Lattice Structures for Attractors III

TL;DR

This paper develops order-theoretic models of dynamical systems using lattice theory, introducing the Conley form to analyze spectral spaces and compute global system characteristics.

Contribution

It introduces the Conley form for bounded, distributive lattices and constructs set-theoretical models of spectral spaces for dynamical systems.

Findings

Conley form effectively models set differences in lattices.

Constructed spectral space representations facilitate dynamical system analysis.

Provides tools for computing global characteristics of dynamical systems.

Abstract

The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of `set-difference' taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretical models of spectral, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations build order-theoretic models of dynamical systems, which are used to develop tools for computing global characteristics of a dynamical system.