Poincaré recurrence for observations

A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan’s work, for a measure preserving system we study Poincaré recurrence f...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2010-11, Vol.362 (11), p.5845-5859
Main Authors: Rousseau, Jérôme, Saussol, Benoît
Format: Article
Language:English
Subjects:
Dynamical systems
Ergodic theory
Exact sciences and technology
Existence theorems
General mathematics
General, history and biography
Global analysis, analysis on manifolds
Hausdorff dimensions
Integers
Lebesgue measures
Lebesgue theorem
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Measure and integration
Multivariate analysis
Probability and statistics
Research article
Sciences and techniques of general use
Statistics
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan’s work, for a measure preserving system we study Poincaré recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relative to the push-forward are equal.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2010-05078-0