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

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In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$?

I am fairly sure this is not the case (unless I am missing something in the definition of $T$) Consider the special case where $T$ is invertible with inverse $T_{\rm inv}$. Then, it holds for any test …
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