A formula of conditional entropy and some applications

Abstract

In this paper we establish a formula of conditional entropy and give two examples of applications of the formula.

Entropy formulae of conditional entropy in mean metrics

“…Then also by the same method of the proof of Theorem 1.2 in [18], we have for any l ≥ N 1 , µ(Q l ) > 1 − ε 1 4 . Clearly, the sets A l are nested, i.e.…”

“…In 2016, Zhou [18] established a conditional version of Brin-Katok formula and obtained a analogous conclusion with (1) for A.…”

“…where χ U δ (ξ) is the characteristic function of the set U δ (ξ). Therefore, by the same method of the proof of Theorem 1.2 in [18], we can choose N 1 ∈ N such that for any l ≥ N 1 , we have…”

“…In this paper, inspired by the ideas of Brin and Katok [1], Zhou, Zhou and Chen [17] and Zhou [18], we construct the Brin-Katok formula of conditional entropy h µ (T |A) for invariant measures of continuous maps on a compact metric space by replacing the Bowen metrics with the corresponding mean metrics. Furthermore, following the idea of Katok [11], we establish the Katok's entropy formula of conditional entropy h µ (T |A) for ergodic measures in the case of mean metrics.…”

Weighted topological and measure-theoretic entropy

“…In 2016, by the measure-theoretic entropy for general Borel probability measures introduced by Feng and Huang in [8], Zhou [21] established an analogous conclusion for arbitrary T -invariant sub-σ-algebra of B µ . More precisely, let µ ∈ M(X, T ), A be a T -invariant sub-σ-algebra of B µ (i.e.…”