Lyapunov Exponents, Entropy and Dimension Page: 23
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CHAPTER 5
ENTROPY
The Lyapunov exponents give one way of quantifying chaotic behavior. Another is entropy.
There are two different versions of the entropy of a transformation: a topological version
and a measure theoretical version. While these differ for most maps the variational principle
gives a relationship for the two.
In this section the standard definitions of measure theoretical entropy, and topological
entropy are given. Several equivalent definitions are also given, along with some pointwise
formulas and approximations of measure theoretical entropy.
5.1 Measure-theoretic Entropy
Let (X, B, ,u) be a probability space, and let T : X - X be a measure preserving
transformation. To define the measure theoretic entropy of T, we will first need to develop
some specified partitions. Let {An} be a finite partition of X such that:
1.) For each Z, Ai is measurable and (Ai) > 0
2.) (UUAi)23
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Williams, Jeremy M. Lyapunov Exponents, Entropy and Dimension, thesis, August 2004; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc4559/m1/29/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .