Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior of a typical point of the system. It was proved in [2] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative, simpler proof of this result. This implies that if also the invariant measure is computable then the pseudorandom points are a set which is dense (hence nonempty) on the support of the invariant measure. consider abstract algorithmic questions concerning the evolution of a dynamical system. In particular, the algorithmic estimation of the speed of convergence of ergodic averages and the recursive construction of points whose dynamics is typical for the system. This latter problem is related to the possibility of computer simulations, as actual computers can only calculate the evolution of computable initial conditions. Let X be a met...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas
• Keywords: Computable, CORR 2010, Education, Ergodic, Invariant Measure