Measurable Dynamical System

Let (Ωμ) be a measure space, and:Ω→ Ω such that for al. Then μ is said to be invariant with respect to if μ)=μ. If is invariant with respect to and is also a probability measure, then the quadruple (Ω) is called a measurable dynamical system. One of the aims of this project is to present a variety of examples of measurable dynamical systems showing their importance and relevance. rnLet (Ω,) be a measurable dynamical system and letdenotes the iterate of. We also define be the identity function on Ω. For ∈Ω, the sequence,… called the orbit of , describes the path of the point as it moves in Ω under iteration of the mapping . The second most important part of this project, called ergodic theory, studies the properties of this sequence. rnIn addition, isomorphism of two measurable dynamical systems is introduced, and the most powerful tool for deciding when two measurable dynamical systems are isomorphic, namely entropy, is presented

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