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2 - Algorithmic randomness in ergodic theory

Published online by Cambridge University Press:  07 May 2020

Johanna N. Y. Franklin
Affiliation:
Hofstra University, New York
Christopher P. Porter
Affiliation:
Drake University, Iowa
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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.

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Algorithmic Randomness
Progress and Prospects
, pp. 40 - 57
Publisher: Cambridge University Press
Print publication year: 2020

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