A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite
binary sequence {\omega} and an effectively open set A of measure less than 1,
some tail of {\omega} is not in A. We first prove several results in the same
spirit and generalize them via an effective version of a weak form of
Birkhoff's ergodic theorem. We then use this result to get a stronger form of
it, namely a very general effective version of Birkhoff's ergodic theorem,
which improves all the results previously obtained in this direction, in
particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.Comment: Improved version of the CiE'10 paper, with the strong form of
  Birkhoff's ergodic theorem for random point

Bienvenu, Laurent

Day, Adam

Hoyrup, Mathieu

Mezhirov, Ilya

Shen, Alexander

English

arXiv

Improved version of the CiE'10 paper, with the strong form of Birkhoff's ergodic theorem for random pointsInternational audienceA theorem of Kucera states that given a Martin-Löf random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas

Hal-Diderot

A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points

AbstractWe prove the effective version of Birkhoffʼs ergodic theorem for Martin-Löf random points and effectively open sets, improving the results previously obtained in this direction (in particular those of Vyugin, Nandakumar and Hoyrup, Rojas). The proof consists of two steps. First, we prove a generalization of Kučeraʼs theorem, which is a particular case of effective ergodic theorem: a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1. Second, we show that the full statement of the effective ergodic theorem can be reduced to this special case. Both steps use the statement of classical ergodic theorem but not its usual classical proof. Therefore, we get a new simple proof of the effective ergodic theorem (with weaker assumptions than before).This result was recently obtained independently by Franklin, Greenberg, Miller and Ng

Day, Adam R.

Elsevier - Publisher Connector 

A constructive version of Birkhoffʼs ergodic theorem for Martin-Löf random points 

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https://hal.inria.fr/hal-00643629/file/ergodic-edited.pdf

A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points

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Hal-Diderot

Elsevier - Publisher Connector

HAL Descartes

INRIA a CCSD electronic archive server

Elsevier - Publisher Connector