Skip to main content
SpringerLink
Log in

Finite-Expansivity and N-Shadowing

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

We prove that every finite-expansive homeomorphism with the shadowing property has a kind of stability. This stability will be good enough to imply both the shadowing property and the denseness of periodic points in the chain recurrent set. Next we analyze the N-shadowing property which is really stronger than the multishadowing property in Cherkashin and Kryzhevich (Topol Methods Nonlinear Anal 50(1): 125–150, 2017). We show that an equicontinuous homeomorphism has the N-shadowing property for some positive integer N if and only if it has the shadowing property.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Compare with Lewowicz Lewowicz (1983).

References

  • Aoki, N., Hiraide, K.: Topological theory of dynamical systems. Recent advances. North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, (1994)

  • Aoki, N.: Homeomorphisms without the pseudo-orbit tracing property. Nagoya Math. J. 88, 155–160 (1982)

    Article  MathSciNet  Google Scholar 

  • Arbieto, A., Morales, C.A.: Topological stability from Gromov–Hausdorff viewpoint. Discrete Contin. Dyn. Syst. 37(7), 3531–3544 (2017)

    Article  MathSciNet  Google Scholar 

  • Arhangel’skii, A., Tkachenko, M.: Topological groups and related structures, Atlantis Studies in Mathematics, 1. Paris; World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ, Atlantis Press (2008)

  • Aubin, J.-P., Frankowska, H., Set-valued analysis. Reprint of the: edition, p. 2009. Modern Birkäuser Classics, Birkhäuser Boston Inc, Boston, MA (1990)

  • Bautista, S., Morales, C.A., Villavicencio, H.: Descriptive set theory for expansive systems. J. Math. Anal. Appl. 461(1), 916–928 (2018)

    Article  MathSciNet  Google Scholar 

  • Bowen, R.: \(\omega \)-limit sets for axiom A diffeomorphisms. J. Differ. Equ. 18(2), 333–339 (1975)

    Article  MathSciNet  Google Scholar 

  • Carvalho, B., Cordeiro, W.: \(N\)-expansive homeomorphisms with the shadowing property. J. Differ. Equ. 261(6), 3734–3755 (2016)

    Article  MathSciNet  Google Scholar 

  • Cherkashin, D., Kryzhevich, S.: Weak forms of shadowing in topological dynamics. Topol. Methods Nonlinear Anal. 50(1), 125–150 (2017)

    MathSciNet  MATH  Google Scholar 

  • Choi, S.K., Chu, C., Lee, K.: Recurrence in persistent dynamical systems. Bull. Aust. Math. Soc. 43(3), 509–517 (1991)

    Article  MathSciNet  Google Scholar 

  • Chung, N.-P., Lee, K.: Topological stability and pseudo-orbit tracing property of group actions. Proc. Am. Math. Soc. 146(3), 1047–1057 (2018)

    Article  MathSciNet  Google Scholar 

  • Das, P., Das, T.: Stable group actions on uniform spaces. Topol. Proc. 56, 71–83 (2020)

    MathSciNet  MATH  Google Scholar 

  • Hemmingsen, E., Reddy, W.L.: Lifting and projecting expansive homeomorphisms. Math. Syst. Theory 2, 7–15 (1968)

    Article  MathSciNet  Google Scholar 

  • Hiraide, K.: Expansive homeomorphisms with the pseudo-orbit tracing property on compact surfaces. J. Math. Soc. Jpn 40(1), 123–137 (1988)

    Article  MathSciNet  Google Scholar 

  • Hu, H., Zhu, Y.: Quasi-stability of partially hyperbolic diffeomorphisms. Trans. Am Math. Soc. 366(7), 3787–3804 (2014)

    Article  MathSciNet  Google Scholar 

  • Hu, H., Zhou, Y., Zhu, Y.: Quasi-shadowing for partially hyperbolic diffeomorphisms. Ergodic Theory Dyn. Syst. 35(2), 412–430 (2015)

    Article  MathSciNet  Google Scholar 

  • Hurley, M.: Consequences of topological stability. J. Differ. Equ. 54(1), 60–72 (1984)

    Article  MathSciNet  Google Scholar 

  • Kato, H.: Continuum-wise expansive homeomorphisms. Can. J. Math. 45, 576–598 (1993)

    Article  MathSciNet  Google Scholar 

  • Kawaguchi, N.: Properties of shadowable points: chaos and equicontinuity. Bull. Braz. Math. Soc. (N.S.) 48, no. 4, 599–622 (2017)

  • Kawaguchi, N.: Quantitative shadowable points. Dyn. Syst. 32(4), 504–518 (2017)

