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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.

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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.

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  • $\begingroup$ Thank you, professor. Can we use Marty criterion to prove it (not use the fact repelling periodic points are dense in Julia sets.)? $\endgroup$
    – yaoxiao
    Commented Sep 11, 2013 at 14:05

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