A Bilateral version of Shannon-Breiman-McMillan Theorem

可数群作用下随机动力系统的Shannon-McMillan-Breiman定理

The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions

本文从可数群作用的角度证明了随机动力系统中的Shannon-McMillan-Breiman定理，为该框架下熵理论的发展提供了推动力。

The Shannon-McMillan-Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko's orbital approach to entropy of a partition, and is motivated by Seward's recent generalization of Rokhlin entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon-McMillan-Breiman poi…

On the Vershik-Kerov Conjecture Concerning the Shannon-McMillan-Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik-Kerov conjecture can be seen as an analogue of the Shannon-McMillan-Breiman Theorem for the non-stationary Markov process of the growth of a Young diagram. The limiting constant is then interpreted as the entropy of the Plancherel measure. The main result of the paper is the proof of the Vershik-Kerov conjecture. The a…

Chained Typical Subspaces - a Quantum Version of Breiman's Theorem

We give an equivalent finitary reformulation of the classical Shannon-McMillan-Breiman theorem which has an immediate translation to the case of ergodic quantum lattice systems. This version of a quantum Breiman theorem can be derived from the proof of the quantum Shannon-McMillan theorem presented in our previous work (math.DS/0207121).

负曲率群中沿几乎测地线的Shannon-McMillan-Breiman定理

Shannon-McMillan-Breiman theorem along almost geodesics in negatively curved groups

本文针对非初等Gromov双曲群的遍历保概率测度作用，构造了沿几乎测地线的有限子集序列，证明了相应的Shannon-McMillan-Breiman定理，并结合Seward不等式得出自由作用下该极限下确界等于Rokhlin熵。

The Almost Sure Invariance Principle for Beta-Mixing measures

The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system of finite entropy the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In addition the measure of $n$-cylinders is in various settings known to be lognormally distributed in the limit. In this paper the logarithm of the measure of $n$-cylinder, the information function, satisfies the almost sure invariance principle in the case in which the measure is $β$-mixing. We get a similar result for the recurrence time. Previous results are due to Ph…