Periodic homotopy and conjugacy idempotents

A self-map $f$ on the CW complex $Z$ is a periodic homotopy idempotent if for some $r\geqslant 0$ and $p>0$ the iterates $f^r$ and $f^{r+p}$ are homotopic. Geoghegan and Nicas defined the rotation index $RI(f)$ of such a map. They proved that for $r=p=1$, the homotopy idempotent $f$ splits if and only if $RI(f)=1$, while for $r=0$, the index $RI(f)$ divides $p^2$. We extend this to arbitrary $p$ and $r$, and generalize various results related to the splitting of homotopy idempotents on CW complexes and conjugacy idempotents on groups.