I'm taking an undergraduate course in nonlinear dynamics, and the idea of topological conjugacy between (one-dimensional) iterated maps was introduced as follows:
Let ${I}$ and ${J}$ be intervals. We have two mapping functions ${f:I\rightarrow I}$ and ${g:J\rightarrow J}$. We say that ${f}$ and ${g}$ are conjugate if $\exists$ a homeomorphism ${h:I\rightarrow J}$ such that ${h}$ satisfies the conjugacy equation $${h \circ f = g \circ h}$$
I checked the Wikipedia article on topological conjugacy and found that the definition given above looks more like that of semiconjugacy. Semiconjugacy requires the map ${h}$ defined above to be a surjection, and not necessarily a homeomorphism. (I have not taken any formal courses in topology, but I understand a homeomorphism to be a continuous invertible map with a continuous inverse). Wikipedia says that for topological conjugacy, we require $${f = h^-1 \circ g \circ h}$$
This notion was introduced to us in order to relate different kinds of maps - if two maps are topologically conjugate, then solving the dynamics of one of them, i.e. finding fixed points, periodic orbits etc. enables us to trivially find those of the other.
My question is: Is topological conjugacy necessary in order to use the solved dynamics of one map to solve those of a conjugate map, or does semiconjugacy suffice for the same purposes? That is to say, if ${f}$ and ${g}$ are semiconjugate and ${f}$ is chaotic, then can I say ${g}$ is chaotic? Or do I need conjugacy to make this statement?
An example given in class was the "conjugacy" between the tent map and the logistic map. The tent map is defined as: $${f(x)= \begin{cases} 2x & 0 \leq x \leq \frac{1}{2} \\ 2(1-x) & \frac{1}{2} \leq x \leq 1 \end{cases} }$$
And the logistic map is: $${g(x)= 4x(1-x), 0 \leq x \leq 1}$$
This was demonstrated by defining ${h:[0,1] \rightarrow [0,1]}$ as:
$${h(x)=\frac{1}{2}(1-cos2 \pi x)}$$
and showing that ${h \circ f = g \circ h}$. Here, ${h}$ is continous and surjective but not injective, and therefore not invertible, and also not a homeomorphism. So we have established semiconjugacy between the logistic and tent maps. It is quite easy to show that the tent map is chaotic, and this is how it is argued that the logistic map, too, exhibits chaos. Can we make this leap from the tent map to the logistic map, given that we have established merely semiconjugacy, and not conjugacy?