A note on periodic points of expanding maps of the interval

Abstract

We sharpen a result of Byers on the existence of periodic points for some continuous expanding maps of the interval and generalize it to some classes of continuous maps of the interval which are not necessarily expanding. We then use these results to construct one-parameter families of continuous maps of the interval which have a bifurcation form fixed points directly to period 3 points together with a series of reverse bifurcations from period 3 points back to fixed points. Consequently, our results also provide examples of one-parameter families of continuous maps of the interval whose topological entropy jumps form zero to some positive number and then changes back to zero as the parameter varies.