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Finite Resolution Dynamics

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

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Author notes

  1. Stefano Luzzatto

    Present address: Abdus Salam, International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy

Authors and Affiliations

  1. Mathematics Department, Imperial College, 180 Queen’s Gate, London, SW7 2AZ, UK

    Stefano Luzzatto

  2. Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Paweł Pilarczyk

Authors
  1. Stefano Luzzatto
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  2. Paweł Pilarczyk
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Correspondence to Paweł Pilarczyk.

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Communicated by Peter Kloeden.

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Luzzatto, S., Pilarczyk, P. Finite Resolution Dynamics. Found Comput Math 11, 211–239 (2011). https://doi.org/10.1007/s10208-010-9083-z

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  • DOI: https://doi.org/10.1007/s10208-010-9083-z

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