PlanetMath: metric entropy

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metric entropy

(Definition)

Let $(X,\mathscr{B},\mu)$ be a probability space, and $T\colon X\to X$ a measure-preserving transformation. The entropy of with respect to a finite measurable partition $\mathcal{P}$ is

$\displaystyle h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right),$

where $H_\mu$ is the entropy of a partition and $\vee$ denotes the join of partitions. The above limit always exists, although it can be $+\infty$ . The entropy of

is then defined as

$\displaystyle h_\mu(T) = \sup_{\mathcal{P}} h_\mu(T,\mathcal{P}),$

with the supremum taken over all finite measurable partitions. Sometimes $h_\mu(T)$ is called the metric or measure theoretic entropy of

, to differentiate it from topological entropy.

Remarks.

There is a natural correspondence between finite measurable partitions and finite sub- $\sigma$ -algebras of $\mathscr{B}$ . Each finite sub- $\sigma$ -algebra is generated by a unique partition, and clearly each finite partition generates a finite $\sigma$ -algebra. Because of this, sometimes $h_\mu(T,\mathcal{P})$ is called the entropy of with respect to the $\sigma$ -algebra $\mathscr{P}$ generated by $\mathcal{P}$ , and denoted by $h_\mu(T,\mathscr{P})$ . This simplifies the notation in some instances.

"metric entropy" is owned by Koro.

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Other names:

entropy, measure theoretic entropy

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Cross-references: generates, generated by, differentiate, metric, supremum, limit, partitions, join, entropy of a partition, measurable partition, finite, measure-preserving transformation, probability space
There are 3 references to this entry.

This is version 3 of metric entropy, born on 2004-08-05, modified 2007-07-03.
Object id is 6077, canonical name is MetricEntropy.
Accessed 4907 times total.

Classification:

AMS MSC:	37A35 (Dynamical systems and ergodic theory :: Ergodic theory :: Entropy and other invariants, isomorphism, classification)
	28D20 (Measure and integration :: Measure-theoretic ergodic theory :: Entropy and other invariants)

Pending Errata and Addenda

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