I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are there any general results on bounds for this sort of measure of non-commutativity of functions that I can look up to see what to expect and how the proofs go?

Under what conditions would repeated application of $f^{-1} \circ g^{-1} \circ f \circ g$ to $x$ revert to $x$ in finitely many steps or converge to it in the limit?

notation: $fx=f(x), gx=g(x)$; then pick a point $(x,0)$ on the $x$-axis; go up/down to $(x,gx)$, turn left/right to $(gx,gx)$, up/down to $(gx,fgx)$, left/right to $(g^{-1}fgx, fgx)$, up/down to $(g^{-1}fgx,g^{-1}fgx)$, left/right to $(f^{-1}g^{-1}fgx,g^{-1}fgx)$, and finally up/down to $(f^{-1}g^{-1}fgx,0)$. Using Geogebra to create some piece-wise quadratic (or linear) $f,g$ one can then modify them and see how the final point moves around. $\endgroup$