain is allowed to vary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The inverse-orbit viewpoint may classify entire families of holomorphic maps by the sequences they can produce.
Similar existence questions could be posed for real-analytic or meromorphic functions on the Riemann sphere.
Explicit constructions for sequences that do work could generate new examples of dynamical systems with prescribed orbit behavior.

Load-bearing premise

The functions under consideration are holomorphic or meromorphic on domains in the complex plane.

What would settle it

Take any concrete sequence such as 0, 1, 2, 3, … and check directly whether a holomorphic f exists satisfying f(n) = n+1 for all natural numbers n.

read the original abstract

In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study. In particular, restricting to the complex plane, we start with a sequence of complex numbers and study the functions (if any) for which this sequence is an orbit under iteration. This gives rise to questions of existence and of uniqueness. We resolve some questions, and show that these issues can be quite delicate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper flips the usual complex dynamics setup to ask which maps make a given sequence an orbit and claims some existence/uniqueness results, but the delicacy only makes sense under an unstated holomorphic restriction.

read the letter

The core move here is reversing the usual question in discrete dynamics on the complex plane. Instead of fixing a function and studying its orbits, they fix a sequence and ask what functions, if any, have that sequence as an orbit under iteration. They tackle existence and uniqueness and note that the answers can be delicate. That reversal appears new relative to the cited literature and gives a clean way to frame the questions. They resolve some of the existence and uniqueness issues, which is the actual new content. The work is careful about the setting once the holomorphic condition is in place, and the abstract is consistent on its own terms. The main soft spot is that the abstract never states the restriction to holomorphic (or meromorphic) maps. Without that, existence is trivial for arbitrary functions—you just define the map on the orbit points and extend arbitrarily—and uniqueness fails immediately. The claim that the issues are “quite delicate” therefore depends on the standard complex-dynamics regularity that is left implicit. If the body makes the restriction explicit and proves the results under it, the gap is minor; if the paper tries to claim the delicacy for general functions, the central assertion does not hold. The paper is aimed at people already working in complex dynamics who want to think about orbit structures from the sequence side. It is not going to change the field or open new technology, but the questions are well-posed and the answers look honest within the subfield. I would send it to peer review because the framing is coherent and the technical questions are substantive enough to deserve referee attention, even if the results stay modest.

Referee Report

2 major / 1 minor

Summary. The paper reverses the standard direction in discrete dynamical systems on the complex plane: given a sequence of complex numbers, it studies the functions (if any) for which the sequence is an orbit under iteration. It resolves some existence and uniqueness questions and shows that these issues can be quite delicate.

Significance. If the results hold under the appropriate regularity conditions, the work offers a novel perspective on inverse problems in complex dynamics by characterizing which sequences can arise as orbits. This could inform the theory of iteration and provide tools for distinguishing orbit sequences from arbitrary ones.

major comments (2)

[Abstract] Abstract: the claim that 'these issues can be quite delicate' is not supported without an explicit restriction on the class of functions. For arbitrary (non-analytic) functions on ℂ, existence is immediate by enumerating the orbit points, setting f(a_n)=a_{n+1}, and extending arbitrarily off the orbit; uniqueness likewise fails. The delicacy assertion therefore depends on an unstated assumption of holomorphicity (or meromorphicity) on domains in ℂ, which must be declared at the outset to make the central claims non-trivial.
[§1] §1 (Introduction): the setting is described only as 'restricting to the complex plane' without specifying the regularity class of the functions under consideration. This omission is load-bearing because the existence/uniqueness questions change character completely once holomorphicity is imposed; the paper must state the precise function space (e.g., holomorphic maps on domains in ℂ) before presenting any resolutions.

minor comments (1)

The abstract supplies no equations, proofs, or concrete examples, so the resolutions of the existence/uniqueness questions cannot be verified from the provided text alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We agree that the manuscript must state the regularity class (holomorphic functions) explicitly at the outset to render the existence and uniqueness questions non-trivial, and we will revise the abstract and introduction accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'these issues can be quite delicate' is not supported without an explicit restriction on the class of functions. For arbitrary (non-analytic) functions on ℂ, existence is immediate by enumerating the orbit points, setting f(a_n)=a_{n+1}, and extending arbitrarily off the orbit; uniqueness likewise fails. The delicacy assertion therefore depends on an unstated assumption of holomorphicity (or meromorphicity) on domains in ℂ, which must be declared at the outset to make the central claims non-trivial.

Authors: We agree. The paper concerns holomorphic functions on domains in the complex plane, but the abstract does not declare this restriction explicitly. We will revise the abstract to state that we consider holomorphic functions, which makes the delicacy of existence and uniqueness non-trivial. revision: yes
Referee: [§1] §1 (Introduction): the setting is described only as 'restricting to the complex plane' without specifying the regularity class of the functions under consideration. This omission is load-bearing because the existence/uniqueness questions change character completely once holomorphicity is imposed; the paper must state the precise function space (e.g., holomorphic maps on domains in ℂ) before presenting any resolutions.

Authors: We concur that the introduction must specify the function class at the outset. We will revise the opening of Section 1 to state explicitly that we restrict attention to holomorphic maps on domains in ℂ. revision: yes

Circularity Check

0 steps flagged

No circularity; theoretical questions resolved without reduction to inputs

full rationale

The paper reverses the usual dynamical systems setup by beginning with a sequence in the complex plane and investigating existence/uniqueness of functions for which the sequence is an orbit under iteration. The provided abstract and context contain no equations, fitted parameters, predictions derived from data subsets, or load-bearing self-citations. The central claims concern delicacy of existence/uniqueness under (implicit) regularity conditions standard to the field, but these are resolved as mathematical questions rather than constructed from prior results or ansatzes within the paper itself. The derivation chain is therefore self-contained and independent of its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No free parameters, invented entities, or ad-hoc axioms are visible. The work rests on standard assumptions of complex analysis and iteration theory.

axioms (1)

domain assumption Functions are holomorphic on domains in the complex plane
Implicit in the setting of complex dynamical systems and the mention of the complex plane.

pith-pipeline@v0.9.0 · 5602 in / 1067 out tokens · 15907 ms · 2026-05-24T16:03:50.747707+00:00 · methodology

Which sequences are orbits?

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)