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About this paper
Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas
An integrated framework combining computability theory, algorithmic randomness, and ergodic theory demonstrates that Martin-Löf random points exhibit typical statistical behavior within computable dynamical systems. The framework establishes that their orbit complexity precisely quantifies the system's entropy measures, specifically Kμ(x, T) = hμ(T) for random points and sup K(x, T) = h(T).
Key insights
- Martin-Löf random points in computable dynamical systems are not only probabilistically typical but also individually exhibit statistical regularities, such as recurrence and Birkhoff typicality.
- The algorithmic information content required to describe an individual random orbit precisely quantifies the system's measure-theoretic entropy (Kolmogorov-Sinai entropy).
- The maximum algorithmic complexity of orbits across the entire space corresponds to the system's topological entropy, offering a purely algorithmic and topological proof of this relationship.
Summary by alphaXiv.
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