대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권4호
  • /
  • Pages.935-955
  • /
  • 2020
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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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)
  • 투고 : 2019.06.30
  • 심사 : 2019.09.16
  • 발행 : 2020.07.01
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초록

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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참고문헌

  1. N. Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math. 6 (1983), no. 2, 329-334. https://doi.org/10.3836/tjm/1270213874
  2. A. Arbieto, Periodic orbits and expansiveness, Math. Z. 269 (2011), no. 3-4, 801-807. https://doi.org/10.1007/s00209-010-0767-5
  3. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, second revised edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.
  4. B. Carvalho and W. Cordeiro, n-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), no. 6, 3734-3755. https://doi.org/10.1016/j.jde.2016.06.003
  5. W. Cordeiro, M. Denker, and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 1941-1957. https://doi.org/10.3934/dcds.2017082
  6. T. Das, K. Lee, D. Richeson, and J. Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl. 160 (2013), no. 1, 149-158. https://doi.org/10.1016/j.topol.2012.10.010
  7. J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301-308. https://doi.org/10.2307/1995906
  8. C. Good and J. Meddaugh, Orbital shadowing, internal chain transitivity and $\omega$-limit sets, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 143-154. https://doi.org/10.1017/etds.2016.30
  9. K. Lee, N.-T. Nguyen, and Y. Yang, Topological stability and spectral decomposition for homeomorphisms on noncompact spaces, Discrete Contin. Dyn. Syst. 38 (2018), no. 5, 2487-2503. https://doi.org/10.3934/dcds.2018103
  10. C. A. Morales and V. F. Sirvent, Expansive measures, Publicacoes Matematicas do IMPA., Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 2013.
  11. P. Oprocha, Shadowing, thick sets and the Ramsey property, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1582-1595. https://doi.org/10.1017/etds.2014.130
  12. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. https://doi.org/10.1090/S0002-9904-1967-11798-1