  We survey an area of recent development, relating dynamics to theoretical
computer science. We discuss the theoretical limits of simulation and
computation of interesting quantities in dynamical systems. We will focus on
central objects of the theory of dynamics, as invariant measures and invariant
sets, showing that even if they can be computed with arbitrary precision in
many interesting cases, there exists some cases in which they can not. We also
explain how it is possible to compute the speed of convergence of ergodic
averages (when the system is known exactly) and how this entails the
computation of arbitrarily good approximations of points of the space having
typical statistical behaviour (a sort of constructive version of the pointwise
ergodic theorem).
