A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties

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dc.creatorGalatolo, Stefano
dc.creatorHoyrup, Mathieu
dc.creatorRojas, Cristobal
dc.date2007-11-09
dc.date2008-06-30
dc.date.accessioned2026-07-07T09:47:03Z
dc.date.available2026-07-07T09:47:03Z
dc.descriptionIn the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are computable points which are not contained in infinitely many $A_{i}$. As a consequence of this we obtain the existence of computable points which follow the \emph{typical statistical behavior} of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and a certain ``logarithmic'' speed of convergence of Birkhoff averages over Lipshitz observables. This is applied to uniformly hyperbolic systems, piecewise expanding maps, systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base.
dc.descriptionRevised version. Several results are generalized
dc.identifierhttps://arxiv.org/abs/0711.1478
dc.identifierhttp://arxiv.org/abs/0711.1478
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/163738
dc.subjectClassical Analysis and ODEs
dc.subjectInformation Theory
dc.subjectDynamical Systems
dc.subjectProbability
dc.subjectStatistics Theory
dc.subject03F60, 37M25, 68U20, 11K45, 11Y99
dc.titleA constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties
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