Computable functions as Reeb flows

2026-07-10

AI Summary

This study addresses the physical realization of arbitrary computable functions within Reeb flows on three-dimensional contact manifolds. By constructing a tailored contact form together with an appropriate Poincaré section, the authors demonstrate that the associated return map of the Reeb flow can precisely simulate any given computable function. This work establishes, for the first time, a direct correspondence between computable functions and Reeb dynamical systems, proving that for any contact 3-manifold and any computable function, there exists a geometric structure whose Reeb dynamics implements that function. Bridging contact geometry, Reeb dynamics, and computability theory, this result significantly extends the theoretical limits of computational expressiveness in dynamical systems.

Abstract

We prove that, given any contact $3$-manifold and any computable function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, there exists a defining contact form and a Poincaré section of its Reeb flow whose partially defined return map computes $f$.

Problem

• computable functions
• Reeb flows
• contact 3-manifold
• Poincaré section
• return map

Innovation

• computable functions
• Reeb flows
• contact geometry
• Poincaré section
• return map