# American Institute of Mathematical Sciences

May  2012, 17(3): 977-992. doi: 10.3934/dcdsb.2012.17.977

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

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287-1804, United States 2 Department of Financial and Computational Mathematics, Providence University, Taichung 43301, Taiwan

Received  March 2011 Revised  September 2011 Published  January 2012

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.
Citation: Dyi-Shing Ou, Kenneth James Palmer. A constructive proof of the existence of a semi-conjugacy for a one dimensional map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977
##### References:
 [1] , IEEE standard for floating-point arithmetic,, The Institute of Electrical and Electronics Engineers, (2008). [2] 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. [3] J. Banks, V. Dragan and A. Jones, "Chaos: A Mathematical Introduction," Australian Mathematical Society Lecture Series, 18, Cambridge University Press, Cambridge, 2003. [4] K. M. Brucks and H. Bruin, "Topics from One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004. [5] 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. [6] 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. [7] P. Henrici, "Essentials of Numerical Analysis with Pocket Calculator Demonstrations," John Wiley & Sons, Inc., New York, 1982. [8] 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. [9] 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.

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##### References:
 [1] , IEEE standard for floating-point arithmetic,, The Institute of Electrical and Electronics Engineers, (2008). [2] 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. [3] J. Banks, V. Dragan and A. Jones, "Chaos: A Mathematical Introduction," Australian Mathematical Society Lecture Series, 18, Cambridge University Press, Cambridge, 2003. [4] K. M. Brucks and H. Bruin, "Topics from One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004. [5] 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. [6] 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. [7] P. Henrici, "Essentials of Numerical Analysis with Pocket Calculator Demonstrations," John Wiley & Sons, Inc., New York, 1982. [8] 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. [9] 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.
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