Proving semi-conjugacy preserves chaotic behavior 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.
 A: Would have been nice if you mentioned what is meant by chaos in this case. I presume you're referring to the three conditions listed on p. 338. In this case a semi-conjugacy certainly preserves the first two, but not necessarily the third. It is the third condition that is most important, in some sense, to produce chaos, and is often referred to as "expansivity." 
Expansivity is not in general preserved by a semi-conjugacy. For example, let for $m\in\mathbb{N}$, $m > 1$, $E_m : S^1 \rightarrow S^1$ be given by $E_m(x) = m x\mod 1$, where $S^1$ is the unit circle $\mathbb{R}/\mathbb{Z}$. Now let $h: S^1 \rightarrow \{0\}$ be the constant map, and let $C: \{0\} \rightarrow \{0\}$ be the constant map. Then certainly $C$ is a factor of $E_m$ under the semi-conjugacy $h$, but there is nothing chaotic about $C$. 
Also, often the first indication of chaos is positivity of topological entropy (at least in compact systems). But topological entropy is not preserved under a semi-conjugacy. For instance, it is possible to find systems with positive entropy that project onto systems with zero entropy (as, for instance, the not particularly exciting example above).
To avoid pathologies as above one must impose more conditions.
