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.
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Definition
A measure-preserving dynamical system 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, \mu)</math>
with the following structure:
- <math>X</math> is a set,
- <math>\mathcal{B}</math> is a σ-algebra over <math>X</math>,
- <math>\mu:\mathcal{B}\rightarrow[0,1]</math> is a probability measure, so that <math>\mu(X)=1</math>, and
- <math>T:X\rightarrow X</math> is a measurable transformation which preserves the measure <math>\mu</math>, i. e. each measurable <math>A\subseteq X</math> satisfies
- <math>\mu(T^{-1}A)=\mu(A).\,</math>
Examples
Examples include:
- For example, μ could be the normalized angle measure dθ/2π on the unit circle, and T a rotation. See equidistribution theorem.
- The Bernoulli scheme.
- With the definition of an appropriate measure, a subshift of finite type.
Discussion
One may wonder why the seemingly simpler identity
- <math>\mu(T(A))=\mu(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.
Homomorphisms
The concept of a homomorphism and an isomorphism may be defined.
Consider two dynamical systems <math>(X, \mathcal{A}, \mu, T)</math> and <math>(Y, \mathcal{B}, \nu, S)</math>. Then a mapping
- <math>\phi:X \to Y</math>
is a homomorphism of dynamical systems if it satisfies the following three properties:
- The map φ is measurable,
- For each <math>B \in \mathcal{B}</math>, one has <math>\mu (\phi^{-1}B) = \nu(B)</math>,
- For μ-almost all <math>x \in X</math>, one has <math>\phi(Tx) = S(\phi x)</math>.
The system <math>(Y, \mathcal{B}, \nu, S)</math> is then called a factor of <math>(X, \mathcal{A}, \mu, T)</math>.
The map φ is an isomoprhism of dynamical systems if, in addition, there exists another mapping
- <math>\psi:Y \to X</math>
that is also a homomorphism, which satisfies
- For μ-almost all <math>x \in X</math>, one has <math>x = \psi(\phi x)</math>
- For ν-almost all <math>y \in Y</math>, one has <math>y = \phi(\psi y)</math>.
Generic points
A point <math>x \in X</math> is called a generic point if the orbit of the point is distributed uniformly according to the measure.
Symbolic names and generators
Let <math>Q=\{Q_1,\ldots,Q_k\}</math> be a partition of X into k measurable pair-wise disjoint pieces. Given a point <math>x \in X</math>, clearly x belongs to only one of the <math>Q_i</math>. Similarly, the iterated point <math>T^nx</math> can belong to only one of the parts as well. The symbolic name of x, with regards to the partition Q, is the sequence of integers <math>\{a_n\}</math> such that
- <math>T^nx \in Q_{a_n}</math>.
The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. A partition Q is called a generator if μ-almost every point x has a unique symbolic name.
Operations on partitions
Given a partition <math>Q=\{Q_1,\ldots,Q_k\}</math> and a dynamical system <math>(X, \mathcal{B}, T, \mu)</math> , we define <math>T</math>-pullback of <math>Q</math> as
- <math> T^{-1}Q = \{T^{-1}Q_1,\ldots,T^{-1}Q_k\}</math>
Further, given two partitions <math>Q=\{Q_1,\ldots,Q_k\}</math> and <math>R=\{R_1,\ldots,Q_m\}</math>, we define their refinement <math> Q \vee R </math> as
- <math> Q \vee R = \{Q_i \cap R_j| i=1,\ldots,k , j=1,\ldots,m , \mu(Q_i \cap R_j) > 0 \} </math>
With these two constructs we may define refinement of an iterated pullback
- <math> \vee_{n=0}^N T^{-n}Q = \{Q_{i_0} \cap T^{-1}Q_{i_1} \cap ... \cap T^{-N}Q_{i_N} | i_l = 1,\ldots,k , l=0,\ldots,N , \mu(Q_{i_0} \cap T^{-1}Q_{i_1} \cap ... \cap T^{-N}Q_{i_N})>0 \} </math>
which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.
Measure-theoretic Entropy
The entropy of a partition Q is defined as
- <math>H(Q)=-\sum_{m=1}^k \mu (Q_m) \log \mu(Q_m)</math>
The measure-theoretic entropy of a dynamical system <math>(X, \mathcal{B}, T, \mu)</math> with respect to a partition <math>Q=\{Q_1,\ldots,Q_k\}</math> is then defined as
- <math>h(T,Q) = \lim_{N \rightarrow \infty} \frac{1}{N} H(\vee_{n=0}^N T^{-n}Q) </math>
Finally, measure-theoretic entropy of a dynamical system <math>(X, \mathcal{B}, T, \mu)</math> is defined as
- <math>h(T) = \sup_{Q} h(T,Q)</math>
where the supremum is taken over all finite measurable partitions. A theorem of Ya. 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 <math>\log 2</math>, since every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals <math>[0,1/2)</math> and <math>[1/2,1]</math>. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of <math>2^nx</math>.
References
- Michael S. Keane, Ergodic theory and subshifts of finite type, (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X (Provides expository introduction, with exercises, and extensive references.)
- Lai-Sang Young, "Entropy in Dynamical Systems", appearing as Chapter 16 in Entropy, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003). ISBN 0-691-11338-6