Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXPANDING MEASURES FOR HOMEOMORPHISMS WITH EVENTUALLY SHADOWING PROPERTY

  • Dong, Meihua (Department of Mathematics College of Science Yanbian University) ;
  • Lee, Keonhee (Department of Mathematics Chungnam National University) ;
  • Nguyen, Ngocthach (Department of Mathematics Chungnam National University)
  • Received : 2019.06.30
  • Accepted : 2019.09.16
  • Published : 2020.07.01
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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 Ω-stability.

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References

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