Let me ask you: In (6), where you rewrite the correlation term with the transfer operator (using its definition), everything works fine integrating against Lebesgue as long as it is an invariant measure. However, it Lebesgue is not invariant, the (relevant) correlation should be expressed as an integral against an invariant measure, while the definition of the transfer operator will still be that integrating against Lebesgue — in which case the rephrasing in (6) may not work anymore.

Is this really it and the writing trick in (6)should be generalized in a different manner when Lebesgue is not invariant?

Best! ]]>

Your post is extremely useful for some research I’m developing. I want to ask you. Do you know if there are results about synchronisation for alpha-beta shifts for alpha different than (0,0,…)? The case for beta shifts is very well known.

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