A Topological Classification of Interval Mappings with Finitely Many Discontinuous Points

• Jinghua Liu ^[1] ; Lin Li ^[2]
  1. [1] Hanshan Normal University

     Hanshan Normal University

     China

     
  2. [2] Jiaxing University

     Jiaxing University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Topological conjugacy plays an important role in the study of dynamical systems andfunctional equations. In this paper, a topological classification for monotone functionswith finitely many discontinuous points is considered. By introducing the definitionof symbolic vector for discontinuous functions, we present necessary and sufficientconditions to determine the conjugate relations between these functions. Moreover,the explicit expressions for those conjugacies are also given. Finally, as an application,our results are applied to the study of classification of generalized Lorenz maps.

• Referencias bibliográficas
  • 1. Baron, K., Jarczyk, W.: Recent results on functional equations in a single variable, perspectives andopen problems. Aequ. Math.60, 1–48...
  • 2. Belitskii, G., Tkachenko, V.: One-Dimensional Functional Equations, Operator Theory: Advances andApplications, vol. 144. Birkhäuser Verlag,...
  • 3. Block, L., Hart, D.: The bifurcation of periodic orbits of one-dimensional maps. Ergod. Theory Dynam.Syst.2, 125–129 (1982)
  • 4. Block, L., Coven, E.M.: Topological conjugacy and transitivity for a class of piecewise monotone mapsof the interval. Trans. Am. Math....
  • 5. Chow, S.N., Li, C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. CambridgeUniversity Press, Cambridge (1994)
  • 6. Collet, P., Eckmann, J.P., Lanford, O.E.: Universal properties of maps on a interval. Commun. Math.Phys.76, 211–254 (1980)
  • 7. Cui, H., Ding, Y.: Renormalization and conjugacy of piecewise linear Lorenz maps. Adv. Math.271,235–272 (2015)
  • 8. Damanik, D., Killip, R.: Almost everywhere positivity of the Lyapunov exponent for the doubling map.Commun. Math. Phys.257, 287–290 (2005)
  • 9. Ding, Y.: Renormalization andα-limit set for expanding Lorenz map. Discrete Dyn. Syst.29, 979–999(2011)
  • 10. Fort, M.K., Jr.: The embedding of homeomorphisms in flows. Proc. Am. Math. Soc.6, 960–967 (1955)
  • 11. Gerard, O., Enric, F., Carles, B.: Bifurcations and chaos in converters. Discontinuous vector fields andsingular Poincaré maps. Nonlinearity13,...
  • 12. Glandinning, P.: Topological conjugation of Lorenz maps byβ-transformation. Math. Proc. Camb.Philos. Soc.107, 401–413 (1990)
  • 13. Glandinning, P., Sparrow, C.: Prime and renormalizable kneading invariants and the dynamics ofexpanding Lorenz maps. Phys. D62, 22–50...
  • 14. Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)
  • 15. Hubbard, J.H., Sparrow, C.T.: The classification of topologically expansive Lorenz maps. Commun.PureAppl.Math.XLIII, 431–443 (1990)
  • 16. Jiang, Y.: On Ulam–von Neumann transformations. Commun. Math. Phys.172, 449–459 (1995)
  • 17. Karlin, S., McGregor, J.: Embedding iterates of analytic functions with two fixed points into continuousgroups. Trans. Am. Math. Soc.132,...
  • 18. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. CambridgeUniversity Press, Cambridge (1995)
  • 19. Kuczma, M.: Functional Equations in a Single Variable. Polish Scientific Publishers, Warszawa (1968)
  • 20. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia of Mathematicsand its Applications, Cambridge University...
  • 21. Labarca, R., Moreira, C.: Essential dynamics for Lorenz maps on the real line and the LexicographicalWorld. Ann. Inst. Henri Poincaré23,...
  • 22. Le ́sniak, Z., Shi, Y.: Topological conjugacy of piecewise monotonic functions of nonmonotonicityheight≥1. J. Math. Anal. Appl.423, 1792–1803...
  • 23. Li, L., Zhang, W.: Conjugacy between piecewise monotonic functions and their iterative roots. Sci.China Ser. A59, 367–378 (2016)
  • 24. Li, S., Shen, W.: Smooth conjugacy betweenS-unimodal maps. Nonlinearity19, 1629–1634 (2006)
  • 25. Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Month.82, 985–992 (1975)
  • 26. Li, W.: Normal Form Theory and Its Application. Science Press, Beijing (2000). (in Chinese)
  • 27. Liu, J., Shi, Y.: Conjugacy problem of strictly monotone maps with only one jump discontinuity. ResultsMath.19, 75–90 (2020)
  • 28. Llibre, J.: Structure of the Set of Periods for the Lorenz Maps. In: Dynamical System and BifurcationTheory. Wiley, New York (1987)
  • 29. Palmore, J.: Chaos, cycles, and Schwarzian derivatives for families of asymmetric maps. Appl. Anal.57, 235–242 (1995)
  • 30. Parry, W.: Symbolic dynamics and transformations of the unit interval. Trans. Am. Math. Soc.122,368–378 (1966)
  • 31. Puph, C.: On a theorem of P. Hartman. Am. J. Math.91, 363–367 (1962)
  • 32. Sarkovskii, A.N.: Coexistence of cycles of a continuous map of a line into itself. Ukr. Math. J.16,61–71 (1964)
  • 33. Shi, Y., Tang, Y.: On conjugacies between asymmetric Bernoulli shifts. J. Math. Anal. Appl.434,209–221 (2016)
  • 34. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, NewYork (1982)
  • 35. Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Applied Mathematical Sciences,Springer, New York (1988)