Effective properties of sets and functions in metric spaces with computability structure
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
An example of a computable absolutely normal number
Theoretical Computer Science
Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
Applications of Effective Probability Theory to Martin-Löf Randomness
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
Information and Computation
Computability of the Radon-Nikodym derivative
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
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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 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.