RELATIVIZED VARIATIONAL PRINCIPLE FOR CONTINUOUS TRANSFORMATIONS
RELATIVIZED VARIATIONAL PRINCIPLE FOR CONTINUOUS TRANSFORMATIONS
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DOI：
10.1112/jlms/s2-16.3.568
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发表时间：
1977-01-01
影响因子：
1.2
通讯作者：
WALTERS, P
中科院分区：
文献类型：
--
作者：
LEDRAPPIER, F;WALTERS, P
设X和Y是紧度量空间，T：X->；X，S：Y->；Y，n：X-*Y是连续映射，使得7t是满射，Nt=Src。换句话说，S(在拓扑上)是图7r的T的一个因子。如果h{T)表示T的拓扑熵，则Bowen[3]证明了(1.1)h(T)(定义见§2)。另一方面，如果Pl是X上的T-不变概率，且HJJT)表示T关于FI的测度论熵，则Pinsker[6]的公式推导出/^(T)-/^0 n-i(S)=h^T/S)，即相对度量熵(见§3)。证明了给定Y上的S不变概率v，则(1.2)sup h^T)=hv(S)+[h(T，n-L(Y))dv(Y)).
Let X and Y be compact metric spaces and let T: X-> X, S: Y-> Y, n: X-* Y be continuous maps such that 7t is surjective and nT= Src. In other words S is (topologically) a factor of T by the map 7r. If h {T) denotes the topological entropy of T then Bowen [3] has shown that (1.1) h (T)(see § 2 for the definitions). On the other hand, if pL is a T-invariant probability on X and hJJT) denotes the measure-theoretic entropy of T with respect to fi then a formula of Pinsker [6] implies that/^(T)-/^ 0 n-i (S)= h^ T/S), the relative metric entropy (see § 3). Here we show that given an S-invariant probability v on Y then (1.2) sup h^ T)= hv (S)+[h (T, n-l (y)) dv (y).