Dynamics of annulus coverings II: periodic points

Let $f$ be a covering map of the open annulus $A=S^1\times (0,1)$ of degree $d$ , $\mathrm{\mid }d\mathrm{\mid }>1$. Assume that $f$ preserves an essential (i.e not contained in a disk of $A$) compact subset $K$. We show that $f$ has at least the same number of periodic points in each period as the map $z^d$ in $S^1\mathrm{.}$