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Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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On the affine property of entropy map

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 …