Period doubling patterns of interval maps

We prove the existence and demonstrate the construction of period doubling patterns centered at periodic orbits of continuous maps on the interval. In particular we prove that f C0(I,I) exhibits a period doubling pattern centered at a fixed point of f if and only if the set of periodic points is not closed. Furthermore, we prove that if f has a periodic orbit of period n>1, which is not a power of two, and n m in Sarkovskii’s ordering, then f exhibits a period doubling pattern centered at a periodic orbit of period m. An analytic configuration of such period doubling patterns is exhibited.