Measure-theoretic Entropy
The entropy of a partition Q is defined as
The measure-theoretic entropy of a dynamical system with respect to a partition Q = {Q1, ..., Qk} is then defined as
Finally, the Kolmogorov–Sinai or metric or measure-theoretic entropy of a dynamical system is defined as
where the supremum is taken over all finite measurable partitions. A theorem of Yakov G. Sinai in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process is log 2, since almost every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals . Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2nx.
If the space X is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.
Read more about this topic: Measure-preserving Dynamical System
Famous quotes containing the word entropy:
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)