Inverse map characterization of asymptotic stability on the line (Q1567158)

This paper studies the asymptotic stability in which continuous mapping \(f\) is compared to its inverse \(f^{-1}\) in a cut-and-paste sort of way. The author presents a necessary and sufficient condition for asymptotic stability. More precisely, the author proves that a fixed point \(x_*\) is asymptotically stable if and only if the inverse image of the part of \(f\) to the right of \(x_*\) lies in the region of the plane that is above both the identity line and the part of \(f\) to the left of \(x_*\). Moreover, the author presents an application of the criterion mentioned above.