    Article  MathSciNet  Google Scholar 

  • Kawaguchi, N.: Topological stability and shadowing of zero-dimensional dynamical systems. Disc. Contin. Dyn. Syst. 39(5), 2743–2761 (2019)

    Article  MathSciNet  Google Scholar 

  • Koo, N., Lee, K., Morales, C.A.: Pointwise topological stability. Proc. Edinb. Math. Soc. (2) 61, no. 4, 1179–1191 (2018)

  • Lee, K., Lee, S.: Continuum-wise expansive homeomorphisms with shadowing. J. Chungcheong Math. Soc. 29(1), 151–155 (2016)

    Article  MathSciNet  Google Scholar 

  • Lee, K., Morales, C.A.: Topological stability and pseudo-orbit tracing property for expansive measures. J. Differ. Equ. 262(6), 3467–3487 (2017)

    Article  MathSciNet  Google Scholar 

  • Lewowicz, J.: Persistence in expansive systems. Ergodic Theory Dyn. Syst. 3(4), 567–578 (1983)

    Article  MathSciNet  Google Scholar 

  • Li, J., Zhang, R.: Levels of generalized expansiveness. J. Dyn. Differ. Equ. 29(3), 877–894 (2017)

    Article  MathSciNet  Google Scholar 

  • Metzger, R., Morales, C.A., Thieullen, P.: Topological stability in set-valued dynamics. Disc. Contin. Dyn. Syst. Ser. B 22(5), 1965–1975 (2017)

    MathSciNet  MATH  Google Scholar 

  • Moothathu, T.K.S.: Implications of pseudo-orbit tracing property for continuous maps on compacta. Topol. Appl. 158(16), 2232–2239 (2011)

    Article  MathSciNet  Google Scholar 

  • Morales, C.A.: Shadowable points. Dyn. Syst. 31(3), 347–356 (2016)

    Article  MathSciNet  Google Scholar 

  • Nitecki, Z.: On semi-stability for diffeomorphisms. Invent. Math. 14, 83–122 (1971)

    Article  MathSciNet  Google Scholar 

  • O’Brien, T.: Expansive homeomorphisms on compact manifolds. Proc. Am. Math. Soc. 24, 767–771 (1970)

    Article  MathSciNet  Google Scholar 

  • Ombach, J.: Consequences of the pseudo-orbits tracing property and expansiveness. J. Aust. Math. Soc. Ser. A 43(3), 301–313 (1987)

    Article  MathSciNet  Google Scholar 

  • Parthasarathy, K.R.: Probability measures on metric spaces, probability and mathematical statistics, vol. 3. Academic Press Inc, New York, London (1967)

    MATH  Google Scholar 

  • Reddy, W.: The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32, 627–632 (1965)

    Article  MathSciNet  Google Scholar 

  • Utz, W.R.: Unstable homeomorphisms. Proc. Am. Math. Soc. 1, 769–774 (1950)

    Article  MathSciNet  Google Scholar 

  • Walters, P.: 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), pp. 231–244, Lecture Notes in Math., 668, Springer, Berlin, (1978)

  • Walters, P.: Anosov diffeomorphisms are topologically stable. Topology 9, 71–78 (1970)

    Article  MathSciNet  Google Scholar 

  • Yano, K.: Topologically stable homeomorphisms of the circle. Nagoya Math. J. 79, 145–149 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Depto. de Mat., Fac. de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepcíon, VIII-Region, Chile

    D. Carrasco-Olivera

  2. Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea

    K. Lee

  3. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    C. A. Morales

  4. Instituto de Matemática y Ciencias Afines, Universidad Nacional de Ingeniería, Lima, Peru

    H. Villavicencio

Authors
  1. D. Carrasco-Olivera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. A. Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. Villavicencio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Villavicencio.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

DCO was partially supported by Agencia Nacional de Investigación y Desarrollo (ANID), Chile, Project FONDECYT 1181061 and by Universidad del Bío-Bío, Chile, Project 196108 GI/C. KL was supported by the NRF Grant funded by the Korea government (MSIT) (NRF-2018R1A2B3001457). CAM was partially supported by CNPq-Brazil and the NRF Brain Pool Grant funded by the Korea Government No. 2020H1D3A2A01085417. HV was partially supported by Fondecyt-Concytec contract 100-2018 and Universidad Nacional de Ingeniería, Peru, Project FC-PF-33-2021.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carrasco-Olivera, D., Lee, K., Morales, C.A. et al. Finite-Expansivity and N-Shadowing. Bull Braz Math Soc, New Series 53, 107–126 (2022). https://doi.org/10.1007/s00574-021-00253-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-021-00253-w

Keywords

Mathematics Subject Classification

Search

Navigation