Measure-preserving dynamical system
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In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory.
It is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system
- <math>(X, \mathcal{B}, T, m)<math>
with the following structure:
- <math>X<math> is a set,
- <math>\mathcal{B}<math> is a <math>\sigma<math>-algebra over <math>X<math>,
- <math>m:\mathcal{B}\rightarrow[0,1]<math> is a probability measure, so that <math>m(X)=1<math>, and
- <math>T:X\rightarrow X<math> is a measurable transformation which preserves the measure <math>m<math>, i. e. each measurable <math>A\subseteq X<math> satisfies
- <math>m(T^{-1}A)=m(A).<math>
For example, m could be the normalised angle measure dθ/2π on the unit circle, and T a rotation.
One may wonder why the seemingly simpler identity
- <math>m(T(A))=m(A)<math>
is not used. Here is the problem: suppose T : [0, 1] → [0, 1] is defined by T(x) = (4x mod 1), i.e., T(x) is the "fractional part" of 4x. Then the interval [0.01, 0.02] is mapped to an interval four times as long as itself, but nonetheless the measure of T −1( [0.04, 0.08] ) = [0.01, 0.02] ∪ [0.251, 0.252] ∪ [0.501, 0.502] ∪ [0.751, 0.752] is no different from the measure of [0.04, 0.08]. That hypothesis suffices for the proofs of ergodic theorems. This transformation is measure-preserving.