09/29/2008, 01:03 AM
As I now see, even if the uniqueness criterion is valid, the precondition:
\( \lim_{n\to\infty} \exp_b^{\circ n}(z)=\infty \) for each \( z\in f(S) \)
can not be true for any \( f \) for \( b=e \) (and probably for each \( b>e^{1/e} \))!
Michał Misiurewicz [1] showed 1981 that the Julia set of \( \exp \) is the whole \( \mathbb{C} \) (that is not directly visible from the \( e^z \) fractal). The Julia set is the boundary of the set \( K \) of all \( z \) such that \( \lim_{n\to\infty} \exp^{\circ n}(z)=\infty \). That means that the set \( K \) and its complement \( \mathbb{C}\setminus K \) is dense in \( \mathbb{C} \). In every neighborhood of any complex number \( z_0 \) there is a complex number \( z \) such that \( \lim_{n\to\infty} \exp^{\circ n}(z)=\infty \) and also a \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \)!
And that implies that \( f(S) \), which contains an open non-empty set, always contains points \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \).
[1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X
\( \lim_{n\to\infty} \exp_b^{\circ n}(z)=\infty \) for each \( z\in f(S) \)
can not be true for any \( f \) for \( b=e \) (and probably for each \( b>e^{1/e} \))!
Michał Misiurewicz [1] showed 1981 that the Julia set of \( \exp \) is the whole \( \mathbb{C} \) (that is not directly visible from the \( e^z \) fractal). The Julia set is the boundary of the set \( K \) of all \( z \) such that \( \lim_{n\to\infty} \exp^{\circ n}(z)=\infty \). That means that the set \( K \) and its complement \( \mathbb{C}\setminus K \) is dense in \( \mathbb{C} \). In every neighborhood of any complex number \( z_0 \) there is a complex number \( z \) such that \( \lim_{n\to\infty} \exp^{\circ n}(z)=\infty \) and also a \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \)!
And that implies that \( f(S) \), which contains an open non-empty set, always contains points \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \).
[1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X