For a semigroup \(\mathcal {S}\) of Markov operators on a space of continuous functions, we use \(\mathcal {S}\)-invariant ideals to describe qualitative properties of \(\mathcal {S}\) such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive\(\mathcal {S}\)-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of \(\mathcal {S}\).

中文翻译：

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半群马尔可夫算子的本原谱

对于连续函数空间上的半群Markov算子\（\ mathcal {S} \），我们使用\（\ mathcal {S} \）-不变理想来描述\（\ mathcal {S} \）的定性性质例如平均遍历及其固定空间的结构。为此，我们专注于原始\（\ mathcal {S} \） -理想，并为这些理想的空间分配适当的拓扑。这种方法受到C *代数表示理论的启发，可以适应我们的动力学环境。在Koopman半群的特别重要的情况下，我们根据\（\ mathcal {S} \）的不变理想结构来描述基本动力系统的吸引中心。