 Semiconjugacy of dynamical systems

A semiconjugacy from $f:X\to X$ to $g:Y\to Y$ is an intertwining map $h:X\to Y$: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 $f(\theta )=2\theta (\mathrm{mod}1)$ on the circle, $g(y)=4y(1-y)$ on $[0,1]$, and $h(\theta )={\sin }^2(\pi \theta )$. ThenThus 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.