FLUCTUATIONS OF THE METRIC ENTROPY FOR MIXING MEASURES

It is known from the theorem of Shannon–McMillan–Breiman that the measure of cylinder sets decays in the limit exponentially with a rate which is given by the metric entropy. Here we prove that the measure of cylinders satisfies a Central Limit Theorem for a wide class of mixing measures which do not necessarily have the Gibbs property. We, moreover, provide the rate of convergence (which is algebraic). As a consequence we can then also prove that the distribution of the first return time in cylinder sets is log-normally distributed and give the speed of convergence. We also show that the weak invariance principle and the law of the iterated logarithm hold for the convergence to the entropy in the Shannon–McMillan–Breiman and for the distribution of the first return time.

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