Metric Entropy of Dynamical System

We shall start by giving the definition of the entropy of dynamical system. Consider dynamical systems with discrete time. The phase space of dynamical system is denoted by M . It is equipped with σ-algebra M and a probability measure μ defined on M. In the general ergodic theory dynamics is given by a measurable transformation T of M onto itself preserving the measure μ. It is enough for many applications to assume that M is the Lebesgue space, i.e., it is isomorphic (as a measure space) to the unit interval with the usual Lebesgue measure. The concept of Lebesgue space was introduced in the works of Halmos, von Neumann and Rokhlin and now is widely used. For many applications of ergodic theory, it is sufficient to consider Lebesgue spaces.