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If $x$ is a fixed point of $f$, thenConversely, applying the same argument to $f={\sigma }^{-1}g\sigma$ transports every fixed point of $g$ back to one of $f$. Thus the bijection $\sigma$ restricts to a bijection between the two fixed-point sets, so conjugate functions have the same number of fixed points.

A semiconjugacy from $f:X\to X$ to $g:Y\to Y$ is a map $h:X\to Y$ satisfying $h\circ f=g\circ h$. It transports forward orbits but need not be invertible, A bijection instead gives conjugate functions.