Computability, Noncomputability, and Hyperbolic Systems

TL;DR

This paper investigates the computability of stable and unstable manifolds in hyperbolic systems, demonstrating local computability but global semi-computability, and establishing the computability of Smale's horseshoe.

Contribution

It provides new insights into the computability properties of hyperbolic invariant sets and manifolds, highlighting differences between local and global computability.

Findings

Locally, stable and unstable manifolds are computable.

Globally, these manifolds are semi-computable but not fully computable.

Smale's horseshoe is shown to be computable.

Abstract

In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot (though we prove they are semi-computable). We also show that Smale's horseshoe, the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit, is computable.