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  • 2 - Algorithmic randomness in ergodic theory

  • Edited by Johanna N. Y. Franklin, Hofstra University, New York, Christopher P. Porter, Drake University, Iowa
  • Book:
    Algorithmic Randomness
    Published online:
    07 May 2020
    Print publication:
    07 May 2020, pp 40-57

  • Summary

    Ergodic theory is concerned with dynamical systems -- collections of points together with a rule governing how the system changes over time. Much of the theory is concerned with the long term behavior of typical points-- how points behave over time, ignoring anomalous behavior from a small number of exceptional points. Computability theory has a family of precise notions of randomness: a point is "algorithmically random'' if no computable test can demonstrate that it is not random. These notions capture something essential about the informal notion of randomness: algorithmically random points are precisely the ones that have typical orbits in computable dynamical systems. For computable dynamical systems with or without assumptions of ergodicity, the measure 0 set of exceptional points for various theorems (such as Poincaré's Recurrence Theorem or the pointwise ergodic theorem) are precisely the Schnorr or Martin-Löf random points identified in algorithmic randomness.