4,235 research outputs found
Randomness and the ergodic decomposition
International audienceThe interaction between algorithmic randomness and ergodic theory is a rich field of investigation. In this paper we study the particular case of the ergodic decomposition. We give several positive partial answers, leaving the general problem open. We shortly illustrate how the effectivity of the ergodic decomposition allows one to easily extend results from the ergodic case to the non-ergodic one (namely Poincaré recurrence theorem). We also show that in some cases the ergodic measures can be computed from the typical realizations of the process
Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning
We introduce a simple analysis of the structural complexity of
infinite-memory processes built from random samples of stationary, ergodic
finite-memory component processes. Such processes are familiar from the well
known multi-arm Bandit problem. We contrast our analysis with
computation-theoretic and statistical inference approaches to understanding
their complexity. The result is an alternative view of the relationship between
predictability, complexity, and learning that highlights the distinct ways in
which informational and correlational divergences arise in complex ergodic and
nonergodic processes. We draw out consequences for the resource divergences
that delineate the structural hierarchy of ergodic processes and for processes
that are themselves hierarchical.Comment: 8 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/soi.pd
A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points
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
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Fully oscillating sequences and weighted multiple ergodic limit
We prove that fully oscillating sequences are orthogonal to multiple ergodic
realizations of affine maps of zero entropy on compact abelian groups. It is
more than what Sarnak's conjecture requires for these dynamical systems
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