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Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible itineraries, that is, $I=\{(i_1,\dots,i_n,\dots)\in k^N;$ there is a $x\in X$, such that $T^n(x)\in P_{i_n}$ for all $n\in\Bbb N$.

Suppose that $I$ is countably infinite.

Is true that the entropy of $T$ with respect to $P$ is $0$ ($h(T,P)=0$)?

  • 2
    Do you mean "denumerable", i.e. countably infinite?2012-12-28
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    If $\nu$ is ergodic, I know that is true.2012-12-28
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    yes, countably infinite2012-12-28
  • 1
    this question has been solved here: http://mathoverflow.net/questions/118758/question-about-entropy2013-01-25

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