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 ${2^k}p$ for some $p$ and all $k \geq 0$. There is a version of the result for a parametrized family $f_t$ which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.