We define a hierarchy of systems with topological completely positive entropy in the context of countable amenable continuous group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.

我们在紧凑度量空间上的可数顺从连续组动作的上下文中定义了具有拓扑完全正熵的系统层次结构。对于每个可数序数，我们在上述层次结构的相应级别上构建一个动态系统，并为前三个级别提供有限类型的子移位。我们通过具有伪轨道追踪性质的系统上的渐近关系给出熵对的充要条件，从而在巴甫洛夫的结果和梅耶罗维奇的结果之间架起了一座桥梁。作为推论，我们否定地回答了巴甫洛夫提出的关于完全正熵的必要条件的开放性问题。