I believe Milnor has something to say about this, I'll try and check his book later.
The gist is, that \( \exp^{\circ n}(z) \) gets arbitrarily close to \( 0 \) for almost all \( z \) (it misses on a Lebesgue measure zero set (which are the periodic points)). I believe Milnor has a section discussing how to estimate the size of \( n \); particularly, if I recall correctly; it'll give a nice bound on the size of \( n \) in relation to its nearness to 0... It's something to do with Julia sets; I can't remember perfectly.
I can't remember if it's Milnor though; or I read it somewhere else. I'll have to look through the book again...
EDIT: ACK! I'm having trouble finding it; but from memory, there's a word for it. Consider \( f:\mathbb{C}\to\mathbb{C} \). Take a neighborhood \( \mathcal{N} \) about a point \( z \in \mathcal{J}(f) \) (the julia set of \( f \)). Then the orbits of \( f \) on \( \mathcal{N} \) are dense in \( \mathcal{J}(f) \). From here you can create a function,
\(
N(\mathcal{N},L,\epsilon)= n\\
\)
Such that some point of \( z' \in \mathcal{N} \) satisfies \( |f^{\circ n}(z') - L| < \epsilon \) for some \( L \in \mathcal{J}(f) \). (or something like this.)
I believe there is a way to estimate this function; I can't seem to find it at the moment. I'll keep looking.
The gist is, that \( \exp^{\circ n}(z) \) gets arbitrarily close to \( 0 \) for almost all \( z \) (it misses on a Lebesgue measure zero set (which are the periodic points)). I believe Milnor has a section discussing how to estimate the size of \( n \); particularly, if I recall correctly; it'll give a nice bound on the size of \( n \) in relation to its nearness to 0... It's something to do with Julia sets; I can't remember perfectly.
I can't remember if it's Milnor though; or I read it somewhere else. I'll have to look through the book again...
EDIT: ACK! I'm having trouble finding it; but from memory, there's a word for it. Consider \( f:\mathbb{C}\to\mathbb{C} \). Take a neighborhood \( \mathcal{N} \) about a point \( z \in \mathcal{J}(f) \) (the julia set of \( f \)). Then the orbits of \( f \) on \( \mathcal{N} \) are dense in \( \mathcal{J}(f) \). From here you can create a function,
\(
N(\mathcal{N},L,\epsilon)= n\\
\)
Such that some point of \( z' \in \mathcal{N} \) satisfies \( |f^{\circ n}(z') - L| < \epsilon \) for some \( L \in \mathcal{J}(f) \). (or something like this.)
I believe there is a way to estimate this function; I can't seem to find it at the moment. I'll keep looking.