| Abstract: |
| The Spectral Decomposition Property (SDP) plays a central role in understanding the structure of nonwandering sets in dynamical systems. In this talk, we extend the classical SDP to a measure-theoretic setting for homeomorphisms on compact metric spaces. We demonstrate that a homeomorphism has the SDP if and only if each Borel probability measure satisfies this property. Additionally, we have the relationship between shadowing properties and spectral decomposition by proving that shadowable measures for expansive homeomorphisms exhibit the SDP. This talk is based on reference \cite{s}.
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\bibitem{s} Shin, Bomi A measurable spectral decomposition. Monatsh. Math. 204 (2024), no. 2, 311--322.
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