In this video the presenter shows that for strictly increasing functions $f(x)=f^{-1}(x) \implies f(x)=x$ because when you iterate $f$ it either increases forever or decreases forever. We cannot solve $f(x)=f^{-1}(x)$ completely in general using $f(x)=x$ for other functions e.g. $f(x)=\frac{-1}{x}$ because when you iterate them you might get a loop.
In this example, $f(x)=\frac{-1}{x}$ forms a loop when you iterate the point $(1,-1)$.
Are there "well-behaved functions" (e.g. polynomials, trig functions, etc, the usual suspects) which can be iterated infinitely such that the values are both bounded and aperiodic?