Efficient automation of index pairs in computational Conley index theory

TL;DR

This paper introduces new automated algorithms for constructing index pairs in discrete Conley index theory, enabling rigorous entropy bounds for the standard map across various perturbations.

Contribution

The paper develops novel algorithms to automate index pair construction, advancing computer-assisted proofs in dynamical systems.

Findings

Rigorous lower bounds for topological entropy of the standard map.

Algorithms successfully applied at different perturbation levels.

Enhanced automation in Conley index computations.

Abstract

We present new methods of automating the construction of index pairs, essential ingredients of discrete Conley index theory. These new algorithms are further steps in the direction of automating computer-assisted proofs of semi-conjugacies from a map on a manifold to a subshift of finite type. We apply these new algorithms to the standard map at different values of the perturbative parameter {\epsilon} and obtain rigorous lower bounds for its topological entropy for {\epsilon} in [.7, 2].