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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.
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참고문헌 (12)
참고문헌 신청
N. Aoki / 1983 / On homeomorphisms with pseudo-orbit tracing property / Tokyo J. Math 6(2):329~334
A. Arbieto / 2011 / Periodic orbits and expansiveness / Math. Z 269(3-4):801~807
R. Bowen / 2008 / Equilibrium states and the ergodic theory of Anosov dieomorphisms / Lecture Notes in Mathematics 470
B. Carvalho / 2016 / n-expansive homeomorphisms with the shadowing prop-erty / J. Dierential Equations 261(6):3734~3755
W. Cordeiro / 2017 / On specication and measure expansiveness / Discrete Contin. Dyn. Syst 37(4):1941~1957
A. Arbieto / 2011 / Periodic orbits and expansiveness / Math. Z 269(3-4):801~807
R. Bowen / 2008 / Equilibrium states and the ergodic theory of Anosov dieomorphisms / Lecture Notes in Mathematics 470
B. Carvalho / 2016 / n-expansive homeomorphisms with the shadowing prop-erty / J. Dierential Equations 261(6):3734~3755
W. Cordeiro / 2017 / On specication and measure expansiveness / Discrete Contin. Dyn. Syst 37(4):1941~1957
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