# Measure-preserving dynamical system

(Redirected from Metric entropy)

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

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

with the following structure:

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

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 Ts : XX parametrized by sZ (or R, or N ∪ {0}, or [0, +∞)), where each transformation Ts satisfies the same requirements as T above. In particular, the transformations obey the rules:

• ${\displaystyle T_{0}=id_{X}:X\rightarrow X}$, the identity function on X;
• ${\displaystyle T_{s}\circ T_{t}=T_{t+s}}$, whenever all the terms are well-defined;
• ${\displaystyle T_{s}^{-1}=T_{-s}}$, whenever all the terms are well-defined.

The earlier, simpler case fits into this framework by definingTs = Ts for sN.

The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.

## Examples

Example of a (Lebesgue measure) preserving map: T : [0,1) → [0,1), ${\displaystyle x\mapsto 2x\mod 1.}$

Examples include:

## Homomorphisms

The concept of a homomorphism and an isomorphism may be defined.

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

${\displaystyle \varphi :X\to Y}$

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

1. The map φ is measurable,
2. For each ${\displaystyle B\in {\mathcal {B}}}$, one has ${\displaystyle \mu (\varphi ^{-1}B)=\nu (B)}$,
3. For μ-almost all xX, one has φ(Tx) = Sx).

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

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

${\displaystyle \psi :Y\to X}$

that is also a homomorphism, which satisfies

1. For μ-almost all xX, one has ${\displaystyle x=\psi (\varphi x)}$
2. For ν-almost all yY, one has ${\displaystyle y=\varphi (\psi y)}$.

Hence, one may form a category of dynamical systems and their homomorphisms.

## Generic points

A point xX is called a generic point if the orbit of the point is distributed uniformly according to the measure.

## Symbolic names and generators

Consider a dynamical system ${\displaystyle (X,{\mathcal {B}},T,\mu )}$, and let Q = {Q1, ..., Qk} be a partition of X into k measurable pair-wise disjoint pieces. Given a point xX, clearly x belongs to only one of the Qi. Similarly, the iterated point Tnx 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 {an} such that

${\displaystyle T^{n}x\in Q_{a_{n}}.\,}$

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 = {Q1, ..., Qk} and a dynamical system ${\displaystyle (X,{\mathcal {B}},T,\mu )}$, we define T-pullback of Q as

${\displaystyle T^{-1}Q=\{T^{-1}Q_{1},\ldots ,T^{-1}Q_{k}\}.\,}$

Further, given two partitions Q = {Q1, ..., Qk} and R = {R1, ..., Rm}, we define their refinement as

${\displaystyle Q\vee R=\{Q_{i}\cap R_{j}\mid i=1,\ldots ,k,\ j=1,\ldots ,m,\ \mu (Q_{i}\cap R_{j})>0\}.\,}$

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

${\displaystyle \bigvee _{n=0}^{N}T^{-n}Q=\left\{Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}{\text{ where }}i_{\ell }=1,\ldots ,k,\ \ell =0,\ldots ,N,\ \mu \left(Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}\right)>0\right\}}$

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[1][2]

${\displaystyle H(Q)=-\sum _{m=1}^{k}\mu (Q_{m})\log \mu (Q_{m}).}$

The measure-theoretic entropy of a dynamical system ${\displaystyle (X,{\mathcal {B}},T,\mu )}$ with respect to a partition Q = {Q1, ..., Qk} is then defined as

${\displaystyle h_{\mu }(T,Q)=\lim _{N\rightarrow \infty }{\frac {1}{N}}H\left(\bigvee _{n=0}^{N}T^{-n}Q\right).\,}$

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

${\displaystyle h_{\mu }(T)=\sup _{Q}h_{\mu }(T,Q).\,}$

where the supremum is taken over all finite measurable partitions. A theorem of Yakov 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 almost 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 2nx.

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