Rényi entropies of aperiodic dynamical systems

In this paper we continue the study of R6nyi entropies of measurepreserving transformations started in [22]. We have established there that for ergodic transformations with positive entropy, the R6nyi entropies of order q, q E R, are equal to either plus infinity (q <1), or to the measure-theoretic (Kolmogorov-Sinai) entropy (q > 1). The answer for non-ergodic transformations is different: the R~nyi entropies of order q > 1 are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition into ergodic components.Thus, it is possible that the R6nyi entropies of order q > 1 are strictly smaller than the measure-theoretic entropy, which is the average value of entropies of ergodic components. This result is a bit surprising: the R~nyi entropies are metric invariants, which axe sensitive to ergodicity. The proof of the described result is based on the construction of partitions with independent iterates. However, these partitions are obtained in different ways depending on q: for q > 1 we use a version of the well-known Sinai theorem on Bernoulli factors for the non-ergodic transformations; for q <1 we use the notion of collections of independent sets in Rokhlin-Halmos towers and their properties.