| Version current |
Version 2 |
| Let $(X,\mathscr{B},\mu)$ be a probability space, and $T\colon X\to X$ a measure-preserving transformation. |
Let $(X,\mathscr{B},\mu)$ be a probability space, and $T\colon X\to X$ a measure-preserving transformation. |
| The entropy of $T$ with respect to a finite measurable partition $\mathcal{P}$ is |
The entropy of $T$ with respect to a finite measurable partition $\mathcal{P}$ is |
| \[h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right),\] |
\[h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right),\] |
| where $H_\mu$ is the entropy of a partition and $\vee$ denotes the join of partitions. |
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 above limit always exists, altough it can be $+\infty$.
|
| The entropy of $T$ is then defined as |
The entropy of $T$ is then defined as |
| \[h_\mu(T) = \sup_{\mathcal{P}} h_\mu(T,\mathcal{P}),\] |
\[h_\mu(T) = \sup_{\mathcal{P}} h_\mu(T,\mathcal{P}),\] |
| with the supremum taken over all finite measurable partitions. |
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. |
Sometimes $h_\mu(T)$ is called the metric or measure theoretic entropy of $T$, to differentiate it from topological entropy. |
|
|
| \textbf{Remarks.} |
\textbf{Remarks.} |
|
|
| \begin{enumerate} |
\begin{enumerate} |
| \item There is a natural correspondence between finite measurable partitions and finite |
\item There is a natural correspondence between finite measurable partitions and finite |
| sub-$\sigma$-algebras of $\mathscr{B}$. Each finite sub-$\sigma$-algebra is |
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. |
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 $T$ with respect to |
Because of this, sometimes $h_\mu(T,\mathcal{P})$ is called the entropy of $T$ with respect to |
| the $\sigma$-algebra $\mathscr{P}$ generated by $\mathcal{P}$, and denoted by $h_\mu(T,\mathscr{P})$. |
the $\sigma$-algebra $\mathscr{P}$ generated by $\mathcal{P}$, and denoted by $h_\mu(T,\mathscr{P})$. |
| This simplifies the notation in some instances. |
This simplifies the notation in some instances. |
| \end{enumerate} |
\end{enumerate} |
