It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}\_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}\_{n=1}^{\infty}$, is normal $\forall m\geq 1$.


For a sequence of integers $\{n\_i\}\_{i=1}^{\infty}$,
There is natural question to ask under which condition:


$\{f^{n\_i}\}\_{i=1}^{\infty}$ is a normal family implies $\{f^{i}\}\_{i=1}^{\infty}$ is a normal family. 


We just know a obvious condition: for $i$ sufficiently large, $\{n\_i\}$ is arithmetic progression. It is natural to ask whether this condition is optimal in some sense?


Any reference and comments will be appreciated.



Normal family and arithmetic progression
This is true for every subsequence. Indeed, if $z$ is on the set of normality
(where $f^n$ is normal), then evidently every subset of $f^n$ is normal. If $z$ is on
the Julia set, then there is a repelling periodic point in every neighborhood of $z$.
At a repelling periodic point, any subsequence of iterates is evidently not normal.


