http://www.math.upatras.gr/~bountis/files/def-eq.pdf
In the above documentation it states "It is easy to check that a semiconjugacy also preserves chaotic behavior on intervals of finite length" on page 341 (PDF page 356) , but I am not seeing how. There is a proof that conjugacy preserves chaotic behavior on the previous page, but it doesn't seem that that argument will translate to the semi-conjugate version. Any help is appreciated.
Thanks
P.S. - The above reference is "Differential Equations, Dynamical Systems, And an Introduction to Chaos", 2nd edition, by Hirsch, Smale and Devaney.