metric entropy

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

$$h_{\mu }(T,𝒫)=\underset{n\to \mathrm{\infty }}{lim}{H}_{\mu }\left(\underset{k=0}{\overset{n-1}{\bigvee }}{T}^{-k}𝒫\right),$$

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 $+\mathrm{\infty }$. The entropy of $T$ is then defined as

$$h_{\mu }(T)=\underset{𝒫}{sup}{h}_{\mu }(T,𝒫),$$

with the supremum taken over all finite measurable partitions. Sometimes $h_{\mu }(T)$ is called the metric or measure theoretic entropy of $T$, to differentiate it from topological entropy.

Remarks.

1. There is a natural correspondence between finite measurable partitions and finite sub-$\sigma$-algebras of $ℬ$. 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 $h_{\mu }(T,𝒫)$ is called the entropy of $T$ with respect to the $\sigma$-algebra $𝒫$ generated by $𝒫$, and denoted by $h_{\mu }(T,𝒫)$. This simplifies the notation in some instances.

Title	metric entropy
Canonical name	MetricEntropy
Date of creation	2013-03-22 14:31:59
Last modified on	2013-03-22 14:31:59
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	6
Author	Koro (127)
Entry type	Definition
Classification	msc 28D20
Classification	msc 37A35
Synonym	entropy
Synonym	measure theoretic entropy