A quantitative approach to transitivity and mixing

We suggest new quantitative characteristics for discrete dynamical systems called degrees of transitivity, weak mixing and strong mixing. These are numbers from [0, 1]. For dynamical systems on a large class of compact metric spaces we show that the system is topologically transitive if and only if its degree of transitivity equals 1 and similarly for weak mixing and strong mixing. On the other hand we construct a simple dynamical system on the unit interval with all degrees positive.