具有全支撑的不变测度与在$C^0$-Gromov--Hausdorff拓扑下由零熵系统逼近
Invariant measures with full support and approximation by zero-entropy systems in the $C^0$-Gromov--Hausdorff topology
AI总结：
本文证明紧度量空间上连续映射在$C^0$-Gromov--Hausdorff拓扑下可由零熵映射逼近，依赖遍历分解定理和正轨道点稠密性，揭示拓扑熵不稳定性及周期点稠密性，最终得出全支撑不变测度动态落入零熵系统逼近范围。
摘要：
本文证明了每个紧度量空间上的连续映射，若存在全支撑不变概率测度，则可在$C^0$-Gromov--Hausdorff拓扑下由零熵连续映射逼近。论证依赖于遍历分解定理和合适遍历分量支持中稠密正轨道点的存在性。因此，拓扑熵在该类中不稳定性。此外，若映射拓扑$GH$-稳定，则其周期点在环境空间中稠密。最后，通过结合先前关于可传递和拓扑$GH$-稳定映射的结果，得出此类动态具有全支撑不变测度，从而落入零熵系统逼近定理范围内。
英文摘要：
In this paper we prove that every homeomorphism of a compact metric space admitting an invariant probability measure with full support can be approximated in the $C^0$-Gromov--Hausdorff topology by homeomorphisms with zero topological entropy. The argument relies on the ergodic decomposition theorem and on the existence of points with dense positive orbit in the supports of suitable ergodic components. As a consequence, topological entropy is not stable under $C^0$-Gromov--Hausdorff perturbations within this class. We also show that if, in addition, the homeomorphism is topologically $GH$-stable, then its periodic points are dense in the ambient space. Finally, by combining this framework with a previous result on transitive and topologically $GH$-stable homeomorphisms, we deduce that every dynamics in this class admits an invariant measure with full support and therefore falls within the scope of the general approximation theorem by zero-entropy systems.
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