Finite Resolution Dynamics

• Published in: Foundations of computational mathematics Vol. 11; no. 2; pp. 211 - 239
• Main Authors: Luzzatto, Stefano, Pilarczyk, Paweł
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.04.2011; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Approximation; Combinatorial analysis; Combinatorics; Computer Science; Dynamical systems; Economics; Equivalence; Finite element analysis; Foundations; Graphs; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Open cover; 37M99; Mixing; Rigorous numerics; 65G20; Dynamical system; Combinatorial dynamics; Transitivity; Algorithm; 65P20; Finite resolution; Directed graph
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-010-9083-z

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 Hénon attractor is mixing at all resolutions coarser than 10 −5 .