标题： 香农权重对于零熵的二元动力学递归源 显示英文标题

标题： Shannon Weights for binary dynamical recurrent sources of zero entropy

摘要： 概率源被定义为无限字母表上的一组无穷词，并赋予概率$\mu$。 本文研究了二元源的一般情况，其中任何符号（0 或 1）的分布可能依赖于之前历史的无界部分。 本文研究了香农权重：虽然经典的香农熵${\cal E}_{\mu}$是发射词中每个符号带来的平均信息量，但香农权重序列则处理由长度为$n$的发射前缀带来的平均信息量$m_{\mu}(n)$。 对于具有非零熵的源，估计$m_{\mu}(n)\sim{\cal E}_{\mu} \cdot n$因此成立。 本文考虑了动力源模型，在该模型中，当赋予概率$\mu$时，源词作为单位区间动力系统的编码轨迹发出。 它关注于零熵的源，并给出了香农权重序列满足$m_{\mu}(n)=o(n)$且具有指定行为的源的显式构造。 在这种情况下，零熵的源导致基于具有漠视不动点的地图的动力系统构建。 这一类显著包含了著名的法里源，该源展示了众所周知的间歇现象。 方法基于解析组合学和生成函数，并在此动力学情况下扩展了动力系统工具（主要为转移算子）。 显示英文摘要

摘要： A probabilistic source is defined as the set of infinite words (over a given denumerable alphabet) endowed with a probability $\mu$. The paper deals with general binary sources where the distribution of any symbol (0 or 1) may depend on an unbounded part of the previous history. The paper studies Shannon weights: whereas the classical Shannon entropy ${\cal E}_{\mu}$ is the average amount of information brought by one symbol of the emitted word, the Shannon weight sequence deals with the average amount of information $m_{\mu}(n)$ that is brought by the emitted prefix of length $n$. For a source with a non zero entropy, the estimate $m_{\mu}(n)\sim{\cal E}_{\mu} \cdot n$ thus holds. The paper considers the model of dynamical sources, where a source word isemitted as an encoded trajectory of a dynamical system of the unit interval, when endowed with probability $\mu$. It focus on sources with zero entropy and gives explicit constructions for sources whose Shannon weight sequence satisfies $m_{\mu}(n)=o(n)$, with a prescribed behaviour. In this case, sources with zero entropy lead to dynamical systems built on maps with an indifferent fixed point. This class notably contains the celebrated Farey source, which presents well-known intermittency phenomena. Methods are based on analytic combinatorics and generating functions, and they are enlarged, in the present dynamical case, with dynamical systems tools (mainly transfer operators).