Theorem 8.1 in Wlaters' book "An introduction to ergodic theory" says the entropy map is affine. Namely, let $T:X\to X$ be a continuous map of a compact metrc sace. If $\mu, m\in M(X,T)$ and $p\in[0,1]$ then $h_{p\mu+(1-p)m}(T)=ph_\mu(T)+(1-p)h_m(T)$.

Can someone know above result still hold for invariant measures more than two under some additional conditions? The remark after Theorem 8.1 tells us that in general one can only get one side inequality. Many thanks!


The result on affinity of the entropy map holds for arbitrary finite convex combinations of invariant measures: if $\mu_1,\dots,\mu_n\in M(X,T)$ are any invariant measures and $a_1,\dots, a_n\in [0,1]$ sum to 1, then $h(\sum a_i \mu_i) = \sum a_i h(\mu_i)$, where I write $h(\mu)$ in place of $h_\mu(T)$.

This can be proved by induction in a pretty standard way. The case $n=2$ is proved already. Now if it holds for some $n$, we prove it for $n+1$: first note that $\sum_{i=1}^n a_i \mu_i = (1-a_{n+1}) \nu$, where $\nu\in M(X,T)$ is a probability measure, and we have $$ h(\sum_{i=1}^{n+1} a_i \mu_i) = h((1-a_{n+1})\nu + a_{n+1}\mu_{n+1}) = (1-a_{n+1})h(\nu) + a_{n+1} h(\mu_{n+1}), $$ and then expanding $h(\nu)$ using the induction hypothesis gives the result we want.

The remark after Theorem 8.1 in Walters' book doesn't say quite what you suggested it does. Quoting the remark: "The first part of the proof shows that if $\mu,m\in M(X)$, $p\in [0,1]$, and $\xi$ is a finite partition then $H_{p\mu + (1-p)m}(\xi) \geq p H_\mu(\xi) + (1-p)H_m(\xi)$." Indeed this only gives an inequality in one direction, but two points must be made:

  1. The measures $\mu,m$ are not assumed to be invariant.
  2. The quantity here is $H_\mu(\xi)$, the entropy of a partition, rather than $h_\mu(T)$, the (Kolmogorov-Sinai) entropy of an invariant measure, which is defined as the (linear) growth rate of the quantities $H_\mu(\xi_n)$ for a particular sequence of partitions $\xi_n$. So the remark is not about the KS entropy $h_\mu(T)$, which is affine over all finite convex combinations, but rather about a specific intermediary quantity which is not generally affine, although it does satisfy a super-affinity inequality. In particular there is no contradiction between the remark and the fact that affinity of $h_\mu$ holds for arbitrary finite combinations.

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