= Semiconjugacy of dynamical systems
= Semiconjugacy
{synonym}
A <semiconjugacy> from $f:X\to X$ to $g:Y\to Y$ is an intertwining map $h:X\to Y$:
$$
h\circ f=g\circ h.
$$
In a topological factor-map convention, $h$ is also continuous and onto. Authors using the term for an arbitrary intertwining map should state that weaker convention. If $h$ is a <homeomorphism>, the relation is a <topological conjugacy>.
The identity transports every forward <orbit>. Induction gives $h(f^n(x))=g^n(h(x))$, so iterating in the larger space and then projecting agrees with projecting first and then iterating. The map $h$ can merge different states, so the projected dynamics need not retain all information about the original <orbit>.
For example, let $f(\theta)=2\theta\pmod1$ on the circle, $g(y)=4y(1-y)$ on $[0,1]$, and $h(\theta)=\sin^2(\pi\theta)$. Then
$$
h(f(\theta))=\sin^2(2\pi\theta)=4\sin^2(\pi\theta)(1-\sin^2(\pi\theta))=g(h(\theta)).
$$
Thus the <doubling map> maps onto the parameter-four <logistic map> by a continuous semiconjugacy. Since $h(\theta)=h(1-\theta)$, it is not one-to-one and does not give a topological conjugacy by this map.
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