Universal uniqueness criterion?

[bo198214]

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And that implies that \( f(S) \), which contains an open non-empty set, always contains points \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \).
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Why do you think that this set is open?

Just take a point \( w \) of the interior of \( S \) such that \( f'(w)\neq 0 \) (if such a point does not exist then \( f \) is constant), then there is a neighborhood \( W \) of \( w \) such that \( f \) is bijective on \( W \). That means \( f \) and \( f^{-1} \) are continuous (because analytic). And the preimage of an open set is open under a continuous map. So the preimage \( f(W) \) of \( W \) under \( f^{-1} \) is open and so we find arbitrary many points \( z \) in \( f(W)\subseteq f(S) \) such that \( \exp^{\circ n}(z)\not\to\infty \).

Do you make any difference between \( \exp_b^{\circ p}(q) \) and \( \exp_b^{p}(q) \) ?

Oh no its both the same here.