In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, \(R_{1}, R_{2}, \ldots , R_{N}\), each with degree \(d_{i};\ 1 \le i \le N\) and with at least one \(R_{i}\) in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbański (equilibrium) measure associated to some real-valued Hölder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

在本文中，我们考虑了在N个符号和允许我们N个有理图\（R_ {1}，R_ { 2}，\ ldots，R_ {N} \），每个度数为\（d_ {i}; \ 1 \ le i \ le N \）且至少具有一个\（R_ {i} \）在度至少为2的集合中。我们获得有关点前图像和周期点在乘积空间子集中（其中乘积-乘积图不正常地表现）的分布的结果。我们进一步探索与偏积图的Julia集上定义的一些实值Hölder连续函数相关的Sumi-Urbański（平衡）测度的遍历性，并获得关于典型轨道行为的平均偏差的估计值，这违反了遍历的必需品。