Weakly chaotic attractors of an intermittent map defined on the complex unit circle arise in an analogy with a recurrence relation of the scattering matrix associated with wave transport through locally periodic structures of consecutive sizes. It is demonstrated that the fixed-point solution (infinite iteration time or scattering structure size) of the relation corresponds to an average of the scattering matrix over a set, or “ensemble”, of systems of all sizes. This ergodic property implies the analyticity of the scattering matrix $S$ and the existence of its “ensemble” average $〈S〉$, called the optical $S$-matrix. We find that the invariant density of the map that governs the sample-to-sample fluctuations of coherent transport is given by the Poisson kernel of potential theory, and consequently the distribution of $S$ is uniquely determined by $〈S〉$ which depends only on the transport properties of a single scattering cell. The theoretical distribution, closely related to the Cauchy distribution, shows perfect agreement with numerical results for a chain of delta potentials. A consequence of our findings is the a priori knowledge of $〈S〉$ without the customary resort to experimental data.

类似地，在复杂单位圆上定义的间歇图的弱混沌吸引子以与通过连续大小的局部周期性结构的波传输相关的散射矩阵的递归关系类推地出现。证明了该关系的定点解（无限迭代时间或散射结构大小）对应于一组或所有大小系统的“集合”上散射矩阵的平均值。这种遍历特性暗示了散射矩阵的解析性$\mathrm{小号}$ 及其“整体”平均值的存在 $〈\mathrm{小号}〉$，称为光学 $\mathrm{小号}$-矩阵。我们发现，控制相干输运的样本间波动的图的不变密度由势能理论的泊松核给出，因此，$\mathrm{小号}$ 由...唯一决定 $〈\mathrm{小号}〉$这仅取决于单个散射池的传输特性。与柯西分布密切相关的理论分布显示出与一系列δ电位的数值结果完全吻合。我们发现的结果是先验知识$〈\mathrm{小号}〉$ 无需惯用实验数据。