Conditional Entropy Formula with Respect to Monotonic Partitions

We investigate conditional entropy with respect to monotonic (invariant, decreasing, or increasing) measurable partitions. In particular, we obtain Brin-Katok’s and Katok’s entropy formula for conditional entropy with respect to invariant, decreasing and a large class of increasing partitions. Our proofs develop new adaptions on applying conditional variants of Shannon-McMillan-Breiman theorems corresponding to the monotonic partitions, to link with the local and global conditional entropy forms. As an application, we obtain a number of reformulations of the unstable entropy for ergodic measures of general \(C^{r}(r>1)\) $C^r(r>1)$ diffeomorphisms.