🤖 AI Summary
This paper investigates the high-dimensional orbit problem: given a linear transformation, an initial point, and a target subspace, determine whether the iterated orbit reaches the subspace; and concurrently, its continuous analogue—whether the trajectory of a linear differential system intersects the target subspace. Both problems generalize the discrete and continuous Skolem problems, long-standing open questions in computability and dynamical systems. The authors innovatively reduce the problems to dynamics on the real projective space, developing a geometric framework that replaces traditional algebraic-number-theoretic approaches. This framework integrates tools from real projective geometry, linear dynamical systems, and algebraic geometry. As a result, they establish a unified decidability criterion for subspace orbit reachability. Their approach significantly simplifies and generalizes several prior results reliant on deep number-theoretic machinery, offering a novel paradigm for decidability analysis in linear dynamical systems.
📝 Abstract
The higher-dimensional version of Kannan and Lipton's Orbit Problem asks whether it is decidable if a target vector space can be reached from a starting point under repeated application of a linear transformation. This problem has remained open since its formulation, and in fact generalizes Skolem's Problem -- a long-standing open problem concerning the existence of zeros in linear recurrence sequences. Both problems have traditionally been studied using algebraic and number theoretic machinery. In contrast, this paper reduces the Orbit Problem to an equivalent version in real projective space, introducing a basic geometric reference for examining and deciding problem instances. We find this geometric toolkit enables basic proofs of sweeping assertions concerning the decidability of certain problem classes, including results where the only other known proofs rely on sophisticated number-theoretic arguments.