Finite Resolution Dynamics

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 H´enon attractor is mixing at all resolutions coarser than 10−5.

• Type: Journal
• Year: 2011
• Where: FOCM
• Authors: Stefano Luzzatto, Pawel Pilarczyk
• Keywords: Combinatorial Multivalued Map, Dynamical System, FOCM 2011, Precise Meaning