Conditional entropy theory in infinite measure and a question of Krengel
Let \(T\) be an infinite measure preserving transformation. In the paper it is introduced in a natural way a conditional entropy \(h(T,{\mathfrak F})\) of \(T\) relative to a \(\sigma\)-finite factor \({\mathfrak F}\). To this end it is applied the orbital approach to conditional entropy. It is shown that \(h(T|{\mathfrak F})\) is a factor orbit invariant as in the finite measure preserving case. This means that the infinite relative entropy, as in finite measure case, depends only on the \((T\uparrow {\mathfrak F})\)-orbit equivalence relation \(\mathcal R\) and the extending (Rokhlin) cocycle of \(\mathcal R\). It appears that the infinite conditional entropy theory is similar (but not identical) to the classical probability case. There are infinite analogues of relative Kolmogorov-Sinai entropy, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition disjointness theorem, etc. In case of \(\mathbb Z\)-action, the concept of relative entropy matches well the entropy \(h_Kr \) introduced by Krengel. Ansering his question and a question of Silva and Thieullen, it is shown that for any non-distal transformation \(S\) there exists an infinite measure preserving transformation \(T\) with \(h_Kr (T\times S)=\infty\) but \(h_kr (T)=0\).
infinite measure preserving transformations
scientific article; zbMATH DE number 5610599
scientific article