Finite Resolution Dynamics

TL;DR

This paper introduces a mathematical framework for analyzing dynamical systems at finite resolutions using combinatorial methods, enabling computable verification of properties like transitivity and mixing.

Contribution

It develops a new finite resolution model for dynamical systems, with algorithms to verify properties such as transitivity and mixing at various scales.

Findings

The Henon attractor is mixing at resolutions coarser than 10^-5.

Provides effective algorithms for property verification.

Uses combinatorial multivalued maps to approximate topology.

Abstract

We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Henon attractor is mixing at all resolutions coarser than 10^-5.