Definition:Kolmogorov-Sinai Entropy

Definition

Let $\struct {X, \BB, \mu}$ be a probability space.

Let $T: X \to X$ be a $\mu$-preserving transformation.

Then the Kolmogorov-Sinai entropy of $T$ is defined as:

$\map h T := \sup \set {\map h {T, \AA}: \text {$\AA$ finite sub-$\sigma$-algebra of $\BB$} }$

where:

$\map h {T, \AA}$ denotes the entropy of $T$ with respect to $\AA$.

Also known as

The Kolmogorov-Sinai entropy is also known as:

metric entropy

measure-theoretic entropy

Kolmogorov entropy

KS entropy.

Examples

Identity Mapping

Let $\struct {X, \BB, \mu}$ be a probability space.

Let $I_X: X \to X$ be the identity mapping.

Then $I_X$ is $\mu$-preserving and:

$ \map h {I_X} = 0$

Also see

• Kolmogorov-Sinai Entropy may be Infinite: $\map h T$ may be $+\infty$.

• Results about Kolmogorov-Sinai entropy can be found here.

Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov and Yakov Grigorevich Sinai.

Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.4$: Entropy of Measure-Preserving Transformation

• Yakov Sinai (2009) Kolmogorov-Sinai entropy. Scholarpedia, 4.(3):2034.