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Measure-preserving_dynamical_system

Measure-preserving dynamical system

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.

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1 Definition
2 Examples
3 Discussion
4 Homomorphisms
5 Generic points
6 Symbolic names and generators
7 Operations on partitions
8 Measure-theoretic entropy
9 References

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

$(X, \mathcal{B}, \mu, T)$

with the following structure:

$X$ is a set,
$\mathcal{B}$ is a σ-algebra over $X$ ,
$\mu:\mathcal{B}\rightarrow[0,1]$ is a probability measure, so that $μ(X) = 1$ , and
$T:X\rightarrow X$ is a measurable transformation which preserves the measure $μ$ , i. e. each measurable $A\subseteq X$ satisfies

$\mu(T^{-1}A)=\mu(A).\,$

This definition can be generalized to the case in which $T$ 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 $T_{s} : X \to X$ parametrized by $s \in \mathbb{Z}$ (or $\mathbb{R}$ , or $\mathbb{N} \cup \{ 0 \}$ , or $[0, + \infty)$ ), where each transformation $T s$ satisfies the same requirements as $T$ above. In particular, the transformations obey the rules

$T_{0} = \mathrm{id}_{X} : X \to X$ , the identity function on $X$ ;
$T_{s} \circ T_{t} = T_{t + s}$ , whenever all the terms are well-defined;
$T_{s}^{-1} = T_{-s}$ , whenever all the terms are well-defined.

The earlier, simpler case fits into this framework by defining $T s : = T s$ for $s \in \mathbb{N}$ .

Examples

Examples include:

μ could be the normalized angle measure dθ/2π on the unit circle, and $T$ a rotation. See equidistribution theorem;

the Bernoulli scheme;

the interval exchange transformation;

with the definition of an appropriate measure, a subshift of finite type;

the base flow of a random dynamical system.

Discussion

One may wonder why the seemingly simpler identity

μ(T (A)) = μ(A)

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 $(X, \mathcal{A}, \mu, T)$ and $(Y, \mathcal{B}, \nu, S)$ . Then a mapping

$\phi:X \to Y$

is a homomorphism of dynamical systems if it satisfies the following three properties:

The map φ is measurable,
For each $B \in \mathcal{B}$ , one has $μ(φ - 1 B) = ν(B)$ ,
For μ-almost all $x \in X$ , one has $φ(T x) = S (φ x)$ .

The system $(Y, \mathcal{B}, \nu, S)$ is then called a factor of $(X, \mathcal{A}, \mu, T)$ .

The map φ is an isomorphism of dynamical systems if, in addition, there exists another mapping

$\psi:Y \to X$

that is also a homomorphism, which satisfies

For μ-almost all $x \in X$ , one has $x = ψ(φ x)$
For ν-almost all $y \in Y$ , one has $y = φ(ψ y)$ .

Generic points

A point $x \in X$ is called a generic point if the orbit of the point is distributed uniformly according to the measure.

Symbolic names and generators

Let $Q=\{Q_1,\ldots,Q_k\}$ be a partition of X into k measurable pair-wise disjoint pieces. Given a point $x \in X$ , clearly x belongs to only one of the $Q i$ . Similarly, the iterated point $T n x$ 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 ${a n}$ such that

$T^nx \in Q_{a_n}$ .

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 or generating partition if μ-almost every point x has a unique symbolic name.

Operations on partitions

Given a partition $Q=\{Q_1,\ldots,Q_k\}$ and a dynamical system $(X, \mathcal{B}, T, \mu)$ , we define $T$ -pullback of $Q$ as

$T^{-1}Q = \{T^{-1}Q_1,\ldots,T^{-1}Q_k\}$

Further, given two partitions $Q=\{Q_1,\ldots,Q_k\}$ and $R=\{R_1,\ldots,R_m\}$ , we define their refinement $Q \vee R$ as

$Q \vee R = \{Q_i \cap R_j| i=1,\ldots,k , j=1,\ldots,m , \mu(Q_i \cap R_j) > 0 \}$

With these two constructs we may define refinement of an iterated pullback

$\vee_{n=0}^N T^{-n}Q = \{Q_{i_0} \cap T^{-1}Q_{i_1} \cap ... \cap T^{-N}Q_{i_N} | i_l = 1,\ldots,k , l=0,\ldots,N , \mu(Q_{i_0} \cap T^{-1}Q_{i_1} \cap ... \cap T^{-N}Q_{i_N})>0 \}$

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

$H(Q)=-\sum_{m=1}^k \mu (Q_m) \log \mu(Q_m)$

The measure-theoretic entropy of a dynamical system $(X, \mathcal{B}, T, \mu)$ with respect to a partition $Q=\{Q_1,\ldots,Q_k\}$ is then defined as

$h_\mu(T,Q) = \lim_{N \rightarrow \infty} \frac{1}{N} H\left(\bigvee_{n=0}^N T^{-n}Q\right)$

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

$h_\mu(T) = \sup_{Q} h_\mu(T,Q)$

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 $log2$ , since every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals $[0,1 / 2)$ and $[1 / 2,1]$ . Every real number x is either less than 1/2 or not; and likewise so is the fractional part of $2 n x$ .

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

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

Category: Entropy and information

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