TY - JOUR

T1 - A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties

AU - Galatolo, Stefano

AU - Hoyrup, Mathieu

AU - Rojas, Cristóbal

N1 - Funding Information:
Partly supported by ANR Grant 05 2452 260 OX. Corresponding author. Tel.: +39 3383360942. E-mail addresses: [email protected] (S. Galatolo), [email protected] (M. Hoyrup), [email protected] (C. Rojas).

PY - 2009/5/17

Y1 - 2009/5/17

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 Ai with effectively summable measures, there are computable points which are not contained in infinitely many Ai. As a consequence of this we obtain the existence of computable points which follow the 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 Lipschitz 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.

AB - 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 Ai with effectively summable measures, there are computable points which are not contained in infinitely many Ai. As a consequence of this we obtain the existence of computable points which follow the 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 Lipschitz 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 - Birkhoff ergodic theorem

KW - Computable analysis

KW - Computable dynamics

KW - Computable probability measures

KW - SRB measure

UR - http://www.scopus.com/inward/record.url?scp=64249152274&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2009.02.010

DO - 10.1016/j.tcs.2009.02.010

M3 - Article

AN - SCOPUS:64249152274

VL - 410

SP - 2207

EP - 2222

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 21-23

ER -