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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.

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 $mu(X)=1$ , and
$T:Xrightarrow X$ is a measurable transformation which preserves the measure $mu$ , i. e. each $Ain mathcal{B}$ 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.

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 $mu (phi^{-1}B) = nu(B)$ ,
For μ-almost all $x in X$ , one has $phi(Tx) = S(phi x)$ .

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 $x = psi(phi x)$
For ν-almost all $y in Y$ , one has $y = phi(psi y)$ .

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^nx

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} Hleft(bigvee_{n=0}^N T^{-n}Qright)

Finally, the Kolmogorov–Sinai or 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 $log 2$ , 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 $2^nx$ .

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