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

@misc{citeulike:3036434, abstract = {In 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.}, author = {Galatolo, Stefano and Hoyrup, Mathieu and Rojas, Cristobal }, citeulike-article-id = {3036434}, eprint = {0711.1478}, keywords = {computability, dynamical-system, metric-space, numerical, probability}, month = {Jun}, posted-at = {2008-07-23 10:02:42}, priority = {2}, title = {A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties}, url = {http://arxiv.org/abs/0711.1478}, year = {2008} }

TY - ELEC ID - citeulike:3036434 L3 - citeulike-article-id:3036434 TI - A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties N2 - In 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. KW - computability KW - dynamical-system KW - metric-space KW - numerical KW - probability AU - Galatolo, Stefano AU - Hoyrup, Mathieu AU - Rojas, Cristobal PY - 2008/Jun/30/ UR - http://arxiv.org/abs/0711.1478 ER -