Journal of the
Korean Mathematical Society JKMS

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

Supported by: This work was supported by research fund of Chungnam National University.