Advanced Search
Article Contents
Article Contents

A constructive proof of the existence of a semi-conjugacy for a one dimensional map

Abstract Related Papers Cited by
  • A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that $f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly) monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a map is topologically semi-conjugate to a piecewise linear map; however here we prove that the topological semi-conjugacy is unique for this class of maps; also our proof is constructive and yields a sequence of easily computable piecewise linear maps which converges uniformly to the semi-conjugacy. We also give equivalent conditions for the semi-conjugacy to be a conjugacy as in Parry's theorem. Related work was done by Fotiades and Boudourides and Banks, Dragan and Jones, who however only considered cases where a conjugacy exists. Banks, Dragan and Jones gave an algorithm to construct the conjugacy map but only for one-hump maps.
    Mathematics Subject Classification: Primary: 37C15, 37E05; Secondary: 37B10, 65D15.


    \begin{equation} \\ \end{equation}
  • [1]

    Ll. Alsedà, J. Llibre and M. Misurewicz, "Combinatorial Dynamics and Entropy in Dimension One," 2nd edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.


    J. Banks, V. Dragan and A. Jones, "Chaos: A Mathematical Introduction," Australian Mathematical Society Lecture Series, 18, Cambridge University Press, Cambridge, 2003.


    K. M. Brucks and H. Bruin, "Topics from One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004.


    R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.


    N. A. Fotiades and M. A. Boudourides, Topological conjugacies of piecewise monotone interval maps, International Journal of Mathematics and Mathematical Sciences, 25 (2001), 119-127.doi: 10.1155/S0161171201004343.


    P. Henrici, "Essentials of Numerical Analysis with Pocket Calculator Demonstrations," John Wiley & Sons, Inc., New York, 1982.


    J. Milnor and W. Thurston, On iterated maps of the interval, in "Dynamical Systems" (ed. J. C. Alexander) (College Park, MD, 1986-87), Lecture Notes in Mathematics, 1342, Springer, Berlin, (1988), 465-563.


    W. Parry, Symbolic dynamics and transformations of the unit interval, Transactions of the American Mathematical Society, 122 (1966), 368-378.doi: 10.1090/S0002-9947-1966-0197683-5.

  • 加载中

Article Metrics

HTML views() PDF downloads(245) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint