An Algorithm to Resolve Dynamics in Outer Approximations

Abstract

Outer approximations present a way to conclude rigorous results about the dynamics of a continuous function f : X → X using combinatorial algorithms. In particular, information about the dynamics is captured by a lattice epimorphism ω from the lattice of forward invariant sets to the lattice of attractors associated with an outer approximation. Given a minimal outer approximation of a continuous function f, we explore the existence of a lift τ of ω. We show that this does not exist in general and introduce an algorithm Resolve-OA that aims to refine the minimal outer approximation to produce an outer approximation that preserves the information about the dynamics and for which a lift τ of ω exists. For simplicity, we focus on continuous functions from the unit cube [0, 1]^d to itself. We introduce the notion of cubed complexes on the unit cube [0, 1]^d and an operation of binary sub-division that allows us to refine the cubed complex. We present Resolve-OA in this context.