Topological conjugacy

Remarks

The relation $h \circ f = g \circ h$ gives $h \circ f^n = g^n \circ h$ for all $n \ge 0$, so $h$ maps orbits of $f$ onto orbits of $g$. In particular $h$ maps fixed points to fixed points and periodic points of minimal period $p$ to periodic points of minimal period $p$. Topological conjugacy is an equivalence relation on continuous self-maps.

Example: the tent map $T(x) = 1 - \abs{2x - 1}$ and the logistic map $f(x) = 4x(1 - x)$ on $[0, 1]$ are conjugate through $h(x) = \sin^2(\pi x / 2)$. The map $h$ is an increasing homeomorphism of $[0, 1]$. One has $f(h(x)) = 4 \sin^2(\pi x/2) \cos^2(\pi x/2) = \sin^2(\pi x)$. Also $h(T(x)) = \sin^2(\pi x)$ for $x \le 1/2$, and $h(T(x)) = \sin^2(\pi - \pi x) = \sin^2(\pi x)$ for $x \ge 1/2$.

Non-example: $f(x) = x/2$ and $g(x) = -x/2$ on $\R$ are not conjugate. A conjugacy $h$ would map the only fixed point $0$ to $0$, so it would map $(0, \infty)$ onto $(0, \infty)$ or onto $(-\infty, 0)$. Since $f$ maps $(0, \infty)$ into itself while $g$ maps each half-line into the other one, $h \circ f = g \circ h$ fails.

For flows, the weaker notion of topological equivalence only asks that $h$ map orbits onto orbits and preserve their direction, not the time parametrisation.