Computable functions as Reeb flows
2026-07-10 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors show that for any special type of 3D space called a contact manifold, and for any computable function (a rule that gives an output for each number input), you can find a way to describe the space so that a certain kind of flow inside it can perform the computation. This flow is related to something called the Reeb flow, and the return map of this flow captures the function's behavior. Essentially, they connect geometric structures with computational abilities within this setting.
contact 3-manifoldcontact formReeb flowPoincaré sectionreturn mapcomputable functiondynamical systemstopologydifferential geometry
Authors
Kai Cieliebak, Ángel González-Prieto, Eva Miranda
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 $f$.