Conditional entropy formula with respect to monotonic partitions (Q6568941)

Let \( X \) be a compact metric space with metric \( d \), and \(T: X \rightarrow X \) be a continuous map. A topological dynamical system is defined as a triple \( (X, d, T) \) (or simply as a pair \( (X, T) \)). Denote the sets of all Borel probability measures on \( X \), \( T \)-invariant measures, and \( T \)-ergodic measures by \( \mathcal{M}(X) \), \( \mathcal{M}(X, T)\), and \( \mathcal{M}^e(X, T) \), respectively. Each topological dynamical system \( (X,T) \), combined with a measure \( \mu \in \mathcal{M}(X, T) \), defines a measure-preserving system \( (X, \mathcal{B}_X, \mu, T)\), where \( \mathcal{B}_X \) represents the \(\sigma\)-algebra of Borel sets in \( X \). The measure-theoretic entropy of \( T \) with respect to \( \mu \) is denoted by \( h_{\mu}(T) \).\N\NLet \(\mu_x^{\mathcal{E}_T} \in \mathcal{M}^e(X, T) \) represent the conditional measure of \( \mu \) at \( x \) with respect to \(\mathcal{E}_T = \{ E \in \mathcal{B}_\mu : T^{-1}E = E \} \). It is well established that the ergodic decomposition of entropy applies. Specifically, this means that\N\[\Nh_\mu(T) = h_\mu(T|\mathcal{E}_T) = \int h_{\mu_x^{\mathcal{E}_T}}(T) \, d\mu(x),\N\]\Nwhere \( h_\mu(T|E_T) \) denotes the conditional entropy with respect to \( \mathcal{E}_T \). The primary result of this paper is to establish an analogous formula to the above equation for conditional entropy, specifically with respect to a monotonic measurable partition \( \xi \). In conclusion, in this study, the authors draw inspiration from the results of previous works to derive Brin-Katok's and Katok's entropy formulas for the conditional entropy with respect to monotonic (invariant, decreasing, or\Nincreasing) measurable partitions.