Undecidability of Finite Orbit Recognition in Polynomial Maps

TL;DR

The paper proves that it is undecidable to determine whether the orbit of a point under a finite set of polynomial maps on $\mathbb{Z}^{\mathbb{N}}$ is finite (i.e., $S$-stable) or eventually periodic. The authors achieve this by encoding Conway's Game of Life, via Rendell's universal TM construction, into a polynomial map on $\mathbb{Z}^{\mathbb{N}}$, and then reducing the undecidability of Life configurations recurrences to the orbit-finiteness problem. This establishes a general undecidability result for orbit structure in infinite-dimensional arithmetic dynamics, linking computability theory to polynomial dynamics. The work thereby highlights intrinsic algorithmic intractability of detecting finite orbits in polynomial maps on $\mathbb{Z}^{\mathbb{N}}$, even for a single map, and complements recent decidability results in finite-dimensional settings.

Abstract

We prove the undecidability of determining whether a Turing machine yields an eventually periodic trajectory. From this, we deduce the undecidability of orbit finiteness in the polynomial dynamical system on infinite tuples of integers.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Whang, whang2023
• Theorem 1.5: Whang, whang2023
• Definition 2.1
• Theorem 2.2: Undecidability of the Halting Problem
• Definition 3.1
• Theorem 3.1
• proof