We describe a framework in which it is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the $L^{1}$ or $L^\infty $ distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations and experimental results on some one dimensional maps

In this paper we prove that the Poincaré map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τ r (x, x 0 ) is the time needed for the orbit of a point x to enter for the first time in a ball B r (x 0 ) centered at x 0 , with small radius r. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the SRB measure at x 0 : for each x 0 such that the local dimension d µ (x 0 ) exists,holds for µ almost each x. In a similar way it is possible to consider a quantitative recurrence indicator quantifying the speed of coming back of an orbit to its starting point. Similar results holds for this recurrence indicator.

Abstract. We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time τ r (x, x 0 ) needed for a typical point x to enter for the first time a ball B(x 0 , r) centered in x 0 , with small radius r scales as the local dimension at x 0 , i.e.This result is obtained by proving a kind of dynamical Borel-Cantelli lemma wich holds also in systems having polinomial decay of correlations.

We consider the question of computing invariant measures from an abstract point of view. We work in a general framework (computable metric spaces, computable measures and functions) where this problem can be posed precisely. We consider invariant measures as fixed points of the transfer operator and give general conditions under which the transfer operator is (sufficiently) computable. In this case, a general result ensures the computability of isolated fixed points and hence invariant measures (in given classes of "regular" measures). This implies the computability of many SRB measures.On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two interesting examples of computable dynamics, one having an SRB measure which is not computable and another having no computable invariant measure at all, showing some subtlety in this kind of problems.

We consider maps preserving a foliation which is uniformly contracting and a onedimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables.We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows. -hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm law. V.A. and M.J.P. were partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ, Balzan Research Project of J.Palis .