×2 and ×3 invariant measures and entropy
×2 and ×3 invariant measures and entropy
复制标题
×2 和 ×3 不变测度和熵
DOI：
10.1017/s0143385700005629
复制
发表时间：
1990
影响因子：
0.9
通讯作者：
D. Rudolph
中科院分区：
文献类型：
--
作者：
D. Rudolph
设p和q是互质自然数。定义T0和S 0分别与p和q（mod 1）相乘，即[0，1）的自同态。设μ是对T0和S 0都是Borel测度不变的，且对它们生成的半群是遍历的.证明了如果μ不是Lebesgue测度，则T0和S 0关于μ都有熵零.等价地，T0和S 0都是μ-几乎必然可逆的。
Abstract Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1). Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.