Symbolic dynamics for nonhyperbolic systems

TL;DR

This paper introduces index systems as a new tool for analyzing nonhyperbolic dynamical systems, enabling symbolic dynamics similar to hyperbolic cases, and is applicable to systems with weak expansiveness.

Contribution

The paper presents index systems that extend symbolic dynamics methods to nonhyperbolic systems, providing a robust approach for computer-assisted analysis.

Findings

Index systems can be constructed for systems with weak expansiveness.

They mimic hyperbolic expansion and contraction in a topologically robust manner.

Index systems facilitate rigorous computer-based analysis of nonhyperbolic dynamics.

Abstract

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.