Skip to main content
SpringerLink
Log in

Conjugacy Problem of Strictly Monotone Maps with Only One Jump Discontinuity

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The conjugacy problem is one of the important questions in iteration theory. As far as we know, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all conjugacies. Furthermore, we present the conditions to guarantee \(C^1\) smoothness of these conjugacies.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Block, L., Coven, E.M.: Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval. Trans. Am. Math. Soc. 300, 297–306 (1987)

    Article  MathSciNet  Google Scholar 

  2. Cui, H., Ding, Y.: Renormalization and conjugacy of piecewise linear Lorenz maps. Adv. Math. 271, 235–272 (2015)

    Article  MathSciNet  Google Scholar 

  3. Glendinning, P.: Topological conjugation of Lorenz maps by \(\beta \)-transformation. Math. Proc. Camb. Philos. Soc. 107, 401–413 (1990)

    Article  MathSciNet  Google Scholar 

  4. Glendinning, P., Sparrow, C.: Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Phys. D. 62, 22–50 (1993)

    Article  MathSciNet  Google Scholar 

  5. Hubbard, J.H., Sparrow, C.: The classfication of topological expansive Lorenz maps. Commun. Pure Appl. Math. 62, 431–443 (1990)

    Article  Google Scholar 

  6. Jiang, Y.: On Ulam–von Neumann transformations. Commun. Math. Phys. 172, 449–459 (1995)

    Article  MathSciNet  Google Scholar 

  7. Leśniak, Z., Shi, Y.: Topological conjugacy of piecewise monotonic functions of nonmonotonicity height \(\ge 1\). J. Math. Anal. Appl. 423, 1792–1803 (2015)

    Article  MathSciNet  Google Scholar 

  8. Li, S., Shen, W.: Smooth conjugacy between \(S\)-unimodal maps. Nonlinearity 19, 1629–1634 (2006)

    Article  MathSciNet  Google Scholar 

  9. Llibre, J.: Structure of the set of periods for the Lorenz maps. Dyn. Syst. Bifur. Theory 2, 277–293 (1987)

    MathSciNet  Google Scholar 

  10. Parry, W.: Symbolic dynamics and transformations of the unit interval. Trans. Am. Math. Soc. 122, 368–378 (1966)

    Article  MathSciNet  Google Scholar 

  11. Pring, S.R., Budd, C.J.: The dynamics of regularized discontinuous maps with applications to impacting systems. SIAM J. Appl. Dyn. Syst. 9, 188–219 (2010)

    Article  MathSciNet  Google Scholar 

  12. Segawa, H., Ishitani, H.: On the existence of a conjugacy between weakly multimodal maps. Tokyo J. Math. 21, 511–521 (1998)

    Article  MathSciNet  Google Scholar 

  13. Shi, Y.: Non-monotonic solutions and continuously differentiable solutions of conjugacy equations. Appl. Math. Comput. 215, 2399–2404 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, Guangdong, People’s Republic of China

    Jinghua Liu

  2. Numerical Simulation Key Laboratory of Sichuan Province, Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641100, Sichuan, People’s Republic of China

    Yong-Guo Shi

Authors
  1. Jinghua Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-Guo Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong-Guo Shi.

Additional information

Publisher's Note

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

The first author is supported by Innovation and Developing Projects of universities in Guangdong Province (Grant No. 2019KZDXM032), Characteristic innovation projects of general colleges and universities in Guangdong Province (Grant No. 2019GKTSCX145), National Natural Science Foundation of China (No. 11771197), the Guangdong Natural Science Foundation of China (Nos. 2017A030313030 and 2018A0303070012). The second author is supported by Scientific Research Fund of SiChuan Provincial Education Department (18ZA0274).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Shi, YG. Conjugacy Problem of Strictly Monotone Maps with Only One Jump Discontinuity. Results Math 75, 90 (2020). https://doi.org/10.1007/s00025-020-01219-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01219-y

Keywords

Mathematics Subject Classification

Search

Navigation