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In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$

Because I'm working with a kind of sequence of shif-invariant measures such that the sequence of entropies is monotone increasing (it could be constant...) and I couldn't find in the literature some good "machinery" to prove that convergence... I've tried, for instance, the $\bar{d}$ distance related to joinings (defined here joining or coupling) but, it didn't work too well...(I mean, I was not able to work with...)

So, any tips, hints?

Thanks for your attention

kind regards

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There is only one measure of maximum entropy (i.e. equal to $\log 2$), the uniform measure (or 1/2,1/2-Bernoulli measure). So your problem translates into checking that $\eta_n$ converges to the uniform measure.

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  • $\begingroup$ Dear @user39115, to be more specific, imagine you have a sequence of shif-invariant measures, say, $\eta_{n}$ which all of them have positive entropy, more than that, imagine the sequence of entropies is monote increasing (could b constant) and the support does not mind to me. In a traditional point view, what do I have to do to guarantee that the entropies are growing to $\log2$? What is common to do to show what I want? Are there examples in the literature? I only know thed-bar distance, but it is tough... thanks for your attention $\endgroup$ – Bruno Brogni Uggioni Oct 28 '15 at 14:44
  • $\begingroup$ Dear Bruno, I improved the previous long comment, hope to have understood your question. $\endgroup$ – user39115 Oct 28 '15 at 14:48
  • $\begingroup$ Dear @user39115, you are right. It's just my problem. To make sure that the sequence of entropy is growing to $\log2$ seems quite difficult... it's much stronger than the weak* convergence. Because of this, I'm stuck... $\endgroup$ – Bruno Brogni Uggioni Oct 28 '15 at 14:56

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