Period doubling and the Lefschetz formula

Abstract

This article gives an application of the Lefschetz fixed point theorem to prove, under certain hypotheses, the existence of a family of periodic orbits for a smooth map. The family has points of periods 2 k p {2^k}p for some p p and all k ≥ 0 k \geq 0 . There is a version of the result for a parametrized family f t f_t which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.