Counting the number of periods in one-dimensional maps with multiple critical points
Consider a continuous and piecewise monotonic mapping \(f\) from the interval \([0,1]\) into itself with \(m\) monotone pieces (laps). Ask about the largest number of orbits of period \(n\) that may exist for any such map. If \(m > 2\) one can also ask restricted questions: what if we prescribe or relate the behavior of two or more turning points, for example requiring that one turning point be mapped to 0. The unrestricted orbit counting problem, as well as many restricted problems, can be solved explicitly (formulas are given) in terms of the solution to one particular restricted problem, in which all minima are mapped to 0 and all maxima are mapped to the same point. For this restricted problem many methods can be applied that were originally developed for the \(m = 2\) (logistic map) problem.
scientific article; zbMATH DE number 606920