대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제61권4호
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  • Pages.1137-1159
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  • 2024
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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PERSISTENCE AND POINTWISE TOPOLOGICAL STABILITY FOR CONTINUOUS MAPS OF TOPOLOGICAL SPACES

  • Shuzhen Hua (Department of Mathematics Nanchang University) ;
  • Jiandong Yin (Department of Mathematics Nanchang University)
  • 투고 : 2023.11.08
  • 심사 : 2024.02.23
  • 발행 : 2024.07.31
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초록

In the paper, we prove that if a continuous map of a compact uniform space is equicontinuous and pointwise topologically stable, then it is persistent. We also show that if a sequence of uniformly expansive continuous maps of a compact uniform space has a uniform limit and the uniform shadowing property, then the limit is topologically stable. In addition, we introduce the concepts of shadowable points and topologically stable points for a continuous map of a compact topological space and obtain that every shadowable point of an expansive continuous map of a compact topological space is topologically stable.

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과제정보

This work was financially supported by the National Natural Science Foundation of China (12061043).

참고문헌

  1. S. A. Ahmadi, Shadowing, ergodic shadowing and uniform spaces, Filomat 31 (2017), no. 16, 5117-5124. https://doi.org/10.2298/fil1716117a 
  2. S. A. Ahmadi, X. Wu, and G. Chen, Topological chain and shadowing properties of dynamical systems on uniform spaces, Topology Appl. 275 (2020), 107153, 11 pp. https://doi.org/10.1016/j.topol.2020.107153 
  3. T. Arai, Devaney's and Li-Yorke's chaos in uniform spaces, J. Dyn. Control Syst. 24 (2018), no. 1, 93-100. https://doi.org/10.1007/s10883-017-9360-0 
  4. A. Arbieto and E. Rego, Positive entropy through pointwise dynamics, Proc. Amer. Math. Soc. 148 (2020), no. 1, 263-271. https://doi.org/10.1090/proc/14682 
  5. B. F. Bryant, On expansive homeomorphisms, Pacific J. Math. 10 (1960), 1163-1167. http://projecteuclid.org/euclid.pjm/1103038056  103038056
  6. T. Ceccherini-Silberstein and M. Coornaert, Sensitivity and Devaney's chaos in uniform spaces, J. Dyn. Control Syst. 19 (2013), no. 3, 349-357. https://doi.org/10.1007/s10883-013-9182-7 
  7. 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 
  8. M. Dong, K. Lee, and C. Morales, Pointwise topological stability and persistence, J. Math. Anal. Appl. 480 (2019), no. 2, 123334, 12 pp. https://doi.org/10.1016/j.jmaa.2019.07.024 
  9. R. Engelking, General Topology, translated from the Polish by the author, second edition, Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, 1989. 
  10. C. Good and S. Macias, What is topological about topological dynamics?, Discrete Contin. Dyn. Syst. 38 (2018), no. 3, 1007-1031. https://doi.org/10.3934/dcds.2018043 
  11. S. Hua and J. Yin, Topological stability and pseudo-orbit tracing property of Borel measures from the viewpoint of open covers, Dyn. Syst. 38 (2023), no. 4, 597-611. https://doi.org/10.1080/14689367.2023.2226384 
  12. A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer, New York, 1995. https://doi.org/10.1007/978-1-4612-4190-4 
  13. J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York, 1995. 
  14. A. G. Khan and T. Das, Stability theorems in pointwise dynamics, Topology Appl. 320 (2022), Art. ID 108218. https://doi.org/10.1016/j.topol.2022.108218 
  15. A. G. Khan and T. Das, Topologically stable and persistent points of group actions, Math. Scand. 129 (2023), no. 1, 60-71. https://doi.org/10.7146/math.scand.a134098 
  16. N. Koo and H. Lee, Topologically stable points and uniform limits, J. Korean Math. Soc. 60 (2023), no. 5, 1043-1055. https://doi.org/10.4134/JKMS.j220595 
  17. N. Koo, K. Lee, and C. A. Morales, Pointwise topological stability, Proc. Edinb. Math. Soc. (2) 61 (2018), no. 4, 1179-1191. https://doi.org/10.1017/s0013091518000263 
  18. N. Koo, H. Lee, and N. Tsegmid, On the almost shadowing property for homeomorphisms, J. Chungcheong Math. Soc. 35 (2022), no. 4, 329-333. https://doi.org/10.14403/jcms.2022.35.4.329 
  19. K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations 262 (2017), no. 6, 3467-3487. https://doi.org/10.1016/j.jde.2016.04.029 
  20. J. Lewowicz, Persistence in expansive systems, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 567-578. https://doi.org/10.1017/S0143385700002157 
  21. C. A. Morales, Shadowable points, Dyn. Syst. 31 (2016), no. 3, 347-356. https://doi.org/10.1080/14689367.2015.1131813 
  22. K. Sakai and H. Kobayashi, On Persistent Homeomorphisms, Dynamical systems and nonlinear oscillations (Kyoto, 1985), 1-12, World Sci. Adv. Ser. Dynam. Systems, 1, World Sci. Publishing, Singapore, 1986. 
  23. W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. https://doi.org/10.2307/2031982 
  24. P. Walters, Anosov diffeomorphisms are topologically stable, Topology 9 (1970), no. 1, 71-78. https://doi.org/10.1016/0040-9383(70)90051-0 
  25. P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 231-244, Lecture Notes in Math., 668, Springer, Berlin. 
  26. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, New York, 1982. 
  27. H. Wang and Y. Zhong, A note on sensitivity in uniform spaces, J. Dyn. Control Syst. 24 (2018), no. 4, 625-633. https://doi.org/10.1007/s10883-017-9375-6 
  28. A. Weil, Sur les espaces a structure uniforme et sur la topologie g'en'erale, Actual. Sci. Ind., No. 551, Hermann et Cie., Paris, 1937. 
  29. S. Willard, General Topology, Addison-Wesley Publishing Co., Reading, MA, 1970. 
  30. K. S. Yan and F. P. Zeng, Topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces, Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 2, 431-442. https://doi.org/10.1007/s10114-021-0232-x 
  31. J. Yin and M. Dong, Topological stability and shadowing of dynamical systems from measure theoretical viewpoint, Topol. Methods Nonlinear Anal. 58 (2021), no. 1, 307-321. https://doi.org/10.12775/tmna.2020.071