# Which sequences are orbits?

## TL;DR

This paper explores the inverse problem in discrete dynamical systems on the complex plane, investigating which sequences can be orbits of some function, and addresses existence and uniqueness issues in this context.

## Contribution

It introduces the inverse orbit problem in complex dynamics, providing new insights into existence and uniqueness of functions generating given sequences as orbits.

## Key findings

- Some sequences are shown to be orbits of specific functions.
- Existence and uniqueness of functions for given sequences are complex and delicately dependent.
- The paper resolves certain cases of the inverse orbit problem.

## 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.