Lyapunov Exponents, Entropy and Dimension Page: 7
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theory. For a more thorough treatment see [Roy] or [Ro]. As a note: Measure spaces are
a collection of a set, X, a measure, p , and a sigma algebra E. In this paper we do not
reference the sigma algebra, as it will always be assumed to be the standard Borel sigma
algebra generated by the open sets on the manifold.
The next few definitions lay out some of the properties which the measures encountered
in this paper will have.
Definition 2.2.1. Let X, p be a measure space. The measure p is said to be a probability
measure if p(X)= 1.
Definition 2.2.2. Let (X, p) be a measure space, and let f : X -- X be measurable. A
measure p is called f-invariant if for every A C X, p(f-1(A)) = p(A). When the function
is unambiguous, we may simply call p an invariant measure
Peterson's derivation of ergodicity. Another property we would like our measures to have
is ergodicity. There are many different equivalent definitions of the word ergodic. For this
paper the following definition will be used.
Definition 2.2.3. Let (X, p) be a measure space, and let f : X -- X be measurable. An
invariant probability measure p is called ergodic if for every A C X such that f-1(A) = A,
either p(A) = 0 or p(A) = 1.
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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/m1/13/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .