Computability of invariant measures: two counter-examples

We are interested in the computability of the invariant measures in a computable dynamical system. We construct two counter-examples. The first one has a unique SRB measure, which is not computable. The second one has no computable invariant measure at all. The systems are topological, i.e. continuous transformations on compact spaces, so they admit invariant measures. A topological dynamical system (X,T ) is given by: • a compact metric space X , • a continuous map T : X → X . The Krylov-Bogolyubov theorem states that every topological system admits an invariant Borel probability measure. This theorem is not constructive. In Section 3 we construct a computable system which admits no computable invariant measure. Remark. The proof of Krylov-Bogolyubov theorem uses the fact that the set of invariant measures is compact for the weak topology over the Borel probability measures. When the system is computable, the set of invariant measures is compact in an effective way, which implies that if the system is moreover uniquely ergodic (i.e. has only one invariant measure) then its unique invariant measure is computable.