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 Expanding measures for homeomorphisms with eventually shadowing property J. Korean Math. Soc.Published online November 7, 2019 Meihua Dong, Keonhee Lee, and Ngocthach Nguyen Yanbian University, Chungnam National University Abstract : In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism $f$ on a compact metric space $X$ is invariantly measure expanding on its chain recurrent set $CR(f)$ and has the eventually shadowing property on $CR(f)$ then $f$ has the spectral decomposition. Moreover we show that $f$ is invariantly measure expanding on $X$ if and only if its restriction on $CR(f)$ is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of $\Omega$-stability. Keywords : Expanding measures, eventually shadowing property, $\Omega$-stability, spectral decomposition MSC numbers : 37Bxx, 37Dxx Full-Text :