A constructive proof of the existence of a semi-conjugacy for a one dimensional map
Dyi-Shing Ou Kenneth James Palmer
Discrete & Continuous Dynamical Systems - B 2012, 17(3): 977-992 doi: 10.3934/dcdsb.2012.17.977
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.
keywords: numerical computation. topological conjugacy Dynamical systems piecewise linear maps

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