Linear integral equations and nonlinear difference-difference equations

This paper studies the dynamics of members of the two-parameter family of maps x → μx(1 ? x^v), emphasizing the evolution from snapback repeller to crisis bifurcations. The example of the square root map $\text{v =}\text{1}\text{2}$ is taken to represent the subfamily where v is fixed and taken from the range $\text{1}\text{2}\text{≤ v ≤ 1}$. A map from such a subfamily is shown to be conjugate with a map with negative Schwarzian derivative. This allows a characterization of crisis as the demise of a snapback repeller on a proper subinterval.