A note on the Turing universality of homogeneous potential wells and geodesible flows

TL;DR

This paper investigates the Turing universality of certain dynamical systems, focusing on flows with strongly adapted 1-forms, and introduces a new class of flows with intermediate properties that preserve computational universality.

Contribution

It demonstrates that homogeneity in dynamical systems creates an intermediate class of flows that maintain Turing universality, expanding understanding of flow classifications.

Findings

Homogeneous potential wells can embed Turing machines.

A new class of flows between adapted and geodesible flows is identified.

Turing universality persists under slight modifications of the system.

Abstract

We explore some properties of flows with strongly adapted 1-forms, originally discovered in (Tao 2017), which can be used to embed Turing machines into dynamical systems. In particular, we discuss some relations to geodesible flows, and show that even a slight modification of the dynamical system, such as homogeneity, can lead to an intermediate class of flows between adapted flows and geodesible flows, while still retaining Turing universality.