Orbit Problem Revisited

TL;DR

This paper revisits the classical orbit problem, extending it to an approximate version, and establishes its decidability status with one exception, contributing to the understanding of computational complexity in dynamical systems.

Contribution

It introduces the approximate orbit problem and determines its decidability, highlighting a key exception not previously identified.

Findings

Approximate orbit problem is decidable in most cases.

Identifies one case where the problem remains undecidable.

Extends classical results on the orbit problem.

Abstract

In this letter, we revisit the {\em orbit problem}, which was studied in \cite{HAR69,SHA79,KL86}. In \cite{KL86}, Kannan and Lipton proved that this problem is decidable in polynomial time. In this paper, we study the {\em approximate orbit problem}, and show that this problem is decidable except for one case.