Relation between different definitions of the entropy of decreasing sequences of partitions; scaling

The present paper is devoted to the study of scaling sequences, which occur in the definition of entropy type invariants. The necessity to distinguish nonstandard sequences with zero entropy leads to a generalization of the entropy of decreasing sequences of measurable partitions.The more refined entropy type invariant – “scaling” entropy is described in [4]. It is based on the notion of $\varepsilon$- entropy of a metric space with measure. In the present work it is shown, that the “scaling” entropy from [4] is a generalization of the entropy of decreasing sequences if taking $2^n$ as the scaling sequence. The scaling entropy of the partition into pasts of one important class of endomorphisms, namely $(T,T^{-1})$-endomorphisms, is calculated.