Expanding measures for homeomorphisms with eventually shadowing property
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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.
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.
참고문헌 (12)
- N. Aoki, "On homeomorphisms with pseudo-orbit tracing property", Tokyo J. Math, 6(2), pp.329~334 (1983)[DOI]
- A. Arbieto, "Periodic orbits and expansiveness", Math. Z, 269(3-4), pp.801~807 (2011)[DOI]
- R. Bowen, "Equilibrium states and the ergodic theory of Anosov dieomorphisms", Lecture Notes in Mathematics, 470 (2008), Springer-Verlag
- B. Carvalho, "n-expansive homeomorphisms with the shadowing prop-erty", J. Dierential Equations, 261(6), pp.3734~3755 (2016)[DOI]
- W. Cordeiro, "On specication and measure expansiveness", Discrete Contin. Dyn. Syst, 37(4), pp.1941~1957 (2017)[DOI]
- T. Das, "Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces", Topology Appl, 160(1), pp.149~158 (2013)[DOI]
- J. Franks, "Necessary conditions for stability of dieomorphisms", Trans. Amer. Math. Soc, 158, pp.301~308 (1971)[DOI]
- C. Good, "Orbital shadowing, internal chain transitivity and !-limit sets", Ergodic Theory Dynam. Systems, 38(1), pp.143~154 (2018)[DOI]
- K. Lee, "Topological stability and spectral decomposition for homeomorphisms on noncompact spaces", Discrete Contin. Dyn. Syst, 38(5), pp.2487~2503 (2018)[DOI]
- C. A. Morales, "Sirvent, Expansive measures" (2013), Publicac~oes Matematicas do IMPA., Instituto Nacional de Matematica Pura e Aplicada (IMPA)
- P. Oprocha, "thick sets and the Ramsey property", Ergodic Theory Dynam. Systems, 36(5), pp.1582~1595 (2016)[DOI]
- S. Smale, "Dierentiable dynamical systems", Bull. Amer. Math. Soc, 73, pp.747~817 (1967)[DOI]