局部紧空间上半群作用的逐点连续遍历性
Continuous pointwise ergodicity for semigroup actions on locally compact spaces
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中文总结 AI 辅助
该研究针对局部紧空间上的半群作用，建立了逐点连续遍历性的算子理论刻画，推广了紧空间群作用的经典结果，且适用于无Følner序列的半群，还给出了相关动力学实例。
中文摘要 AI 辅助
我们研究可分局部紧度量空间上任意半群的真作用，其中点轨道允许逃逸至无穷。当每个紧轨道闭包恰好支撑一个不变概率测度，且非紧轨道闭包不支撑任何不变概率测度时，该作用是逐点唯一遍历的。对应的遍历映射因此将选定的概率测度分配给非逃逸点，将零次概率分配给逃逸点。在紧轨道闭包至少容许一个不变测度的假设下，我们证明该遍历映射的弱*连续性与无穷远消失条件等价于Koopman表示在无穷远消失的连续函数空间上的平均遍历性。由此，每个此类函数和每个有限符号测度可唯一分解为不变分量与上边缘极限，对应的投影通过对遍历映射积分得到。由于该算子理论刻画避免了显式平均方案，它甚至适用于没有Følner序列的半群。当限制于可数、离散、双可消且左可和的半群时，这些性质被证明等价于Følner平均的一致收敛及其对偶极限的弱*连续性，推广了紧空间上群作用的经典结果。此外，我们将遍历测度空间等同于紧化的遍历商，证明不变测度单纯形是Bauer单纯形，并表明这些结构性质可通过真因子映射传递。该理论框架辅以动力学实例，包括一个沿遍历映射呈现不连续熵的逐点连续遍历子转移。
英文摘要
We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.
