Kolmogorov–Sinai entropy

Remarks

The limit in clause 1 exists because the sequence $a_n = H(\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P})$ is subadditive, $a_{m+n} \le a_m + a_n$.

Example: let $T(x) = 2x \bmod 1$ on $[0,1)$ with Lebesgue measure, and $\mathcal{P} = \set{[0, 1/2), [1/2, 1)}$. The join $\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P}$ is the partition into the $2^n$ dyadic intervals of length $2^{-n}$, so its entropy is $n \log 2$, and $h(T, \mathcal{P}) = \log 2$. This partition generates the Borel sets under $T$, and the Kolmogorov–Sinai generator theorem gives $h_\mu(T) = \log 2$.

Non-example of positive entropy: the identity map has $\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P} = \mathcal{P}$, so $h(T, \mathcal{P}) = \lim H(\mathcal{P})/n = 0$ for every $\mathcal{P}$, and $h_\mu(\mathrm{id}) = 0$. Rotations of the circle also have entropy $0$.

Isomorphic systems have the same entropy. With the natural logarithm the unit is the nat; with $\log_2$ it is the bit.