    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 & Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977
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show all references

##### References:
  , IEEE standard for floating-point arithmetic,, The Institute of Electrical and Electronics Engineers, (2008).   Google Scholar  Ll. Alsedà, J. Llibre and M. Misurewicz, "Combinatorial Dynamics and Entropy in Dimension One,", 2nd edition, 5 (2000). Google Scholar  J. Banks, V. Dragan and A. Jones, "Chaos: A Mathematical Introduction,", Australian Mathematical Society Lecture Series, 18 (2003). Google Scholar  K. M. Brucks and H. Bruin, "Topics from One-Dimensional Dynamics,", London Mathematical Society Student Texts, 62 (2004). Google Scholar  R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", 2nd edition, (1989). Google Scholar  N. A. Fotiades and M. A. Boudourides, Topological conjugacies of piecewise monotone interval maps,, International Journal of Mathematics and Mathematical Sciences, 25 (2001), 119.  doi: 10.1155/S0161171201004343.  Google Scholar  P. Henrici, "Essentials of Numerical Analysis with Pocket Calculator Demonstrations,", John Wiley & Sons, (1982). Google Scholar  J. Milnor and W. Thurston, On iterated maps of the interval,, in, 1342 (1988), 1986. Google Scholar  W. Parry, Symbolic dynamics and transformations of the unit interval,, Transactions of the American Mathematical Society, 122 (1966), 368.  doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar
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