The Ergodic Theorem
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چکیده
Measure-preserving systems arise in a variety of contexts, such as probability theory, information theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X → R, the series of values f(x), f(Tx), f(T x)... are the values of a physical observable at certain time intervals. Of importance in statistical mechanics is the long-term average of these observables:
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