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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 dynamical systems, and ergodic theory in particular.


Contents

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}, \mu, T)</math>

with the following structure:

<math>\mu(T^{-1}A)=\mu(A).\,</math>

This definition can be generalized to the case in which <math>T</math> is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group) of transformations <math>T_{s} : X \to X</math> parametrized by <math>s \in \mathbb{Z}</math> (or <math>\mathbb{R}</math>, or <math>\mathbb{N} \cup \{ 0 \}</math>, or <math>[0, + \infty)</math>), where each transformation <math>T_{s}</math> satisfies the same requirements as <math>T</math> above. In particular, the transformations obey the rules

The earlier, simpler case fits into this framework by defining <math>T_{s} := T^{s}</math> for <math>s \in \mathbb{N}</math>.

Examples

Examples include:

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] ∪ [0.26, 0.27] ∪ [0.51, 0.52] ∪ [0.76, 0.77] 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:

  1. The map φ is measurable,
  2. For each <math>B \in \mathcal{B}</math>, one has <math>\mu (\phi^{-1}B) = \nu(B)</math>,
  3. 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 isomorphism of dynamical systems if, in addition, there exists another mapping

<math>\psi:Y \to X</math>

that is also a homomorphism, which satisfies

  1. For μ-almost all <math>x \in X</math>, one has <math>x = \psi(\phi x)</math>
  2. 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,R_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_\mu(T,Q) = \lim_{N \rightarrow \infty} \frac{1}{N} H(\vee_{n=0}^N T^{-n}Q) </math>

Finally, the measure-theoretic entropy of a dynamical system <math>(X, \mathcal{B}, \mu, T)</math> is defined as

<math>h_\mu(T) = \sup_{Q} h_\mu(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>.

If the space X is endowed with a metric, then the topological entropy may also be defined.

References

Categories


Dynamical systems | Entropy

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